\section{Introduction}\label{S:0}
\noindent  Approximation of bandlimited functions as limits of polynomials has a long history, as the following question illustrates: if $(\mathrm{sinc}\pi(\cdot -t_n))_{n \in \mathbb{Z}}$ forms a Riesz basis for $PW_{[-\pi,\pi]}$, what are the canonical product expansions of the biorthogonal functions for this Riesz basis? The first results along these lines were given by Paley and Wiener in \cite{PW}, and improved upon by Levinson in \cite[pages 47-67]{L}), while Levin extends these results to different classes of Riesz bases in \cite{Lev}.  A complete solution is given by Lyubarskii and Seip in \cite{LS} and Pavlov in \cite{P}.  In particular, they prove the following theorem:
\begin{thm}\label{lyuseippav}
Let $(t_n)_n \subset \mathbb{R}$, where $t_n \neq 0$ when $n \neq 0$, be a sequence such that the family of functions $(\mathrm{sinc} \pi(\cdot -t_n))_n$ is a Riesz basis for $PW_{[-\pi,\pi]}$, then the function
\begin{equation}
S(z) = \lim_{r \rightarrow \infty} (z-t_0) \prod_{\{ t_n \ : \ |t_n|<r, n \neq 0 \}} \Big(1-\frac{z}{t_n}  \Big)\nonumber
\end{equation}
is entire, where convergence is uniform on compacta,  and the biorthogonal functions $(G_n)_n$ of $$(\mathrm{sinc} \pi((\cdot) -t_n))_n$$ are given by
\begin{equation}
G_n(z) = \frac{S(z)}{(z-t_n)S'(t_n)}.\nonumber
\end{equation}
\end{thm}

\noindent The following is a readily proven corollary of Theorem \ref{lyuseippav}:
\begin{cor}\label{lyuseippavcor}
Let $(t_n)_n\subset \mathbb{R}$ and $(G_k)_k$ be defined as in Theorem \ref{lyuseippav}, then for each $k$, there exists a sequence of polynomials $(\Phi_{N,k})_N$ such that\\

\noindent 1) $\Phi_{N,k} (t_n) = G_k (t_n)$ when $|t_n|<N$.\\

\noindent 2) $\lim_{N \rightarrow \infty} \Phi_{N,k} = G_k$ uniformly on compacta.

\end{cor}

\noindent Corollary \ref{lyuseippavcor} raises two questions:\\

\noindent 1) Does every multivariate bandlimited function, (not just biorthogonal functions associated with a particular exponential Riesz basis), have a corresponding sequence of polynomial interpolants?\\

\noindent 2) If such polynomial interpolants for a multivariate bandlimited function exist, can these interpolants be used to be approximate the function in some simple and straightforward way?\\

\noindent  Let $(t_n)_{n \in \mathbb{Z}^d} \subset \mathbb{R}^d$ be a sequence such that the family of exponentials $\big( e^{i \langle \cdot , t_n \rangle} \big)_{n \in \mathbb{Z}^d}$ is a uniformly invertible Riesz basis for $L_2([-\pi,\pi]^d)$ (defined in section 4).  Under this condition, Theorem \ref{maintheorem} answers the first question affirmatively by showing that multivariate bandlimited functions can be approximated globally, both uniformly and in $L_2$, by a rational function times a multivariate sinc function.  Stated informally,
\begin{equation}\label{firstblip}
f(t) \simeq  \Psi_\ell(t) \frac{\mathrm{SINC} (\pi t)}{Q_{d,\ell} (t)},\quad \ell > 0,
\end{equation}
where $(\Psi_\ell))_{\ell \in \mathbb{N}}$ is a particular sequence of interpolating polynomials and $(Q_{d,\ell})_{d,\ell}$ is a sequence of polynomials which removes the zeros of the SINC function.  This gives a partial answer to the second question, but the fraction in the approximants above becomes more complex as $\ell$ increases.  Theorem \ref{simplethm} gives a more satisfactory answer to question 2) by using $$e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}, \quad \ell>0$$ in lieu of the SINC function in expression (\ref{firstblip}).  The exponent in the above expression is simply a rational function of $\ell$.  This simplicity necessitates replacing global $L_2$ and uniform convergence with a more local (though not totally local) convergence.  Corollary \ref{simplecormod} is of particular interest as a multivariate analogue of Theorem \ref{lyuseippav}, stately informally as
\begin{equation}
f(t) \simeq  \Psi_\ell(t),\quad \ell > 0.\nonumber
\end{equation}
The author is unaware of any other multivariate polynomial approximation theorem which applies to exponential Riesz bases which are not necessarily tensor products of single-variable Riesz bases, or that demonstrate convergence stronger than uniform convergence on compacta.  As a note, Theorems \ref{maintheorem}, \ref{simplethm}, and Corollary \ref{simplecormod} do not, at this point, recover Corollary \ref{lyuseippavcor} in its generality of allowable sequences $(t_n)_n\subset \mathbb{R}$; however, the comments above show that their value is primarily due to their multidimensional nature and convergence properties.\\

\noindent This paper is outlined as follows.  Section 2 covers the necessary preliminary and background material regarding bandlimited functions, and section 3 outlines some basic properties of uniformly invertible operators.  Theorems \ref{maintheorem} and Theorem \ref{simplethm} are proven in sections 4 and 5 respectively, along with pertinant corollaries.  Section 6 gives explicit examples of sequences $(t_n)_{n \in \mathbb{Z}^d}$ to which Theorems \ref{maintheorem} and \ref{simplethm} apply.  Section 7 (as an appendix) addresses the optimality of growth rates appearing in Theorem \ref{simplethm}.

\section{Preliminaries}\label{S:1}

\begin{defin}
A reproducing kernel Hilbert space is a Hilbert space $H$ of functions on $X$ such that there exists $K:X\times X\rightarrow \mathbb{C}$ satisfying the following:\\
\noindent 1) For all $y \in X$, $K(\cdot,y) \in H$.\\
\noindent 2) $f(x) = \langle f(\cdot) , K(\cdot,x) \rangle$ for all $x \in X$ and $f\in H.$
\end{defin}

\begin{defin}
A Riesz basis for a Hilbert space $H$ is a sequence $(f_n)_{n \in \mathbb{N}}$ which is isomorphically equivalent to an orthonormal basis of $H$.  Equivalently, a Riesz basis is an unconditional Schauder basis.\\
\end{defin}

\noindent If $(f_n)_{n \in \mathbb{N}}$ is a Schauder (Riesz) basis for a Hilbert space $H$, then there exists a unique set of functions $(f_n^*)_{n \in \mathbb{N}}$, (the biorthogonals of $(f_n)_{n \in \mathbb{N}}$) such that $\langle f_n , f_m^* \rangle  = \delta_{nm}.$ The biorthogonals also form a Schauder (Riesz) basis for $H$. Note that biorthogonality is preserved under a unitary transformation.\\

\noindent We use the $d$-dimensional $L_2$ isometric Fourier transform
\begin{equation}
\mathcal{F}(f)(\cdotp) = \mathrm{P.V. \ }\frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} f(\xi)e^{-i \langle \cdotp , \xi \rangle} d\xi, \quad f  \in L_2(\mathbb{R}^d),\nonumber
\end{equation}
where the inverse transform is given by 
\begin{equation}
\mathcal{F}^{-1}(f)(\cdotp) = \mathrm{P.V. \ } \frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d} f(\xi)e^{i \langle \cdotp , \xi \rangle} d\xi , \quad f  \in L_2(\mathbb{R}^d).\nonumber
\end{equation}

\begin{defin}
We define $PW_{[-\pi,\pi]^d} := \{f \in L_2(\mathbb{R}^d) \arrowvert \mathrm{supp}(\mathcal{F}^{-1}(f)) \subset [-\pi,\pi]^d \}$, with the inherited $L_2 (\mathbb{R}^d)$ norm.  Functions in $PW_{[-\pi,\pi]^d}$ are also called bandlimited functions.
\end{defin}

\noindent Here are facts concerning $PW_{[-\pi,\pi]^d}$ which will be used ubiquitously.\\

\noindent 1) $PW_{[-\pi,\pi]^d}$ is isometric to $L_2([-\pi,\pi]^d)$ by way of the Fourier transform.\\

\noindent 2) $PW_{[-\pi,\pi]^d}$ consists of entire functions, though in this paper we restrict the domain to $\mathbb{R}^d$.\\

\noindent 3) $PW_{[-\pi,\pi]^d}$ is a reproducing kernel Hilbert space with reproducing kernel $$K(x,y) = \mathrm{SINC}\pi(x-y)$$ where $$\mathrm{SINC}(x) := \mathrm{sinc}(x(1))\cdot\ldots\cdot\mathrm{sinc}(x(d)),\quad \mathrm{sinc}(x) := \frac{\sin(x)}{x}.$$

\noindent 4) $\big(\mathrm{SINC}\pi((\cdot)-n)\big)_{n \in \mathbb{Z}^d}$ is an orthonormal basis for $PW_{[-\pi,\pi]^d}$.  This follows from $$\mathcal{F}\big( \frac{1}{\sqrt{2\pi}}e^{i \tau (\cdot)} \big)(t) = \mathrm{sinc}\pi(t-\tau).$$

\noindent 5) \noindent In $PW_{[-\pi,\pi]^d}$, $L_2$ convergence implies uniform convergence.\\

\noindent 6) If $f\in PW_{[-\pi,\pi]^d}$, then $$\lim_{\|x\|_\infty \rightarrow \infty} f(x)=0.$$  This follows from the $d$-dimensional Riemann-Lebesgue Lemma.\\

\noindent 7)  The following result \cite[Theorem 19.3]{W} due to Paley and Wiener characterizes single-variable bandlimited functions.
\begin{thm}\label{PaleyWiener}
A function $f$ is in $PW_{[-\pi,\pi]}$ if and only if the following statements hold.\\
1) $f$ is entire.\\
2) There exists $M \geq 0$ such that $|f(z)| \leq M e^{\pi |z|}$ for $z \in \mathbb{C}$.\\
3) $f\big|_\mathbb{R} \in L_2(\mathbb{R})$.
\end{thm}

\section{Uniform invertibility of operators and Riesz bases}\label{S:2}

\begin{defin} Let $A:\ell_2(\mathbb{N}) \rightarrow \ell_2(\mathbb{N})$ be an onto isomorphism. Regard $A$ as a matrix map with respect to the unit vector basis of $\ell_2(\mathbb{N})$.  Let $\pi_k$ be the orthogonal projection onto the span of the first $k$ terms of the unit vector basis.  If 
\begin{equation}\label{matunif}
\sup_{j \in \mathbb{N}} \| (\pi_{k_j} A \pi_{k_j})^{-1}\|< \infty,
\end{equation}
for an increasing sequence $(k_j)_{j \in \mathbb{N}},$ then $A$ is said to be uniformly invertible as a matrix map with respect to the projections $(\pi_{k_j})_{j \in \mathbb{N}}$.  The terms in inequality (\ref{matunif}) should be interpreted as standard matrix norms and inverses of finite dimensional matrices.
\end{defin}

\noindent Let $S$ be an orthonormal basis for a Hilbert space $H$. Let $(S_n)_{n \in \mathbb{N}}$ be a sequence of sets such that\\

\noindent 1) $\emptyset \neq S_1 \subsetneq S_2 \subsetneq \cdots \subset S$, and\\
\noindent 2) $\bigcup_{n=1}^{\infty} S_n = S$.\\

\noindent Define $P_\ell$ to be the orthogonal projection onto $\mathrm{span}\{e_k\}_{e_k \in S_\ell}$.  Note that
\begin{equation}
\lim_{\ell \rightarrow \infty} P_\ell x = x, \quad x \in H.
\end{equation}
Linearly order $S = (e_n)_{n \in \mathbb{N}}$ such that, if $e_n \in S_k \setminus S_{k-1}$, and $e_m \in S_{k-1}$, then $m < n$.\\

\begin{defin}
Let $(v_k)_{k \in \mathbb{N}}$ be the unit vector basis for $\ell_2(\mathbb{N})$ and define $\phi$ by $\phi e_k = v_k$.  Let $L:H \rightarrow H$ be an onto isomorphism. $L$ is said to be uniformly invertible with respect to the projections $(P_\ell)_{\ell \in \mathbb{N}}$ if $\phi L \phi^{-1}$ is uniformly invertible as a matrix map with respect to the projections $(\pi_{|S_\ell|})$.
\end{defin}

\noindent We define the following notation:
\begin{equation}\label{abusivenot}
(P_\ell L P_\ell)^{-1} := (\pi_{|s_\ell|} (\phi L \phi^{-1}) \pi_{|s_\ell|})^{-1}.
\end{equation}
By saying $P_\ell L P_\ell$ is invertible, we mean that the right hand side of equation (\ref{abusivenot}) is well defined.  If $L$ is defined on $\mathrm{span}(e_n)_{n \in \mathbb{N}}$, but perhaps not on $H$, we define ``$P_\ell L P_\ell$ is invertible'' in the same way.\\

\noindent If $L$ is an operator on $H$ (perhaps densely defined), and $(P_\ell)_{\ell \in \mathbb{N}}$ is a sequence of projections defined above, define the operator $L_\ell = L P_\ell + I-P_\ell.$\\

\begin{defin}
Let $(v_k)_{k \in \mathbb{N}}$ be a Riesz basis for $H$.  We define $(v_k)_{k \in \mathbb{N}}$ to be uniformly invertible with respect to the projections $(P_\ell)_{\ell \in \mathbb{N}}$ if the corresponding isomorphism $L e_k = v_k$ is uniformly invertible with respect to the projections $(P_\ell)_{\ell \in \mathbb{N}}$.
\end{defin}

\noindent We can now state and prove the following lemmas:

\begin{lem}\label{firstone}
Let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis for $H$,  let $(f_n)_{n \in \mathbb{N}} \subset H$, and let $P_\ell$ be the orthogonal projection onto $\mathrm{span}(e_n)_{n \leq \ell}$.  Define $L: \mathrm{span}\{e_n\}_{n \in \mathbb{N}} \rightarrow H$ by $Le_n= f_n$.  For each $\ell>0$, the following statements are equivalent:\\

\noindent 1) $(f_n)_{n \leq \ell} \cup (e_n)_{n > \ell}$ is a Riesz basis for $H$.\\
\noindent 2) $P_\ell L P_\ell$ is invertible.
\end{lem}
\begin{proof}[Proof of Lemma \ref{firstone}]
\noindent 1) $\Longrightarrow$ 2): From the definition of $L_\ell$ we know that it is an onto isomorphism. This yields $P_\ell = P_\ell L P_\ell L_\ell^{-1}$, implying $P_\ell = (P_\ell L P_\ell)(P_\ell L_\ell^{-1} P_\ell)$.\\ 

\noindent 2) $\Longrightarrow$ 1):  Let $A_\ell$ be the unique square matrix such that $P_\ell L P_\ell A_\ell = A_\ell P_\ell L P_\ell = P_\ell$. We need to show that $L_\ell$ is an onto isomorphism.\\

\noindent First we show that $L_\ell$ is one to one.  Say $ 0 = L_\ell x = L P_\ell x +(I-P_\ell)x$, then $0= P_\ell L P_\ell x$, so that $0 = A_\ell P_\ell L P_\ell x = P_\ell x$.  We conclude that $x = (I- P_\ell)x$. This implies $0 = L_\ell (I-P_\ell)x = (I- P_\ell)x =x $.\\

\noindent Next we show that $L_\ell$ is onto. Note $L_\ell (I-P_\ell)x = (I- P_\ell)x$, so we only need to show that for all $x$, $P_\ell x$ is in the range of $L_\ell$.  Define $$y = P_\ell A_\ell P_\ell x +P_\ell x-L P_\ell A_\ell P_\ell x.$$  We have $L_\ell ( P_\ell A_\ell P_\ell x) = L P_\ell A_\ell P_\ell x$ and
\begin{eqnarray}
L_\ell (P_\ell x-L P_\ell A_\ell P_\ell x) & = & L P_\ell x-L(P_\ell L P_\ell)(A_\ell P_\ell)x+P_\ell x - L P_\ell A_\ell P_\ell x\\
& = & P_\ell x - L P_\ell A_\ell P_\ell x,\nonumber
\end{eqnarray}
from which $L_\ell y = P_\ell x$.\\

\noindent $L_\ell$ is a continuous bijection between Hilbert spaces, and hence is an onto isomorphism by the open mapping theorem.
\end{proof}

\begin{lem}\label{secondone}
Define $L$ as in Lemma \ref{firstone}. For all $\ell>0$, $L_\ell$ is an onto isomorphism iff it is one to one.
\end{lem}
\begin{proof}[Proof of Lemma \ref{secondone}]
We only need to show that $P_\ell L P_\ell$ is one to one on $P_\ell H$ when $L_\ell$ is one to one on $H$, and apply Lemma \ref{firstone}.  Let $(P_\ell L P_\ell) P_\ell x =0.$  We have
\begin{eqnarray}
L_\ell [ P_\ell x - (I-P_\ell)L P_\ell x)] & = & L_\ell P_\ell x -L_\ell (I-P_\ell)L P_\ell x \nonumber\\
& = & L_\ell P_\ell x - (I-P_\ell)L P_\ell x\nonumber\\
& = & L_\ell P_\ell x-L P_\ell x = 0,\nonumber
\end{eqnarray}
where the last equality follows from $L_\ell P_\ell = L P_\ell$.  Since $L_\ell$ is one to one, we have that $P_\ell x = (I-P_\ell)LP_\ell x$, so that $P_\ell x = 0.$ 
\end{proof}

\begin{lem}\label{thirdone}
Let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis for $H$, and $(f_n)_{n \in \mathbb{N}}$ be a Riesz basis for $H$, and $(k_\ell)_{\ell \in \mathbb{N}}\subset \mathbb{N}$ be an increasing sequence. Let $P_\ell$ be the orthogonal projection onto $\mathrm{span}\{ e_n\}_{n \leq k_\ell}$, then the following are equivalent:\\

\noindent 1) The operator $L$ is uniformly invertible with respect to $(P_\ell)_{\ell \in \mathbb{N}}$.\\
\noindent 2) For all $\ell >0$, $L_\ell$ is an onto isomorphism, and $\sup_{\ell \in \mathbb{N}} \|L_\ell^{-1} \| < \infty.$
\end{lem}

\begin{proof}[Proof of Lemma \ref{thirdone}]
\noindent 1) $\Longrightarrow$ 2): By Lemma \ref{firstone}, we only need to show that that $$\sup_{\ell \in \mathbb{N}} \|L_\ell^{-1} \| < \infty.$$  This follows from the identity\\
\begin{equation}\label{ident}
L_\ell^{-1} = [I-(I-P_\ell)L](P_\ell L P_\ell)^{-1}+I-P_\ell,
\end{equation}
which is hereby demonstrated:
\begin{eqnarray}
[I-(I-P_\ell)L](P_\ell L P_\ell)^{-1}+I-P_\ell & = & [I-(I-P_\ell)L]P_\ell L_\ell^{-1} P_\ell+I-P_\ell\\
& = & P_\ell L_\ell^{-1} P_\ell -L P_\ell L_\ell^{-1} P_\ell+I\nonumber\\
& = & (I-L)P_\ell L_\ell^{-1} P_\ell +I.\nonumber
\end{eqnarray}
We have $(I-L)P_\ell = I -L_\ell$, so
\begin{eqnarray}
[I-(I-P_\ell)L](P_\ell L P_\ell)^{-1}+I-P_\ell & = & (I-L) L_\ell^{-1} P_\ell +I\\
& = & L_\ell^{-1}P_\ell -P_\ell +I.\nonumber
\end{eqnarray}
Noting that $L_\ell (I-P_\ell) = I-P_\ell$, we obtain $L_\ell^{-1}P_\ell -P_\ell +I = L_\ell^{-1}$, which proves the identity.\\

\noindent 2) $\Longrightarrow$ 1): Noting that $(P_\ell L P_\ell)^{-1} = P_\ell L_\ell^{-1} P_\ell$ yields the result.\\
\end{proof}

\section{The first main result}\label{S:3}

\noindent We begin with some necessary definitions:\\

\begin{defin}\label{impdef}
Define $C_{\ell,d} = \{-\ell,\ldots,\ell\}^d$, and $e_n(x) = \frac{1}{(2\pi)^{d/2}}e^{i\langle x , n \rangle}$ for $n \in \mathbb{Z}^d$.  Let $P_\ell: L_2([-\pi,\pi]^d) \rightarrow L_2([-\pi,\pi]^d)$ be the orthogonal projection from $ L_2([-\pi,\pi]^d)$ onto $\mathrm{span}(e_n)_{n \in C_{\ell,d}}$.
\end{defin}

\noindent Let $(f_n)_{n \in \mathbb{Z}^d}$ be an exponential Riesz basis.  In the following sections, we abbreviate the statement ``$(f_n)_{n \in \mathbb{Z}^d}$ is a uniformly invertible Riesz basis with respect to the projections $(P_\ell)_{\ell \in \mathbb{N}}$ defined in definition \ref{impdef}'' by ``$(f_n)_{n \in \mathbb{Z}^d}$ is a uniformly invertible Riesz basis''.\\ 

\noindent To avoid confusion of indices, we write $t \in \mathbb{R}^d$ as $t=(t(1),\cdots,t(d))$.\\

\noindent For $\ell, d \in \mathbb{N}$ define the multivariate polynomial $$Q_{d,\ell}(t) = \prod_{k_1 =1}^\ell \Big(1-\frac{t(1)^2}{k_1^2}\Big)\cdot\ldots\cdot\prod_{k_d =1}^\ell \Big(1-\frac{t(d)^2}{k_d^2}\Big), \quad t = (t(1),\cdots , t(d)).$$

\noindent Here is the first main result of this paper.

\begin{thm}\label{maintheorem}
Let $(t_n)_{n \in \mathbb{Z}^d} \subset \mathbb{R}^d$, and define $f_n(x) = \frac{1}{(2\pi)^{d/2}}e^{i\langle x , t_n \rangle}$ for $n \in \mathbb{Z}^d$.  Let $(f_n)_{n \in \mathbb{Z}^d}$ be a uniformly invertible Riesz basis, then for all $f \in PW_{[-\pi,\pi]^d}$, there exists a unique sequence of polynomials $(\Psi_\ell)_{\ell \in \mathbb{N}}$,  $\Psi_\ell :\mathbb{R}^d \rightarrow \mathbb{R},$ such that\\

\noindent (a) $\Psi_\ell$ has coordinate degree at most $2 \ell$.\\
\noindent (b) $\Psi_\ell (t_n) = f(t_n)$ for all $n \in C_{\ell,d}$.\\
\noindent (c) $f(t) = \lim_{\ell \rightarrow \infty} \Psi_\ell(t) \frac{\mathrm{SINC} (\pi t)}{Q_{d,\ell} (t)}$, where the limit is both $L_2$ and uniform.
\end{thm}

\noindent Note: The expression in statement (c) of Theorem \ref{maintheorem} has removable singularities, but these can be evaluated by 
\begin{equation}
\lim_{t \rightarrow n} \frac{\mathrm{sinc} (\pi t)}{Q_{1,\ell} (t)} = \frac{(\ell!)^2}{(\ell+n)!(\ell-n)!},\quad n \in \{-\ell,\ldots \ell \}.\nonumber
\end{equation}

\noindent The proof of Theorem \ref{maintheorem} requires several lemmas, beginning with the following equivalence between the existence of particular Riesz bases and a polynomial interpolation condition:

\begin{lem}\label{othermain}
Let $(t_n)_{n \in \mathbb{Z}^d} \subset \mathbb{R}^d$ where  $f_n(x) = \frac{1}{(2\pi)^{d/2}}e^{i\langle x , t_n \rangle}$ and $e_n(x) = \frac{1}{(2\pi)^{d/2}}e^{i\langle x , n \rangle}$.  The sequence $ (f_{\ell,n})_{n \in \mathbb{Z}^d} := (f_n)_{n \in C_{\ell,d}} \cup (e_n)_{n \notin C_{\ell,d}}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$ iff the following conditions hold.\\

\noindent 1) For all $n \in C_{\ell,d}$, $t_n \notin (\mathbb{Z}\setminus \{-\ell,\cdots,\ell\})^d$.\\
\noindent 2) For any sequence $(c_k)_{k \in C_{\ell,d}}$, there exists a unique polynomial $\Psi_\ell$ with coordinate degree at most $2\ell$ such that $\Psi_\ell(t_k)= c_k$ for $k \in C_{\ell,d}$.\\
\end{lem}

\begin{proof}[Proof of Lemma \ref{othermain}]
Suppose that the sequence $(f_n)_{n \in \mathbb{Z}^d}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$.  We compute the biorthogonal functions of $(f_{\ell,n})_{n \in \mathbb{Z}^d}$ when $n \in C_{\ell,d}$:
\begin{eqnarray}
f_{\ell,n}^*  & = & \sum_{k \in \mathbb{Z}^d} \langle f_{\ell,n}^* , e_k \rangle e_k = \sum_{k \in C_{\ell,d}} \langle f_{\ell,n}^* , e_k \rangle e_k + \sum_{k \notin C_{\ell,d}}  \langle f_{\ell,n}^* , e_k \rangle e_k\nonumber\\
& = & \sum_{k \in C_{\ell,d}} \langle f_{\ell,n}^* , e_k \rangle e_k.\nonumber
\end{eqnarray}
Passing to the Fourier transform and defining $G_{\ell, n} = \mathcal{F}(f_{\ell,n}^*)$, we have
\begin{eqnarray}\label{firstsum}
G_{\ell,n}(t) & = & \sum_{k \in C_{\ell,d}} G_{\ell,n}(k)\mathrm{SINC}\pi(t-k)\\
& = & \bigg( \sum_{k \in C_{\ell,d}}\frac{G_{\ell,n}(k)(-1)^{k(1)+\ldots+k(d)} t(1)\cdot\ldots\cdot t(d)}{(t(1)-k(1))\cdot \ldots\cdot(t(d)-k(d))} \bigg)\mathrm{SINC}(\pi t), \quad t \in \mathbb{R}^d.\nonumber
\end{eqnarray}
Denote the $k^{th}$ summand in equation (\ref{firstsum}) by $A_k$, then
\begin{eqnarray}
A_{\ell,n,k}\!\! & = &\!\! A_{\ell,n,k} \frac{\prod_{j_1 = -\ell \atop j_1 \neq k(1)}^\ell (t(1)-j_1) \cdot \ldots \cdot \prod_{j_d = -\ell \atop j_d \neq k(d)}^\ell (t(d)-j_d)}{\prod_{j_1 = -\ell \atop j_1 \neq k(1)}^\ell (t(1)-j_1) \cdot \ldots \cdot \prod_{j_d = -\ell \atop j_d \neq k(d)}^\ell (t(d)-j_d)}\nonumber\\
& = &\!\! \frac{G_{\ell,n}(k)(-1)^{k(1)+\ldots+k(d)}t(1)\cdot\ldots\cdot t(d) \prod_{j_1 = -\ell \atop j_1 \neq k(1)}^\ell (t(1)-j_1) \cdot \ldots \cdot \prod_{j_d = -\ell \atop j_d \neq k(d)}^\ell (t(d)-j_d)}{\prod_{j_1 = -\ell}^\ell (t(1)-j_1) \cdot \ldots \cdot \prod_{j_d = -\ell}^\ell (t(d)-j_d)}\nonumber\\
& = &\!\! \frac{G_{\ell,n}(k)\frac{1}{(\ell!)^2}(-1)^{k(1)+\ldots+k(d) +\ell d}\prod_{j_1 = -\ell \atop j_1 \neq k(1)}^\ell (t(1)-j_1) \cdot \ldots \cdot \prod_{j_d = -\ell \atop j_d \neq k(d)}^\ell (t(d)-j_d)}{\prod_{j_1 = 1}^\ell \Big(1-\frac{t(1)^2}{j_1^2}\Big) \cdot \ldots \cdot \prod_{j_d = 1}^\ell \Big(1-\frac{t(d)^2}{k_d^2}\Big)}\nonumber\\
& = &\!\! \frac{p_{\ell,n,k} (t)}{Q_{d,\ell} (t)},\nonumber
\end{eqnarray}
where $p_{\ell,n,k}$ is some polynomial with coordinate degree at most $2\ell$. Substituting into equation ({\ref{firstsum}}), we obtain
$$G_{\ell,n}(t) = \Big(\sum_{k \in C_{\ell,d}} p_{\ell,n,k}(t) \Big)\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell} (t)}:= \phi_{\ell,n} (t)\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell} (t)},$$ where $\phi_{\ell,n}$ is a polynomial having coordinate degree at most $2 \ell$.  The fact that each zero of $\mathrm{sinc}(\pi z)$ has multiplicity one implies that the zero set of $\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell} (t)}$ (which is entire) is $(\mathbb{Z}\setminus \{-\ell,\cdots, \ell\})^d \subset \mathbb{C}$. Using that $\mathrm{SINC}\pi(x-y)$ is the reproducing kernel for $PW_{[-\pi,\pi]^d}$, we see that  $$G_{\ell,n}(t_m) = \delta_{nm},$$ for all $n,m \in C_{\ell,d}$. This yields that $$1 = \phi_{\ell,n} (t_n)\Big(\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell} (t)}\Big)\Big|_{t_n}.$$ This shows $\phi_{\ell,n} (t_n) \neq 0$ and $\frac{\mathrm{SINC}(\pi t_n)}{Q_{d,\ell} (t_n)}\neq0$ for $n \in C_{\ell,d}$, and that $t_n \notin (\mathbb{Z}\setminus \{-\ell,\cdots, \ell\})^d$ (statement 1) of Lemma \ref{othermain}).\\

\noindent For $n,m \in C_{\ell,d}, n\neq m, $ $$0 = G_{\ell,n}(t_m) = \phi_{\ell,n} (t_m)\Big(\frac{\mathrm{SINC}(\pi t)}{Q_{d,\ell} (t)}\Big)\Big|_{t_m},$$ We conclude that 
\begin{displaymath}
   \phi_{\ell,n}(t_m) = \left\{
     \begin{array}{lr}
       \frac{Q_{d,\ell}(t_n)}{\mathrm{SINC}\pi t_n} \neq 0,  &  n = m\\
       0, & n \neq m
     \end{array}
   \right.
\end{displaymath}
for $n,m \in C_{\ell,d}$ From this, the ``existence'' part of statement 2) in Lemma \ref{othermain} readily follows.  Restated, the evaluation map taking the space of all polynomials of coordinate degree at most $2 \ell$ to $\mathbb{R}^{(2\ell+1)^d}$ is onto.  These spaces have the same dimension, hence the evaluation map is a bijection, which completes the proof of statement 2).\\

\noindent Suppose that 1) and 2) hold.  For $n \in C_{\ell,d}$, let $p_{\ell,n}$ be the unique polynomial of coordinate degree at most $2\ell$ such that $p_{\ell,n} (t_m) = \delta_{nm}$ for $m \in C_{\ell,d}$.  Define 
\begin{equation}\label{biorth}
\Phi_{\ell,n} (t) = \frac{Q_{d,\ell} (t_n) \mathrm{SINC}\pi t}{Q_{d,\ell}(t) \mathrm{SINC}\pi t_n}p_{\ell,n}(t).
\end{equation}
Partial fraction decomposition can be used to show that $\Phi_{\ell,n} \in PW_{[-\pi,\pi]^d}$.  For $n,m \in C_{\ell,d}$, we therefore have $$\delta_{n,m} = \langle \Phi_{\ell,n} (\cdot) , \mathrm{SINC}\pi((\cdot)-t_m) \rangle = \langle \mathcal{F}^{-1}(\Phi_{\ell,n}) , f_m \rangle.$$
Let $L_\ell$ be defined as before.  Let $f=\sum_{n \in \mathbb{Z}^d} c_n e_n$ such that $L_\ell (f) = 0$, then $$0 = \sum_{n \in C_{\ell,d}} c_n f_n+\sum_{n \notin C_{\ell,d}}c_n e_n.$$  If, for each $n \in C_{\ell,d}$ we integrate the above equation against $\mathcal{F}^{-1}(\Phi_{\ell,n})$, we see that $c_k = 0$ for $k \in C_{\ell,d}$, so that $c_k =0$ for all $k \in \mathbb{Z}^d$.  $L_\ell$ is one to one, so by Lemma \ref{secondone}, it is an onto isomorphism from $L_2([-\pi,\pi]^d)$ to itself.
\end{proof}

\begin{proof}[Proof of (a) and (b) of Theorem \ref{maintheorem}]
Lemmas \ref{thirdone} and \ref{othermain} imply the existence of a unique sequence of polynomials satisfying statements (a) and (b) of Theorem \ref{maintheorem}, namely,
$$\Psi_\ell(t) = \sum_{n \in C_{\ell,d}} f(t_n) p_{\ell,n}(t),$$ where $p_{\ell,n}$ is defined as in the proof of Lemma \ref{othermain}.\\
\end{proof}

\noindent It remains to show that this sequence of polynomials satisfies the statement (c) of Theorem \ref{maintheorem}.

\begin{prop}\label{fourthone}
\noindent The following are equivalent.\\

\noindent 1) $L$ is uniformly invertible with respect to the projections $(P_\ell)_{\ell \in \mathbb{N}}$.\\
\noindent 2) For all $x \in H$, $\lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1}(I-P_\ell)x=0$.
\end{prop}

\begin{proof}[Proof of Proposition \ref{fourthone}]
\noindent It is clear that 1) implies 2). For the other direction, note that the equality $L_\ell^* = P_\ell L^* +I - P_\ell$ implies that
\begin{equation}\label{prebiorthapprox}
I = (L_\ell^*)^{-1} P_\ell L^* +(L_\ell^*)^{-1}(I-P_\ell).
\end{equation}
This implies that $(L_\ell^*)^{-1}P_\ell$ is pointwise bounded.  Together with the assumption in 2), this implies $(L_\ell^*)^{-1}$ is pointwise bounded, hence norm bounded by the uniform boundedness principle.  This yields uniform invertibility of $L$.
\end{proof}

\begin{lem}\label{blip}
\noindent The following are equivalent:\\

\noindent 1) For all $g \in L_2([-\pi,\pi]^d)$, we have
\begin{equation}\label{biorthapprox}
g = \lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1} P_\ell L^* g.
\end{equation}
\noindent 2) $L$ is uniformly invertible.
\end{lem}
\begin{proof}[Proof of Lemma \ref{blip}]
Recall equation (\ref{prebiorthapprox}) and apply Proposition \ref{fourthone}.
\end{proof}

\begin{prop}\label{blip2}
Statement (c) of Theorem \ref{maintheorem} is true iff
\begin{eqnarray}\label{equivlimit}
0 = \lim_{\ell \rightarrow \infty} \sum_{n \in C_{\ell,d}} |f(t_n)|^2\bigg[1- \frac{\mathrm{SINC} \pi t_n}{Q_{d,\ell} (t_n)} \bigg]^2:=  \lim_{\ell \rightarrow \infty} S_{\ell,d}, \quad f\in PW_{[-\pi,\pi]^d}.
\end{eqnarray}
\end{prop}

\begin{proof}[Proof of Proposition \ref{blip2}]
\noindent Note that $Le_n = f_n$ implies that $f_n^* = (L^*)^{-1}e_n.$  Similarly, $f_{\ell,n}^* = (L_\ell^*)^{-1}e_n$.  Given $f \in PW_{[-\pi,\pi]^d}$, let $g= \mathcal{F}^{-1}(f)$.  Equation (\ref{biorthapprox}) shows:
\begin{eqnarray}
\mathcal{F}^{-1}(f) & = & \lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1} \sum_{n \in C_{\ell,d}} \langle L^* g, e_n \rangle e_n = \lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1} \sum_{n \in C_{\ell,d}} \langle g, f_n \rangle e_n\nonumber\\
& = & \lim_{\ell \rightarrow \infty} \sum_{n \in C_{\ell,d}} \langle g, f_n \rangle f_{\ell,n}^* = \lim_{\ell \rightarrow \infty} \sum_{n \in C_{\ell,d}} f(t_n) f_{\ell,n}^*.\nonumber
\end{eqnarray}
Passing to the Fourier transform, we have
\begin{equation}\label{blah}
f = \lim_{\ell \rightarrow \infty} \sum_{n \in C_{\ell,d}} f(t_n) \mathcal{F}(f_{\ell,n}^*), \quad f \in PW_{[-\pi,\pi]^d},
\end{equation}
where convergence in both $L_2$ and uniform.  The values of a function in $PW_{[-\pi,\pi]^d}$ on the set $(t_n)_{n \in \mathbb{Z}^d}$ uniquely determine the function.  This and equation (\ref{biorth}) show that $$\mathcal{F}(f_{\ell,n}^*)(t) = G_{\ell,n}(t) = \frac{Q_{d,\ell} (t_n) \mathrm{SINC}\pi t}{Q_{d,\ell}(t) \mathrm{SINC}\pi t_n}p_{\ell,n}(t), \quad n \in C_{\ell,d}.$$
This implies that
\begin{eqnarray}
\Psi_\ell(t) \frac{\mathrm{SINC} (\pi t)}{Q_{d,\ell} (t)} & = & \Big(\sum_{n \in C_{\ell,d}} f(t_n) p_{\ell,n}(t)\Big)  \frac{\mathrm{SINC} (\pi t)}{Q_{d,\ell} (t)}\nonumber\\
& = & \sum_{n \in C_{\ell,d}} f(t_n) \frac{\mathrm{SINC} \pi t_n}{Q_{d,\ell} (t_n)}\mathcal{F}(f_{\ell,n}^*)(t).\nonumber
\end{eqnarray}
Combined with equation (\ref{blah}), we see that statement (c) of Theorem \ref{maintheorem} holds iff
\begin{eqnarray}
0  =  \lim_{\ell \rightarrow \infty} \sum_{n \in C_{\ell,d}} f(t_n)\bigg[1- \frac{\mathrm{SINC} \pi t_n}{Q_{d,\ell} (t_n)} \bigg] \mathcal{F}(f_{\ell,n}^*),  \quad f\in PW_{[-\pi,\pi]^d},\nonumber
\end{eqnarray}
where the limit is in the $L_2$ sense. Passing to the inverse Fourier transform, statement (c) holds iff
\begin{eqnarray}\label{blip3}
0 = \lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1}\bigg( \sum_{n \in C_{\ell,d}} f(t_n)\bigg[1- \frac{\mathrm{SINC} \pi t_n}{Q_{d,\ell} (t_n)} \bigg] e_n \bigg), \quad f\in PW_{[-\pi,\pi]^d}.
\end{eqnarray}
In addition to having uniformly norm bounded inverses, $L_\ell$ is pointwise bounded, so there exists $m, M>0$ such that for all $\ell \in \mathbb{N}$, $f \in PW_{[-\pi,\pi]^d}$, $m\| f\| \leq \|(L_\ell^*)^{-1} f \| \leq M\| f \|$.  This, combined with equation (\ref{blip3}), proves the proposition.
\end{proof}

\begin{prop}\label{sinclimitstuff}
\noindent The following are true:\\

\noindent 1) $\sup_{x \in \mathbb{R}} \sup_{\ell \in \mathbb{N}} \Big|\frac{\mathrm{sinc}(\pi x)}{Q_{1,\ell}(x)} \Big|=1.$\\

\noindent 2) Define $\Delta_{\ell,d} = \Big\{ n \in \mathbb{Z}^d \big| \Big\| \frac{t_n}{\ell+1} \Big\|_{\infty} < \frac{1}{\ell^{2/3}} \Big\}$ for $\ell \in \mathbb{N}$, then
\begin{eqnarray}
0 \leq 1-\frac{\mathrm{SINC}(\pi t_n)}{Q_{d,\ell} (t_n)} < 1-e^{\frac{-d(\ell+2)}{\ell^{4/3}-1}}, \quad n \in \Delta_{\ell,d}.
\end{eqnarray}
\end{prop}

\begin{proof}[Proof of Proposition \ref{sinclimitstuff}]
\noindent For 1), the identity $$\mathrm{sinc}(\pi t) = \prod_{k=1}^\infty \Big( 1-\frac{t^2}{k^2} \Big)$$ implies
\begin{equation}
\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell} (t)} = \prod_{k=\ell+1}^\infty \Big(1-\frac{t^2}{k^2} \Big),
\end{equation}
where convergence is uniform on compact subsets of $\mathbb{C}$.  Fix $\ell \in \mathbb{N}$.  If $t \in [0,\ell+1]$, then $\Big|\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)} \Big| \leq1.$  Note that $|Q_{1,\ell} (t)| = \prod_{k=1}^\ell \Big(\frac{t^2}{k^2} -1 \Big)$ is increasing on $(\ell+1, \infty)$.  If $t \in (\ell+1, \infty)$, then
\begin{equation}
\Big|\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)} \Big| = \Big|\frac{\sin(\pi t)}{\pi t Q_{1,\ell}(t)} \Big| < \frac{1}{\pi(\ell+1)| Q_{1,\ell} (\ell+1)|}.\nonumber
\end{equation}
Computation yields
\begin{equation}
| Q_{1,\ell} (\ell+1)| = \frac{(2\ell+1)!}{\ell! (\ell+1)!},\nonumber
\end{equation}
so $$ \Big|\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)} \Big| < \frac{(\ell!)^2}{\pi(2\ell+1)!} <1 .$$
Observing that $\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)}$ is even proves 1).\\

\noindent For 2), let $t \in \mathbb{R}$ such that $\Big| \frac{t}{\ell+1} \Big| < \frac{1}{\ell^{2/3}}$, then $0 < \frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)}$, and
\begin{eqnarray}\label{logstuff}
-\log\Big( \frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)} \Big) &  = & -\sum_{k= \ell+1}^\infty \log\Big(1- \frac{t^2}{k^2} \Big) = \sum_{k= \ell+1}^\infty \sum_{j=1}^\infty \frac{t^2}{j k^{2j}}\\
& = & \sum_{j=1}^\infty \frac{1}{j} \Big(\sum_{k= \ell+1}^\infty \frac{1}{k^{2j}} \Big) t^{2j}.\nonumber
\end{eqnarray}
Basic calculus shows that $$\sum_{k= \ell+1}^\infty \frac{1}{k^{2j}} <\frac{1}{(\ell+1)^{2j}}+\frac{1}{(2j-1)(\ell+1)^{2j-1}}.$$  Equality (\ref{logstuff}) implies
\begin{eqnarray}\label{morelogstuff}
-\log\Big( \frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell}(t)} \Big) & < &\sum_{j=1}^\infty \frac{1}{j}\Big(\frac{t}{\ell+1}\Big)^{2j}+  (\ell+1)\sum_{j=1}^\infty \frac{1}{j(2j-1)}\Big(\frac{t}{\ell+1}\Big)^{2j}\\
& < & (\ell+2)\sum_{j=1}^\infty \Big(\frac{t}{\ell+1}\Big)^{2j} < \frac{\ell+2}{\ell^{4/3}-1}.\nonumber
\end{eqnarray}
If $n \in \Delta_{\ell,d}$, then for each $1 \leq k \leq d$, $\Big| \frac{t_n (k)}{\ell+1} \Big| >\frac{\ell+2}{\ell^{4/3}-1}$, so that
\begin{equation}
\log \Big(\frac{\mathrm{SINC}(\pi t_n)}{Q_{d,\ell}(t_n)} \Big)  = \sum_{k=1}^d \log \Big( \frac{\mathrm{sinc}(\pi t_n(k))}{Q_{1,\ell}(t_n(k))} \Big) > -\frac{d(\ell+2)}{\ell^{4/3}-1}.\nonumber
\end{equation}
Statement 2) of Proposition \ref{sinclimitstuff} follows readily.
\end{proof}

\begin{proof}[Proof of statement (c) in Theorem \ref{maintheorem}]
Proposition \ref{sinclimitstuff} gives the following:\\
\begin{eqnarray}
S_{\ell,d} & \leq & \Big(\sum_{n \in \Delta_{\ell,d}} + \sum_{n \in \mathbb{Z}^d \setminus\Delta_{\ell,d}}\Big) |f(t_n)|^2\bigg[1- \frac{\mathrm{SINC} \pi t_n}{Q_{d,\ell} (t_n)} \bigg]^2 \nonumber\\
& \leq &  \bigg( 1-e^{\frac{-d(\ell+2)}{\ell^{4/3}-1}} \bigg)^2 \sum_{n \in \mathbb{Z}^d} |f(t_n)|^2 + \sum_{n:\ \frac{\ell+1}{\ell^{2/3}}\leq \| t_n \|_\infty} 4 | f(t_n) |^2.
\end{eqnarray}
Now $(f(t_n))_{n \in \mathbb{Z}^d} \in \ell_2(\mathbb{Z}^d)$ implies that $\lim_{\ell \rightarrow \infty} S_{\ell,d} =0$, so that by Proposition \ref{blip2}, statement (c) in Theorem \ref{maintheorem} is true.
\end{proof}

\section{The second main result}\label{S:4}

\noindent Theorem \ref{maintheorem} can be simplified.  The function $$t \mapsto \frac{\mathrm{SINC} (\pi t)}{Q_{d,\ell} (t)}$$ becomes more computationally complex for large values of $\ell$.  If, at the cost of global $L_2$ and uniform convergence, we adopt an approximation 
\begin{equation}
\mathrm{SINC} (\pi t) \simeq Q_{d,\ell} (t) e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}},
\end{equation}
we bypass this difficulty as the exponent of the above quantity is simply a rational function of $\ell >0$.  This is precisely quantified in Theorem \ref{simplethm}, which is the second main result of this paper.

\begin{thm}\label{simplethm}
Let $(t_n)_{\mathbb{Z}^d} \subset \mathbb{R}^d$ be a sequence such that the associated exponential functions are a uniformly invertible Riesz basis for $L_2([-\pi,\pi]^d)$.  For $N \in \{0,1,2,\ldots \}$, $A>0$, let $E_{\ell,N,A} = [-A (\ell+1/2)^\frac{2N+1}{2N+2},A (\ell+1/2)^\frac{2N+1}{2N+2}]$.  Let $f \in PW_{[-\pi,\pi]^d}$ where $(\Psi_\ell)_\ell$ is the sequence of interpolating polynomials from Theorem \ref{maintheorem}, and let $N \geq 0$.  Let $f \in PW_{[-\pi,\pi]^d}$, and define $$I_{f,\ell} (t) = \Psi_\ell(t) e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}},$$ then
\begin{equation}\label{simplecor2}
\lim_{\ell \rightarrow \infty} \Big\| f(t) - I_{f,\ell}(t) \Big\|_{L_2((E_{\ell,N,A})^d)}=0,
\end{equation}
and
\begin{equation}\label{simplecor1}
\lim_{\ell \rightarrow \infty} \Big\| f(t) - I_{f,\ell} (t) \Big\|_{L_\infty((E_{\ell,N,A})^d)}=0.
\end{equation}
\end{thm}

\noindent If $N=0$ in Theorem \ref{simplethm}, we have the following extension of Corollary \ref{lyuseippavcor} to arbitrary multivariate bandlimited functions (at the expense of introducing uniform invertibility):
\begin{cor}\label{simplecormod}
For all $f \in PW_{[-\pi,\pi]^d}$, we have
\begin{equation
\lim_{\ell \rightarrow \infty} \Big\| f(t) - \Psi_\ell(t)\Big\|_{L_2([-A(\ell+1/2)^{1/2},A(\ell+1/2)^{1/2}]^d)}=0,
\end{equation}
and
\begin{equation}\label{simplecor1mod}
\lim_{\ell \rightarrow \infty} \Big\| f(t) - \Psi_\ell(t)\Big\|_{L_\infty([-A(\ell+1/2)^{1/2},A(\ell+1/2)^{1/2}]^d)}=0.
\end{equation}
\end{cor}
\noindent It is evident that if $(t_n)_{n \in \mathbb{Z}^d}\subset \mathbb{R}^d$ is any subset such that the associated sequence of exponentials $(f_n)_{n \in \mathbb{Z}^d}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$, then the map $f \mapsto (f(t_n))_{n \in \mathbb{Z}^d}$ is a bijection from $PW_{[-\pi,\pi]^d}$ to $\ell_2(\mathbb{Z}^d)$.  This allows for a nice interpretation of Corollary \ref{simplecormod}.  Given a sequence $(t_n)_{n \in \mathbb{Z}^d}\subset \mathbb{R}^d$ (subject to the hypotheses of Theorem \ref{simplethm}), and sampled data $\big( (t_n,c_n) \big)_{n \in \mathbb{Z}^d}$ where $(c_n)_{n \in \mathbb{Z}^d} \in \ell_2(\mathbb{Z}^d)$, then a unique sequence of Lagrangian polynomial interpolants exists, and in the appropriate limit, converges to the unique band-limited interpolant of the same data.\\

\noindent When $N=1$, we have a sampling theorem with a Gaussian multiplier: $$ f(t) \simeq \Psi_\ell (t) e^{-\frac{\| t\|_2^2}{(\ell+1/2)}}, \quad f \in PW_{[-\pi,\pi]^d}.$$  Compare Theorem \ref{simplethm} with Theorem 2.6 in \cite{BSS}, which is a multivariate sampling theorem with a Gaussian multipler with global $L_2$ and uniform convergence.  Also compare Theorem \ref{simplethm} with Theorem 2.1 in \cite{SS}, which, when $d=1$ and the data nodes are equally spaced, gives another recovery formula involving a Gaussian mulitplier in the context of over-sampling.\\

\noindent The proof of Theorem \ref{simplethm} relies on two lemmas:

\begin{lem}\label{techlem1}
Let $d \in \mathbb{N}$, $N \in \{0,1,2,\ldots \}$, and $A > 0$.  There exists $M >0$ such that for sufficiently large $\ell$, and any $t \in (E_{\ell,N,A})^d$, we have
\begin{eqnarray}
& &\Big|Q_{d,\ell}(t)  e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} - e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)} \mathrm{SINC}(\pi t)\Big|\nonumber\\
& &\quad\leq M(\ell+1/2)^{-\frac{1}{N+1}}|\mathrm{SINC}(\pi t)|.\nonumber
\end{eqnarray}
\end{lem}

\begin{lem}\label{techlem2}
For all $f \in PW_{[-\pi,\pi]^d}$ and $N \in \{0,1,2,\ldots \}$, we have
\begin{equation}
\lim_{\ell \rightarrow \infty} \sup_{t \in (E_{\ell,N,A})^d} \Big| \Big(e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}  -1 \Big)f(t) \Big| = 0.\nonumber
\end{equation}
\end{lem}

\noindent The proof of Lemma \ref{techlem1} relies on the following proposition.

\begin{prop}\label{blip5}
If $f:(0,\infty) \rightarrow (0,\infty)$ is convex, decreasing, differentiable, and integrable away from $0$, then
$$\frac{1}{4}f'(\ell+1/2)\leq \sum_{k=\ell+1}^\infty f(k) -\int_{\ell+1/2}^\infty f(x) dx \leq 0.$$
\end{prop}
\begin{proof}[Proof of Proposition \ref{blip5}]
The proof follows naturally from geometric considerations.
\end{proof}

\begin{proof}[Proof of Lemma \ref{techlem1}]
Letting $|t| < \ell+1/2$ and recalling equation (\ref{logstuff}), we see that
\begin{eqnarray}\label{blork2}
& &-\log\Big(\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell} (t)}\Big)-\sum_{k=1}^{\infty} \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}\\
& & \quad = \sum_{k=1}^\infty \Big[ \sum_{j=\ell+1}^\infty \frac{1}{j^{2k}} -\frac{1}{(2k-1)(\ell+1/2)^{2k-1}} \Big] \frac{t^{2k}}{k}.\nonumber
\end{eqnarray}
Applying Proposition \ref{blip5} to the function $f(t) = \frac{1}{t^{2k}}$ when $k \geq 1$, we obtain
\begin{equation}
\frac{-k}{2(\ell+1/2)^{2k+1}} \leq  \sum_{j=\ell+1}^\infty \frac{1}{j^{2k}} -\frac{1}{(2k-1)(\ell+1/2)^{2k-1}} \leq 0.\nonumber
\end{equation}
Equation (\ref{blork2}) becomes
\begin{eqnarray}
\frac{-1}{2(\ell+1/2)}\sum_{k=1}^{\infty} \Big(\frac{t}{\ell+1/2}\Big)^{2k} \leq -\log\Big(\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell} (t)}\Big)-\sum_{k=1}^{\infty} \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}} \leq 0.\nonumber
\end{eqnarray}
Restated,
\begin{eqnarray}
& &-\frac{1}{2(\ell+1/2)}\frac{\Big( \frac{t}{\ell+1/2} \Big)^2}{1-\Big(\frac{t}{\ell+1/2} \Big)^2}+\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}\\
& &\quad\leq -\log\Big(\frac{\mathrm{sinc}(\pi t)}{Q_{1,\ell} (t)}\Big)-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}\nonumber\\
& &\quad\leq  \sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}.\nonumber
\end{eqnarray}
Exponentiating,
\begin{eqnarray}\label{blork3}
& & e^{\bigg(-\frac{1}{2(\ell+1/2)}\frac{\big( \frac{t}{\ell+1/2} \big)^2}{1-\big(\frac{t}{\ell+1/2} \big)^2}\bigg)} e^{\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}\\
& &\quad \leq\frac{Q_{1,\ell} (t)e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{sinc}(\pi t)} \leq  e^{\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}.\nonumber
\end{eqnarray}
Let $\ell$ be chosen large enough so that $A (\ell+1/2)^\frac{2N+1}{2N+2} <\ell+1/2$. If $\ell$ is large enough, then for any $t \in E_{\ell N,A}$, $t=c (\ell+1/2)^\frac{2N+1}{2N+2}$ for some $c \in [-A,A]$.  For such $t$, inequality (\ref{blork3}) implies
\begin{eqnarray}
& & e^{\bigg(-\frac{1}{2(\ell+1/2)^\frac{N+2}{N+1}}\frac{c^2}{1-c^2(\ell+1/2))^\frac{-1}{N+1}}\bigg)} e^{\sum_{k=N+1}^\infty \frac{c^{2k}}{k(2k-1)}(\ell+1/2)^{\big(1-\frac{k}{N+1}\big)}}\nonumber\\
& & \quad\leq \frac{Q_{1,\ell} (t)e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{sinc}(\pi t)} \leq  e^{\sum_{k=N+1}^\infty \frac{c^{2k}}{k(2k-1)}(\ell+1/2)^{\big(1-\frac{k}{N+1}\big)}}.\nonumber
\end{eqnarray}
If $t \in (E_{\ell,N,A})^d$, then $t=c (\ell+1/2)^\frac{2N+1}{2N+2}$ for some $c \in [-A,A]^d$.  For any such $t$, we have
\begin{eqnarray}\label{blork4}
& & e^{\bigg(-\frac{d}{2(\ell+1/2)^\frac{N+2}{N+1}}\frac{A^2}{1-A^2(\ell+1/2))^\frac{-1}{N+1}}\bigg)} e^{\sum_{k=N+1}^\infty \frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big(1-\frac{k}{N+1}\big)}}\\
& & \quad\leq\frac{Q_{d,\ell} (t)e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{SINC}(\pi t)} \leq  e^{\sum_{k=N+1}^\infty \frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big(1-\frac{k}{N+1}\big)}}.\nonumber
\end{eqnarray}
On one hand,
\begin{eqnarray}\label{upperinequall}
e^{\sum_{k=N+1}^\infty \frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big(1-\frac{k}{N+1}\big)}} \leq e^{\Big( \frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)} + O\big( (\ell+1/2)^\frac{-1}{N+1} \big) \Big)}
\end{eqnarray}
where the ``big O'' constant is independent of $c \in [-A,A]^d$.
\noindent On the other hand,
\begin{eqnarray}\label{lowerinequal}
e^{\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}} \leq e^{\sum_{k=N+1}^\infty \frac{\|c\|_{2k}^{2k}}{k(2k-1)}(\ell+1/2)^{\big(1-\frac{k}{N+1}\big)}}.
\end{eqnarray}
Inequality (\ref{blork4}) yields
\begin{eqnarray}\label{blork5}
& & \bigg(e^{-\frac{d}{2(\ell+1/2)^\frac{N+2}{N+1}}\frac{A^2}{1-A^2(\ell+1/2))^\frac{-1}{N+1}}}-1\bigg)e^\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}\\
& &\quad\leq \frac{Q_{d,\ell} (t)e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{SINC}(\pi t)}-e^\frac{\|c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}\nonumber\\
& &\quad\leq e^\frac{d A^{2(N+1)}}{(N+1)(2N+1)} \Big(e^{O\bigg( \frac{1}{(\ell+1/2)^\frac{1}{N+1}}\bigg)}-1 \Big).\nonumber
\end{eqnarray}
The left most side of inequality (\ref{blork5}) is of the order $O((\ell+1/2)^{-\frac{N+2}{N+1}})$, and the right most side of inequality (\ref{blork5}) is of the order $O((\ell+1/2)^{-\frac{1}{N+1}})$, where the ``big O'' constants are independent of $c\in [-A,A]^d$.  The lemma follows readily.
\end{proof}

\begin{proof}[Proof of Lemma \ref{techlem2}]
Equivalently, we need to show $$\lim_{\ell \rightarrow \infty} \sup_{c \in [-A,A]^d} \Big| \Big(e^\frac{\| c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}  -1 \Big)f\big(c (\ell+1/2)^\frac{2N+1}{2N+2}\big) \Big| = 0.$$  Suppose the contrary.  Let $c_\ell \in [-A,A]^d$ be a value that maximizes the $\ell$-th term in the above limit.  There exists $(\ell_k)_{k \in \mathbb{N}}$, and $\epsilon>0$ such that for all $k \in \mathbb{N}$,
\begin{eqnarray}
\epsilon  &\leq&  \sup_{c \in [-A,A]^d} \Big| \Big(e^\frac{\| c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}  -1 \Big)f\big(c (\ell_k+1/2)^\frac{2N+1}{2N+2}\big) \Big|\nonumber\\ &\leq& \Big(e^\frac{d A^{2(N+1)}}{(N+1)(2N+1)}-1\Big)\big| f\big(c_{\ell_k}(\ell_k+1/2)^\frac{2N+1}{2N+2}\big) \big|,\nonumber
\end{eqnarray}
so that the sequence $\big( f\big(c_{\ell_k} (\ell_k+1/2)^\frac{2N+1}{2N+2}\big) \big)_{k \in \mathbb{N}}$ is bounded away from $0$.  This implies there exists $\delta >0$ such that $\big\| c_{\ell_k} (\ell_k+1/2)^\frac{2N+1}{2N+2} \big\|_{2(N+1)} \leq \delta$ for $k \in \mathbb{N}$, that is, $\| c_{\ell_k} \|_{2(N+1)} \leq \delta (\ell_k+1/2)^{-\frac{2N+1}{2N+2}}$.  This forces
\begin{eqnarray}
\epsilon &\leq&  \sup_{c \in [-A,A]^d} \Big| \Big(e^\frac{\| c\|_{2(N+1)}^{2(N+1)}}{(N+1)(2N+1)}  -1 \Big)f\big(c (\ell_k+1/2)^\frac{2N+1}{2N+2}\big) \Big|\nonumber\\  &\leq & \Big(e^\frac{\delta^{2(N+1)}}{(\ell_k+1/2)^{2N+1}(N+1)(2N+1)} -1 \Big)\| f \|_\infty.\nonumber
\end{eqnarray}
The last term in the above inequality has limit $0$ as $\ell \rightarrow \infty$, which is a contradiction.
\end{proof}

\noindent Now we can prove Theorem \ref{simplethm}.\\

\begin{proof}[Proof of Theorem \ref{simplethm}]
If $f \in PW_{[-\pi,\pi]^d}$, Theorem \ref{maintheorem} states that $$f(t) = \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} +\xi_\ell (t)$$ where $\xi_\ell \rightarrow 0$ on $\mathbb{R}^d$ both in $L_2$ and $L_\infty$ senses.  By Lemma \ref{techlem1}, we have
\begin{eqnarray}\label{firstpart}
& &\sup_{t \in (E_{\ell,N,A})^d} \Big| \Psi_\ell(t)  e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} - e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)} \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} \Big|\\
& &\quad\quad\leq M (\ell+1/2)^{-\frac{1}{N+1}}\sup_{t \in (E_{\ell,N,A})^d}(|f(t)| - |\xi_\ell(t)|),\nonumber
\end{eqnarray}
the right side of which has zero limit.  Also,
\begin{eqnarray}\label{secondpart}
& &\sup_{t \in (E_{\ell,N,A})^d}\quad\Big| \Big(e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}  -1 \Big) \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} \Big|\\
& &\quad \leq \sup_{t \in (E_{\ell,N,A})^d} \Big| \Big(e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}  -1 \Big)f(t) \Big|+\nonumber\\
& & \quad\quad\Big(e^\frac{d A^{2(N+1)}}{(N+1)(2N+1)} -1 \Big) \sup_{t \in (E_{\ell,N,A})^d} |\xi_\ell(t) |,\nonumber
\end{eqnarray}
whose right hand side, by Lemma \ref{techlem2}, also has zero limit.  Combining inequalities (\ref{firstpart}) and (\ref{secondpart}), we obtain
\begin{equation}
\lim_{\ell \rightarrow \infty} \Big\|  \Psi_\ell(t)  e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} - \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} \Big\|_{L_\infty((E_{\ell,N,A})^d)}=0.\nonumber
\end{equation}
Equation (\ref{simplecor1}) follows by a final application of Theorem \ref{maintheorem}.\\

\noindent Now we prove equation (\ref{simplecor2}).  Lemma \ref{techlem1} and Theorem \ref{maintheorem} imply
\begin{eqnarray}\label{firstpartl2}
 & &\Big\| \Psi_\ell(t)  e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} - e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)} \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} \Big\|_{L_2((E_{\ell,N,A})^d)}\\
& &\quad\leq M (\ell+1/2)^{-\frac{1}{N+1}} \|f + \xi_\ell\|_{L_2((E_{\ell,N,A})^d)},\nonumber
\end{eqnarray}
the right hand side of which has zero limit.  Also,
\begin{eqnarray}\label{secondpartl2}
& &\Big\| \Big(e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}  -1 \Big) \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} \Big\|_{L_2((E_{\ell,N,A})^d)}\\
& &\quad\leq \Big\| \Big(e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}  -1 \Big)f(t) \Big\|_{L_2((E_{\ell,N,A})^d)}+\nonumber\\
& & \quad\quad\Big\| \Big(e^\frac{\| t\|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}  -1 \Big)\xi_\ell(t) \Big\|_{L_2((E_{\ell,N,A})^d)}.\nonumber
\end{eqnarray}
The second term in the right hand side of inequality (\ref{secondpartl2}) is bounded from above by $$\Big(e^\frac{d A^{2(N+1)}}{(N+1)(2N+1)} -1 \Big) \|\xi_\ell\|_{L_2((E_{\ell,N,A})^d)},$$ which has zero limit.  The integrand of the first term in the right hand side of inequality (\ref{secondpartl2}) (as a function over $\mathbb{R}^d$), converges uniformly to zero by Lemma \ref{techlem2}, and is bounded from above by $$\Big(e^\frac{d A^{2(N+1)}}{(N+1)(2N+1)} -1 \Big) |f(t)|^2 \in L_1(\mathbb{R}^d),$$ so this term has zero limit by the Dominated Convergence Theorem. Combining equations (\ref{firstpartl2}) and (\ref{secondpartl2}) yields
\begin{equation}
\lim_{\ell \rightarrow \infty} \Big\|  \Psi_\ell(t)  e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} - \frac{\Psi_\ell(t)}{Q_{d,\ell} (t)}\mathrm{SINC(\pi t)} \Big\|_{L_2((E_{\ell,N,A})^d)}=0.\nonumber
\end{equation}
Equation (\ref{simplecor2}) follows by a final application of Theorem \ref{maintheorem}.
\end{proof}

\noindent The optimal growth for any $(E_{\ell,N,k})_\ell$ such that Theorem \ref{simplethm} holds is not known, but an upper bound for the rate is established in the Appendix.

\section{Examples of uniformly invertible exponential Riesz Bases}\label{S:6}

\noindent  Theorems \ref{maintheorem} and \ref{simplethm} both require uniform invertibility of $(f_n)_{n \in \mathbb{Z}^d}$.  Fortunately, there are significant classes of exponential Riesz bases  $(f_n)_{n \in \mathbb{Z}^d}$ which have this property.  Consider the  exponential Riesz bases described given by Theorem \ref{kadeclike} (Corollary 6.1 in \cite{BB}) and Theorem \ref{sunzhou} (Theorem 1.3 in \cite{SZ}).

\begin{thm}\label{kadeclike}
Let $(t_k)_{k \in \mathbb{Z}^d} \subset \mathbb{R}^d$ such that
\begin{equation}
\sup_{n \in \mathbb{Z}^d} \Arrowvert n - t_n \Arrowvert_\infty = L < \frac{\ln(2)}{\pi d},\nonumber
\end{equation}
then the sequence $(f_k)_{k \in \mathbb{Z}^d}$ defined by $f_k (x) = \frac{1}{(2\pi)^{d/2}}e^{i \langle x , t_k \rangle}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$.
\end{thm}

\begin{thm}\label{sunzhou}
For $d \geq 1$, define $$D_d(x) := \Big(1-\cos \pi x+\sin \pi x  +\frac{\sin \pi x}{\pi x} \Big)^d -\Big( \frac{\sin \pi x}{\pi x} \Big)^d,$$ and let $x_d$ be the unique number such that $0 < x_d \leq 1/4$ and $D_d(x_d)=1$.  Let $(t_k)_{k \in \mathbb{Z}^d} \subset \mathbb{R}^d$ such that  
\begin{equation}
\sup_{n \in \mathbb{Z}^d} \| n - t_n \|_\infty< x_d,\nonumber
\end{equation}
then the sequence $(f_k)_{k \in \mathbb{Z}^d}$  defined by $f_k (x) = \frac{1}{(2\pi)^{d/2}}e^{i \langle x , t_k \rangle}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$.
\end{thm}

\noindent It is worth noting that when $d=1$, Theorem \ref{sunzhou} reduces to the classical Kadec's 1/4 Theorem, first proven in \cite{Kad}. A proof of Kadec's 1/4 Theorem can also be found in \cite[Theorem 14, page 36]{Y}.    The proofs of the above theorems show that the map $L e_n = f_n$ satisfies $\| I- L  \| = \delta < 1$, from which we see that $L$ is invertible.  Uniform invertibility is readily verified, or can be seen as a consequence of the following more general proposition.

\begin{prop}\label{normpert}
Let $L: H \rightarrow H$ be a uniformly invertible operator with respect to $(P_\ell)_{\ell \in \mathbb{N}}$, where $$\limsup_{\ell \rightarrow \infty}\| (P_\ell L P_\ell)^{-1} \| = M <\infty.$$  If $A$ is an operator such that
\begin{equation}\label{neumann}
\| L- A \| =\frac{\gamma}{M}
\end{equation}
for some $\gamma <1$, then there exists $N \in \mathbb{N}$ such that $A$ is uniformly invertible with respect to $(P_\ell)_{\ell \geq N}$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{normpert}]
Using uniform invertiblity of $L$, and noting that $(L_\ell^*)^{-1} -(L^*)^{-1} = (L_\ell^*)^{-1}(L^* - L_\ell^*)(L^*)^{-1}$ and $\lim_{\ell \rightarrow \infty} L_\ell^* x=L^* x$ for all $x \in H$, we see that $ \lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1}x = (L^*)^{-1}x$ for all $x \in H$.  Equation (\ref{ident}) implies that $\lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1} x - (P_\ell L^* P_\ell)^{-1} x =0$ for all $x \in H$. Together we have $$\lim_{\ell \rightarrow \infty} (P_\ell L^* P_\ell)^{-1} x  = (L^*)^{-1}x ,\quad x\in H.$$ General principles imply
\begin{equation}
\| L^{-1} \| \leq \liminf_{\ell \rightarrow \infty} \| (P_\ell L P_\ell)^{-1} \| \leq M.\nonumber
\end{equation}
This yields $\| L - A \| \leq \frac{\gamma}{\| L^{-1}\|}$, so that $A$ is invertible by usual Neumann series manipulation.
Equation (\ref{neumann}) yields that $\| P_\ell L P_\ell- P_\ell A P_\ell \| \leq \frac{\gamma}{M}$, where the norm is now the standard  matrix norm on the set of matrices of  dimension $\mathrm{dim}(\mathrm{ran}{P_\ell})$.  This implies $$ \| P_\ell - (P_\ell L P_\ell)^{-1} (P_\ell A P_\ell)\| \leq \frac{\gamma}{M} \| (P_\ell L P_\ell)^{-1} \|,\quad \ell \in \mathbb{N}.$$  Choose $N$ large enough so that $\| (P_\ell L P_\ell)^{-1} \| \leq \frac{\gamma+1}{2\gamma} M $ when $\ell \geq N$.  This yields $$ \| P_\ell - (P_\ell L P_\ell)^{-1} (P_\ell A P_\ell)\| \leq \frac{\gamma +1}{2}, \quad \ell \geq N.$$
Standard manipulation shows that $P_\ell A P_\ell$ is invertible for $\ell \geq N$, and $$ \sup_{\ell \geq N}\| (P_\ell A P_\ell)^{-1} \| \leq \frac{\gamma+1}{\gamma(1-\gamma)}M.$$
\end{proof}

\noindent Note that in the previous proof, if $M$ is redefined to be $\sup_{\ell \in \mathbb{N}} \| (P_\ell L P_\ell)^{-1} \|$, then $A$ is uniformly invertible with respect to $(P_\ell)_{\ell \in \mathbb{N}}$.\\

\noindent The following proposition shows that  compact perturbations (of arbitrary norm), of a uniformly invertible operator also gives a uniformly invertible operator.

\begin{prop}\label{compactpertthm}
Let $L:H \rightarrow H$ be uniformly invertible with respect to $(P_\ell)_{\ell \in \mathbb{N}}$.  If $\Delta: H \rightarrow H$ is compact such that $\tilde{L} = L + \Delta$ is an onto isomorphism, then there exists $N$ such that $\tilde{L}$ is uniformly invertible with respect to $(P_\ell)_{\ell \geq N}$.
\end{prop}

\begin{proof}[Proof of Proposition \ref{compactpertthm}]
From the definition of $L_\ell$, we have $I = (I-P_\ell)L_\ell^{-1}+L P_\ell L_\ell^{-1}$, so that $$L^{-1}(P_\ell-I)L_\ell^{-1} = P_\ell L_\ell^{-1} -L^{-1}.$$  This implies 
\begin{equation}\label{thisone}
(L_\ell^*)^{-1}P_\ell -(L^*)^{-1} = (L_\ell^*)^{-1}(P_\ell -I)(L^*)^{-1}.
\end{equation}
As $\ell \rightarrow \infty$, the right hand side of equation (\ref{thisone}) has 0 limit pointwise.  Combined with the compactness of $\Delta$, we obtain $$\lim_{\ell \rightarrow \infty} (L_\ell^*)^{-1}P_\ell \Delta = (L^*)^{-1} \Delta$$ where limit is in the operator norm topology.  This yields
\begin{equation}\label{thatone}
\lim_{\ell \rightarrow \infty} I+\Delta P_\ell L_\ell^{-1} = I+\Delta L^{-1} = (L+\Delta)L^{-1},
\end{equation}
where the limit is also in the operator norm topology.  The right had side of equation (\ref{thatone}) is invertible, so there exists $N$ such that $\ell \geq N$ implies $(I+\Delta P_\ell L_\ell^{-1})^{-1}$ exists, and 
\begin{equation}\label{newsup}
\sup_{\ell \geq N} \| (I+\Delta P_\ell L_\ell^{-1})^{-1} \| < \infty.
\end{equation}
From the definition of $\tilde{L}_\ell$, we obtain
\begin{eqnarray}
\tilde{L}_\ell = L_\ell +\Delta P_\ell = (I+\Delta P_\ell L_\ell^{-1})L_\ell.\nonumber
\end{eqnarray}
When $\ell \geq N$, we have $$\tilde{L}_\ell^{-1} = L_\ell^{-1} (I+\Delta P_\ell L_\ell^{-1})^{-1},$$ and equation (\ref{newsup}) implies
\begin{equation}
\sup_{\ell \geq N} \|\tilde{L}_\ell^{-1} \| \leq \sup_{\ell \geq N} \| L_\ell^{-1} \| \sup_{\ell \geq N} \| (I+\Delta P_\ell L_\ell^{-1})^{-1} \| < \infty.
\end{equation}
This completes the proof.
\end{proof}


\noindent The following lemma holds:

\begin{lem}\label{mainlemma}
Choose $(t_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$ such that $(h_k)_{k \in \mathbb{N}} :=  \big(\frac{1}{(2\pi)^{d/2}}e^{i\langle (\cdot) , t_k \rangle}\big)_{k \in \mathbb{N}}$ satisfies $$\Big\| \sum_{k=1}^n a_k h_k \Big\|_{L_2 [-\pi,\pi]^d} \leq B \Big(\sum_{k=1}^n |a_k|^2 \Big)^{1/2}, \quad \mathrm{for \ all} \quad (c_k)_{k=1}^n \subset \mathbb{C}.$$  If $(\tau_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$, and $(f_k)_{k \in \mathbb{N}} := \big(\frac{1}{(2\pi)^{d/2}} e^{i \langle (\cdot) , \tau_k \rangle}\big)_{k \in \mathbb{N}}$, then for all $r,s\geq 1$ and any finite sequence $(a_k)_k$, we have
\begin{equation
\Bigg\Arrowvert \sum_{k=r}^s a_k ( h_k - f_k ) \Bigg\Arrowvert_{L_2 [-\pi,\pi]^d} \\
{} \leq B\Big( e^{\pi d \big({\sup \atop {r \leq k \leq s}} \Arrowvert \tau_k-t_k \Arrowvert_\infty\big)}-1 \Big) \Big(\sum_{k=r}^s |a_k|^2\Big)^\frac{1}{2}.\nonumber
\end{equation}
\end{lem}

\noindent This lemma is a slight generalization of Lemma 5.3, found in \cite{BB} using simple estimates.  Lemma \ref{mainlemma} is proven similarly.  A consequence of Lemma \ref{mainlemma} is the following corollary.

\begin{cor}\label{compactcor}
Let $(t_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$ such that $(f_k)_{k \in \mathbb{N}}$ (the usual exponential sequence defined in terms of $(t_n)_n$) is a Riesz basis for $L_2([-\pi,\pi]^d)$.  Let $(\tau_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$, and define $(g_k)_{k \in \mathbb{N}}$ by $g_k(x) = \frac{1}{(2\pi)^{d/2}}e^{i \langle x , \tau_k \rangle}$.  Let $(b_k)_{k \in \mathbb{Z}}$ be an orthonormal basis for $L_2([-\pi,\pi]^d)$.  If  $$\lim_{k \rightarrow \infty} \| t_k -\tau_k \|_\infty=0,$$ then the operator defined by $b_k \mapsto f_k - g_k$ is compact.
\end{cor}

\noindent The proof of Corollary \ref{compactcor} is similar to the proof of Corollary 5.5 in \cite{BB}, so it is omitted.

\begin{cor}\label{multidimpert}
Let $(t_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$. Let $(f_k)_{k \in \mathbb{N}}$ be a  Riesz basis for $L_2([-\pi,\pi]^d)$ which is uniformly invertible with respect to the projections $(P_\ell)_{\ell \in \mathbb{N}}$ defined at the beginning of section 4.  If $(\tau_k)_{k \in \mathbb{N}} \subset \mathbb{R}^d$, and $(g_k)_{k \in \mathbb{N}}$ are as in Corollary (\ref{compactcor}), and additionally, $(g_k)_{k \in \mathbb{N}}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$, then $(g_k)_{k \in \mathbb{N}}$ is uniformly invertible with respect to a sequence of projections $(P_\ell)_{\ell \geq N}$ for some $N >0$.
\end{cor}

\begin{proof}[Proof of Corollary \ref{multidimpert}]
Apply Corollaries \ref{compactpertthm} and \ref{compactcor}.
\end{proof}

\noindent This corollary relates to Theorem \ref{maintheorem} in the following way.  Let $(g_k)_{k \in \mathbb{N}}$ is as in the preceding theorem.  Usage of Corollary \ref{compactpertthm} in the proof of Corollary \ref{multidimpert} does not ensure that low order polynomial interpolants will exist; however, they will existence for sufficiently large $\ell$.  Simple examples show that in Corollary \ref{multidimpert}, the additional assumption that $(g_k)_{k \in \mathbb{N}}$ is a Riesz basis for $L_2([-\pi,\pi]^d)$ cannot be dropped when $d \geq 2$.  Example:  The standard exponential orthonormal basis $(e_n)_{n \in \mathbb{Z}^d}$ is of course uniformly invertible, but the set $$\Big(\frac{1}{(2\pi)^{d/2}}e^{i\langle (\cdot) , (1,1/2,0,\cdots,0) \rangle}\Big)\cup(e_n)_{n \neq 0}$$ is not a Riesz basis.  However, this condition can be dropped when $d=1.$  This follows from Corollary \ref{compactcor} and the following theorem.\\

\begin{thm}\label{pakshinlike}
Let $(t_n)_{n \in \mathbb{Z}}\ \subset \mathbb{R}$ be a sequence such that $(f_n)_{n \in \mathbb{Z}}$ (defined as before) is a Riesz basis for $L_2[-\pi,\pi]$.  If $(\tau_n)_{n \in \mathbb{Z}} \subset \mathbb{R}$ is a sequence of distinct points such that $$\lim_{|n| \rightarrow \infty} |t_n-\tau_n|=0,$$ then $(g_n)_{n \in \mathbb{Z}}$ (defined as before) is a Riesz basis for $L_2[-\pi,\pi].$
\end{thm}

\noindent The proof of Theorem \ref{pakshinlike} relies on Lemma \ref{rbreplace} below, which originally appears as Lemma 3.1 in \cite{PS}.  The proof of Lemma \ref{rbreplace} found in \cite{PS} itself relies on a citation, so for the sake of completeness it is presented here with a self-contained proof.

\begin{lem}\label{rbreplace}
Let $(f_n)_{n \in \mathbb{Z}}$ be an exponential Riesz basis for $L_2[-\pi,\pi]$.  If $(g_n)_{|n| \leq \ell}$ is a sequence of complex exponentials such that
\begin{equation}
(g_n)_{|n| \leq \ell} \cup (f_n)_{|n|>{\ell_0}}\nonumber
\end{equation}
consists of distinct functions, then this set is a Riesz basis for $L_2[-\pi,\pi]$.
\end{lem}

\begin{proof}[Proof of Lemma \ref{rbreplace}]
If we can prove the case when $\ell = 0$, the general result follows inductively.  Let $f_n(\cdot) = \frac{1}{\sqrt{2\pi}}e^{i \langle \cdot , t_n \rangle}$ for $n \neq 0$, and $g_0(\cdot) =  \frac{1}{\sqrt{2\pi}}e^{i \langle \cdot , \tau_0 \rangle}$ where $\tau_0 \in \mathbb{R}$ and $\tau_0 \neq t_n$ for $n \neq 0$.  Let $(e_n)_{n \in \mathbb{Z}}$ be an orthonormal basis for $L_2[-\pi,\pi]$.  Lemma \ref{secondone} shows that $(g_0) \cup (f_n)_{n \neq 0}$ is a Riesz basis if and only if the map defined by $$ e_0 \mapsto g_0, \quad e_k \mapsto f_k, \ \mathrm{for} \ k \neq 0 $$ is one to one.  This is readily seen to be equivalent to $\langle g_0, f_1^* \rangle \neq 0$, or by passing to the Fourier transform, to $G_0 (\tau_0)\neq 0$, (recall that $G_0 = \mathcal{F}(f_0^*)$).  If we can show that the only zeros of $G_0$ in $\mathbb{R}$ are $(t_n)_{n \neq 0}$, we are done.\\

\noindent Suppose there exists $\lambda \in \mathbb{R}$, $\lambda\notin (t_n)_{n \neq 0}$.  Such that $G_0 (\lambda) = 0$ with multiplicity $m$.  Define the entire function $$ H(t) = \frac{(t_0 -\lambda)^m}{(t-\lambda)^m}G_0 (t).$$  Note that $H|_\mathbb{R} \in L_2(\mathbb{R})$, and $H$ is of exponential type $\pi$, so $H \in PW_{[-\pi,\pi]}$ by the Theorem \ref{PaleyWiener}.  The expansion $$H(t) = \sum_{n \in \mathbb{Z}} H(t_n)G_n(t),$$ combined with $H(t_n) = \delta_{n,0}$, shows that $H(t) = G_0(t)$ for all $t \in \mathbb{R}$, an immediate contradiction.  We conclude that $G_0 (\lambda) \neq 0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{pakshinlike}]
\noindent Define $L e_n = f_n$ and $\tilde{L}e_n = g_n$.  By Corollary \ref{compactcor}, $\tilde{L}$ is bounded linear and $\tilde{L} = L+\Delta$ for some compact operator $\Delta$.  Define the operator \begin{displaymath}
   R_\ell e_n = \left\{
     \begin{array}{lr}
       f_n,  &  |n| \leq \ell\\
       g_n & |n| > \ell
     \end{array}
   \right..
\end{displaymath}
Rewritten, we have
\begin{equation}
R_\ell = L P_\ell +(L+\Delta)(I-P_\ell) = L+\Delta(I-P_\ell).\nonumber
\end{equation}
Compactness of $\Delta$ implies that $\lim_{\ell \rightarrow \infty} R_\ell = L$ in the operator norm topology.  We conclude that $R_{\ell_0}$ is an onto isomorphism for some $\ell_0$ sufficently large; that is, the set 
\begin{equation}\label{dumb}
(f_n)_{|n| \leq \ell_0} \cup (g_n)_{|n|>{\ell_0}}
\end{equation}
is a Riesz basis for $L_2[-\pi,\pi]$.  If we apply Lemma \ref{rbreplace}, by replacing $(f_n)_{|n| \leq \ell_0}$ with $(g_n)_{|n| \leq \ell_0}$ in expression (\ref{dumb}), we have that $(g_n)_{n \in \mathbb{Z}}$ is a Riesz basis for $L_2[-\pi,\pi].$
\end{proof}


\section{Appendix: Comments regarding the optimality of Theorem \ref{simplethm}}\label{S:5}

\noindent In the statement of Theorem \ref{simplethm}, it is not apparent whether or not $(E_{\ell,N,k})_\ell$ can be replaced with a more rapidly growing sequence of intervals; however, Proposition \ref{sinctheorem} shows that if $f(t) = \mathrm{SINC}(\pi t)$, equations (\ref{simplecor1}) and (\ref{simplecor2}) can hold for a sequence of intervals $(E_{\ell,N})_\ell$ which grow faster than $(E_{\ell,N,A})_\ell$.  Propositions \ref{l2bound} and \ref{linfbound} show that growth bounds of the intervals in Proposition \ref{sinctheorem} are optimal for the conclusion of said proposition to hold.  Thus, the bounds in Proposition \ref{sinctheorem} provide upper bounds for the growth of \textit{any} sequence $(E_{\ell,N,A})_\ell$ such that either equation (\ref{simplecor1}) or equation (\ref{simplecor2}) hold for general multivariate bandlimited functions.

\begin{prop}\label{sinctheorem}
Define 
\begin{eqnarray}
C_{\ell,N} & = & \Big(\frac{1}{4}(2N+1)^2 (\ell+1/2)^{2N+1} \log(\ell+1/2)\Big)^\frac{1}{2(N+1)}, \ \mathrm{and}\nonumber\\
D_{\ell,N} & = & \Big(\frac{1}{2}(2N+1)^2 (\ell+1/2)^{2N+1} \log(\ell+1/2)\Big)^\frac{1}{2(N+1)},\nonumber
\end{eqnarray}
then
\begin{equation}\label{sinctheorem1}
\lim_{\ell \rightarrow \infty} \Big\| \mathrm{SINC}(\pi t) - I_{\mathrm{SINC}\pi(\cdot), \ell}(t)  \Big\|_{L_2([-C_{\ell,N},C_{\ell,N}]^d)}=0,
\end{equation}
and
\begin{equation}\label{sinctheorem2}
\lim_{\ell \rightarrow \infty} \Big\| \mathrm{SINC}(\pi t) - I_{\mathrm{SINC}\pi(\cdot), \ell}(t)  \Big\|_{L_\infty([-D_{\ell,N},D_{\ell,N}]^d)}=0.
\end{equation}
\end{prop}

\noindent The proof of equation (\ref{sinctheorem1}) requires the following two propositions.

\begin{prop}\label{auxprop1}
\begin{equation}\label{firstbit}
\lim_{\ell \rightarrow \infty} \Big\|\Big( e^{ \frac{\| t \|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big) \mathrm{SINC}(\pi t) \Big\|_{L_2([-C_{\ell,N},C_{\ell,N}]^d)} = 0.
\end{equation}
\end{prop}
\begin{proof}[Proof of Proposition \ref{auxprop1}]
Let $t = \alpha C_{\ell,N}$ where $\alpha \in [-1,1]^d$. Noting that $$e^{ \frac{\| t \|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}} = \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}},$$ the quantity in equation (\ref{firstbit}) becomes
\begin{eqnarray}
& &\bigg(\int_{[-C_{\ell,N},C_{\ell,N}]^d} \bigg|\Big(\Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}}-1\Big) \mathrm{SINC}(\pi t)\bigg|^2 dt \bigg)^{1/2}\nonumber\\
& &\quad\leq\frac{1}{C_{\ell,N}^{d/2}}\bigg( \int_{[-1,1]^d} \bigg|\Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}}-1\bigg|^2 d \alpha \bigg)^{1/2} \nonumber\\
& &\quad\leq \frac{2^{d/2} \Big( \ell+\frac{1}{2} \Big)^{d\frac{2N+1}{4(N+1)}} +2^{d/2}}{\big(\frac{1}{4}(2N+1)^2\big)^\frac{d}{4(N+1)}\big(\log(\ell+1/2) \big)^\frac{d}{4(N+1)}(\ell+1/2)^{d\frac{2N+1}{4(N+1)}}}.\nonumber
\end{eqnarray}
The last term in the above inequality has limit $0$ as $\ell \rightarrow \infty$.  This proves the proposition.
\end{proof}

\begin{prop}\label{auxprop2}
\begin{eqnarray}
\lim_{\ell \rightarrow \infty} \Big\| Q_{d,\ell}(t) e^{-\!\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\!\!-\!e^{ \frac{\| t \|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi t)  \Big\|_{L_2([-C_{\ell,N},C_{\ell,N}]^d)}\!=\!0.\nonumber
\end{eqnarray}
\end{prop}

\begin{proof}[Proof of Proposition \ref{auxprop2}]
If $t \in \mathbb{R}^d$ and $\|t \|_\infty <\ell+1/2$, then equation (\ref{blork3}) implies
\begin{eqnarray}\label{needsone}
& & \bigg(e^{\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \bigg)\prod_{k=1}^d e^{\bigg(-\frac{1}{2(\ell+1/2)}\frac{\big( \frac{t}{\ell+1/2} \big)^2}{1-\big(\frac{t}{\ell+1/2} \big)^2}\bigg)}\\
& &\quad \leq\frac{Q_{d,\ell} (t)e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}}{\mathrm{SINC}(\pi t)} \leq  e^{\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}.\nonumber
\end{eqnarray}
Let $t \in [-C_{\ell,N},C_{\ell,N}]^d$ where $t= \alpha C_{\ell,N}$, $\alpha \in [-1,1]$.  Consider the right hand side of inequality (\ref{needsone}) for such $t$.
\begin{eqnarray}
& & e^{\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \leq \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}} e^{(\ell+1/2) O \Big( \big\| \frac{t}{\ell+1/2} \big\|_{2(N+2)}^{2(N+2)} \Big)}\\
& &\quad\leq \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}} e^{M (\ell+1/2)^{-\frac{1}{N+1}} (\log(\ell+1/2) )^\frac{N+2}{N+1} \| \alpha \|_{2(N+2)}^{2(N+2)} }.\nonumber
\end{eqnarray}
for some constant $M$. Noting that $$ \frac{t^2}{(\ell+1/2)^3} = \frac{\| \alpha \|_2^2 \big(\frac{1}{4}(2N+1)^2\big)^\frac{1}{N+1}\big( \log(\ell+1/2)\big)^\frac{1}{N+1}}{(\ell+1/2)^\frac{N+2}{N+1}},$$ we can bound the left hand side of inequality (\ref{needsone}) from below as follows:
\begin{eqnarray}\label{blahblah}
& & e^{\Big(-m  \frac{\| \alpha \|_2^2 \big( \log(\ell+1/2)\big)^\frac{1}{N+1}}{(\ell+1/2)^\frac{N+2}{N+1}} \Big)} \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}}\\
& &\quad\leq \bigg(e^{\sum_{k=N+1}^\infty \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \bigg)\prod_{k=1}^d e^{\bigg(-\frac{1}{2(\ell+1/2)}\frac{\big( \frac{t}{\ell+1/2} \big)^2}{1-\big(\frac{t}{\ell+1/2} \big)^2}\bigg)},\nonumber
\end{eqnarray}
where $m>0$ is chosen independently of $\ell$.  Relations (\ref{needsone}) through (\ref{blahblah}) imply
\begin{eqnarray}
& & \bigg( e^{\Big(-m  \frac{(\log(\ell+1/2))^\frac{1}{N+1}\| \alpha \|_2^2}{(\ell+1/2)^\frac{N+2}{N+1}} \Big)}-1\bigg) \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}} |\mathrm{SINC}(\pi t)| \nonumber\\
& &\quad\leq \bigg| Q_{d,\ell}(t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{ \frac{\| t \|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi t)  \bigg| \nonumber\\
& &\quad\leq \bigg( e^{\Big(M \frac{ (\log(\ell+1/2) )^\frac{N+2}{N+1} \| \alpha \|_{2(N+2)}^{2(N+2)}}{(\ell+1/2)^\frac{1}{N+1}}\Big)}-1 \bigg) \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}}|\mathrm{SINC}(\pi t)|.\nonumber
\end{eqnarray}
Further simplification implies (for appropriate constants $C$, $C'$, and $C''$) that
\begin{eqnarray}
& & \Big\| Q_{d,\ell}(t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}-e^{ \frac{\| t \|_{2(N+1)}^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{SINC}(\pi t)  \Big\|_{L_2([-C_{\ell,N},C_{\ell,N}]^d)} \nonumber\\
& &\quad\leq C\frac{(\log(\ell+1/2))^\frac{N+2}{N+1}}{(\ell+1/2)^\frac{1}{N+1}} \Bigg(\int_{[-C_{\ell,N},C_{\ell,N}]^d} \Bigg| \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}} \| \alpha \|_2^2 \mathrm{SINC}(\pi t)  \Bigg|^2  dt \Bigg)^{1/2}\nonumber\\
& &\quad =C'\frac{(\log(\ell+1/2))^\frac{N+2}{N+1}}{(\ell+1/2)^\frac{1}{N+1}} \Bigg(\int_{[-1,1]^d} \frac{\Bigg| \Big( \ell+\frac{1}{2} \Big)^{\frac{2N+1}{4(N+1)}\|\alpha \|_{2(N+1)}^{2(N+1)}} \| \alpha \|_2^2 \mathrm{SINC}(\pi t)  \Bigg|^2}{(\log(\ell+1/2))^\frac{d}{2(N+1)}\big(\ell+\frac{1}{2} \big)^{\frac{2N+1}{2(N+1)}d}}  d\alpha \Bigg)^{1/2}\nonumber\\
& &\quad\leq  C''\frac{(\log(\ell+1/2))^\frac{N+2}{N+1}}{(\ell+1/2)^\frac{1}{N+1} (\log(\ell+1/2))^\frac{d}{4(N+1)}},\nonumber
\end{eqnarray}
after the change in variable $t = \alpha C_{\ell,N}$ and simple estimates.  This proves the proposition.
\end{proof}

\begin{proof}[Proof of equation (\ref{sinctheorem1})]
Equation (\ref{sinctheorem1}) follows immediately from Propositions \ref{auxprop1} and \ref{auxprop2}.
\end{proof}

\noindent The proof of equation (\ref{sinctheorem2}) requires the following two propositions.

\begin{prop}\label{auxprop3}
\begin{equation}\label{thirdbit}
\lim_{\ell \rightarrow \infty} \Big\|\Big( e^{ \frac{ t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}-1\Big) \mathrm{sinc}(\pi t) \Big\|_{L_\infty [-D_{\ell,N},D_{\ell,N}]} = 0.
\end{equation}
\end{prop}

\begin{proof}[Proof of Proposition \ref{auxprop3}]
Let $t \in [-D_{\ell,N},D_{\ell,N}]$, then $t = \alpha D_{\ell,N}$ for $\alpha \in [-1,1]$. Simplification shows that equation (\ref{thirdbit}) holds if
\begin{equation}\label{sam}
\lim_{\ell \rightarrow \infty} \sup_{\alpha \in [0,1]} \bigg| \frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha \big(\log(\ell+1/2)\big)^\frac{1}{2(N+1)} (\ell+1/2)^\frac{2N+1}{2(N+1)}} \bigg| = 0.
\end{equation}
Note that for large $\ell$,
\begin{equation}\label{sam2}
\sup_{\alpha \in [1/2,1]} \Bigg| \frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha \big(\log(\ell+1/2)\big)^\frac{1}{2(N+1)} (\ell+1/2)^\frac{2N+1}{2(N+1)}} \Bigg| \leq \frac{2}{\big(\log(\ell+1/2)\big)^\frac{1}{2(N+1)}}.
\end{equation}
\noindent Let $0  < \alpha \leq 1/2$.  The Mean Value Theorem implies
\begin{equation}
\bigg| \frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha} \bigg| \leq (2N+1)(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}} \alpha^{2N+1} \log(\ell+1/2).
\end{equation}
This yields
\begin{eqnarray}
\sup_{\alpha \in [0,1/2]} \bigg| \frac{(\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}-1}{\alpha \big(\log(\ell+1/2)\big)^\frac{1}{2(N+1)} (\ell+1/2)^\frac{2N+1}{2(N+1)}} \bigg|\leq M \frac{\big(\log(\ell+1/2)\big)^\frac{2N+1}{2(N+1)}}{(\ell+1/2)^{\frac{2N+1}{2(N+1)}(1-(1/2)^{2(N+1)})}}\nonumber
\end{eqnarray}
for some constant $M$. Combined with inequality (\ref{sam2}), we have equation (\ref{sam}), which proves the proposition.
\end{proof}

\begin{prop}\label{blip6}
\begin{equation}\label{fourthbit}
\lim_{\ell \rightarrow \infty} \Big\| Q_{1,\ell}(t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}-e^{ \frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{sinc}(\pi t)  \Big\|_{L_\infty [-D_{\ell,N},D_{\ell,N}]}=0.\nonumber
\end{equation}
\end{prop}

\begin{proof}[Proof of Proposition \ref{blip6}]
Let $t \in [-C_{\ell,N},C_{\ell,N}]$ where $t=\alpha C_{\ell,N}$, $\alpha \in [-1,1]$.  Proceeding in the same manner as in the proof of Proposition \ref{auxprop2}, we see (for appropriate constants $C$ and $C'$) that
\begin{eqnarray}
& & \bigg| Q_{1,\ell}(t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}-e^{ \frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1)}}\mathrm{sinc}(\pi t)  \bigg|_{L_\infty([-C_{\ell,N},C_{\ell,N}])}\nonumber\\
& &\quad\leq \frac{C (\ell+1/2)^{\alpha^{2(N+1)}\frac{2N+1}{2(N+1)}}\alpha^2 (\log(\ell+1/2))^\frac{N+2}{N+1}|\sin(\pi t)|}{\alpha (\ell+1/2)^\frac{1}{N+1} (\log(\ell+1/2))^\frac{1}{2(N+1)}(\ell+1/2)^\frac{2N+1}{2(N+1)}}\nonumber\\
& &\quad\leq \frac{C' (\log(\ell+1/2))^\frac{2N+3}{2(N+1)}}{(\ell+1/2)^\frac{1}{N+1}}.\nonumber
\end{eqnarray}
This proves the proposition.
\end{proof}

\begin{proof}[Proof of equation (\ref{sinctheorem2})]
\noindent The previous two propositions prove equation when $d=1$.  The multidimensional case follows inductively.
\end{proof}

\begin{prop}\label{l2bound}
Let $N \geq 0$.  If $(M_{\ell,N})_\ell$ is a sequence of positive numbers such that (\ref{sinctheorem1}) holds when $(C_{\ell,N})_\ell$ is replaced by $(M_{\ell,N})_\ell$, then
\begin{equation}\label{l2boundeq}
\limsup_{\ell \rightarrow \infty} \frac{M_{\ell,N}}{C_{\ell,N}} \leq 1.
\end{equation}
\end{prop}

\noindent The proof of Proposition \ref{l2bound} requires the following simple estimate:

\begin{prop}\label{blork6}
Let $a>1/2$, $\epsilon>0$, $0 < \omega <1$, then $$\int_a^{(1+\epsilon)a} \frac{\sin^2 \pi x}{x^{1+\omega}}dx >\frac{\epsilon}{2 a^\omega (1+\epsilon)^\omega}-\frac{a}{2(a-1/2)^{1+\omega}}.$$
\end{prop}
\begin{proof}[Proof of Proposition \ref{blork6}]
Let $b=(1+\epsilon)a$.  We have
$$\int_a^b \frac{\sin^2 \pi x}{x^{1+\omega}}dx + \int_a^b \frac{\cos^2 \pi x}{x^{1+\omega}}dx = \frac{1}{\omega}\Big(\frac{1}{a^\omega}-\frac{1}{b^\omega} \Big)$$
and
$$\int_a^b \frac{\cos^2 \pi x}{x^{1+\omega}}dx = \int_{a-1/2}^{b-1/2} \frac{\sin^2 \pi x}{(x+1/2)^{1+\omega}}dx < \int_{a-1/2}^{b-1/2} \frac{\sin^2 \pi x}{x^{1+\omega}}dx.$$  This yields $$2 \int_a^b \frac{\sin^2 \pi x}{x^{1+\omega}}dx - \int_{b-1/2}^b \frac{\sin^2 \pi x}{x^{1+\omega}}dx+ \int_{a-1/2}^a \frac{\sin^2 \pi x}{x^{1+\omega}}dx > \frac{1}{\omega}\Big(\frac{1}{a^\omega}-\frac{1}{b^\omega} \Big),$$ so that $$\int_a^b \frac{\sin^2 \pi x}{x^{1+\omega}}dx >\frac{1}{2\omega}\Big(\frac{1}{a^\omega}-\frac{1}{b^\omega} \Big) -\frac{1}{2(a-1/2)^{1+\omega}}.$$ Noting that $$\frac{1}{2\omega}\Big(\frac{1}{a^\omega}-\frac{1}{b^\omega} \Big) =\frac{\epsilon}{2\omega a^\omega (1+\epsilon)^\omega} \frac{(1+\epsilon)^\omega -1}{\epsilon}>\frac{\epsilon}{2a^\omega (1+\epsilon)^\omega}$$ proves the proposition.
\end{proof}

\begin{proof}[Proof of Proposition \ref{l2bound}]  Fix $N\geq0$, and define $c=\frac{2N+1}{2N+4}+\delta/2$ where $0<\delta$ is small enough so that $c<1/2$.  Define $$A_\ell = (c(N+1)(2N+1)\log(\ell+1/2))^\frac{1}{2(N+1)}(\ell+1/2)^\frac{-1}{2(N+1)}$$ and $$\epsilon_\ell = (\ell+1/2)^{1-2c} A_\ell.$$  Note that $\lim_{\ell \rightarrow \infty} \epsilon_\ell = 0.$  Let $t \in [A_\ell (\ell+1/2), (1+\epsilon_\ell)A_\ell(\ell+1/2)],$ then $t=\alpha(\ell +1/2)$ for some $\alpha \in [A_\ell,(1+\epsilon_\ell)A_\ell]$. For large $\ell$, note that inequality (\ref{blork3}) implies
\begin{eqnarray}
\frac{1}{2\pi}e^\frac{(\ell+1/2)\alpha^{2(N+1)}}{(N+1)(2N+1)}\frac{|\sin \pi\alpha(\ell+1/2)|}{\alpha (\ell+1/2)} \leq \Big| Q_{1,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}} \Big|.
\end{eqnarray}
Moving to the multivariate case, if $t \in [A_\ell (\ell+1/2), (1+\epsilon_\ell)A_\ell(\ell+1/2)]^d$, then $t = \alpha (\ell+1/2)$ for some $\alpha \in [A_\ell, (1+\epsilon_\ell)A_\ell]^d$.  This yields
\begin{eqnarray}
\prod_{i=1}^d \frac{1}{2\pi\alpha_i^c} \frac{|\sin \pi\alpha_i(\ell+1/2)|}{(\alpha_i(\ell+1/2))^{1-c}} \leq  \Big| Q_{d,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \Big|.\nonumber
\end{eqnarray}
For sufficiently large $\ell$, we can conclude
\begin{eqnarray}
& &\bigg[\frac{1}{9 \pi^2 A_\ell^{2c}} \int_{A_\ell(\ell+1/2)}^{(1+\epsilon_\ell)A_\ell(\ell+1/2)} \frac{\sin^2 \pi x}{x^{2-2c}}dx\bigg]^d\nonumber\\
& &\quad\leq \int_{[A_\ell(\ell+1/2),(1+\epsilon_\ell)A_\ell(\ell+1/2)]^d}  \Big| Q_{d,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \Big|^2 dt.\nonumber
\end{eqnarray}

\noindent Applying Proposition \ref{blork6} for $a= A_\ell (\ell+1/2)$, $\epsilon = \epsilon_\ell,$ and $\omega = 1-2c$, and using the definition of $\epsilon_\ell$, we obtain
\begin{eqnarray}
& &\bigg[\frac{1}{9\pi^2}\Big[\frac{1}{2(1+\epsilon_\ell)^{1-2c}} - \frac{1}{2 A_\ell^{2c}(A_\ell(\ell+1/2)-1)^{2-2c}} \Big]\bigg]^d\nonumber\\
& &\quad\leq \int_{[A_\ell(\ell+1/2),(1+\epsilon_\ell)A_\ell(\ell+1/2)]^d}  \Big| Q_{d,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \Big|^2 dt.\nonumber
\end{eqnarray}
The first term in the brackets in the previous equation has limit $1/2$, while the second term has limit $0$. We conclude there exists a constant $\beta >0$ such that
\begin{equation}\label{blork7}
\beta \leq \int_{[A_\ell(\ell+1/2),(1+\epsilon_\ell)A_\ell(\ell+1/2)]^d}  \Big| Q_{d,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \Big|^2 dt, \quad \ell >0.
\end{equation}
If $M_{\ell,N} \geq (\ell+1/2)(1+\epsilon_\ell)A_\ell$ for infinitely many $\ell$, there exists a subsequence $(\ell_k)_{k \in \mathbb{N}}$ such that (in particular),
\begin{eqnarray}
\lim_{\ell_k \rightarrow \infty}\! \Big\| \mathrm{SINC}(\pi t) - Q_{d,\ell_k}(t) e^{-\sum_{k=1}^{N} \frac{1}{k(2k-1)}\frac{\| t \|_{2k}^{2k}}{(\ell_k+1/2)^{2k-1}}}\Big\|_{L_2([A_{\ell_k} (\ell_k+1/2)), A_{\ell_k} (\ell_k+1/2)(1+\epsilon_{\ell_k})]^d)}\!\!=\!0.\nonumber
\end{eqnarray}
This contradicts inequality (\ref{blork7}).  This yields that for sufficiently large $\ell$, 
\begin{eqnarray}
 M_{\ell,N} &<& (\ell+1/2)(1+\epsilon_\ell)A_\ell\nonumber\\
& = & (1+\epsilon_\ell)\Big(\Big(\frac{2N+1}{4N+4}+\delta/2 \Big)(N+1)(2N+1)(\ell+1/2)^{2N+1}\log(\ell+1/2)\Big)^\frac{1}{2(N+1)}.\nonumber
\end{eqnarray}
Note that since $\epsilon_\ell \rightarrow 0$, for large $\ell$, the quantity $(1+\epsilon_\ell)\Big(\frac{2N+1}{4N+4}+\delta/2 \Big)^\frac{1}{2(N+1)}$ is less than, (and bounded away from) the quantity $\Big(\frac{2N+1}{4N+4}+\delta \Big)^\frac{1}{2(N+1)}$.  We conclude that for any $\delta>0$, there exists $\ell_{N,\delta}$ such that $$ \sup_{\ell >\ell_{N,\delta}} \frac{M_{\ell,N}}{((N+1)(2N+1)\log(\ell+1/2))^\frac{1}{2(N+1)}(\ell+1/2)^\frac{2N+1}{2(N+1)}} < \Big( \frac{2N+1}{4N+4} +\delta \Big)^\frac{1}{2(N+1)}.$$  Proposition \ref{l2bound} follows.
\end{proof}

\begin{prop}\label{linfbound}
Let $N \geq 0$.  If $(M_{\ell,N})_\ell$ is a sequence of positive numbers such that equation (\ref{sinctheorem2}) holds when $(D_{\ell,N})_\ell$ is replaced by $(M_{\ell,N})_\ell$, then
\begin{equation}
\limsup_{\ell \rightarrow \infty} \frac{M_{\ell,N}}{D_{\ell,N}} \leq 1.
\end{equation}
\end{prop}

\noindent The proof of Proposition \ref{linfbound} requires the following fact:

\begin{prop}\label{sinprop}
Let $0 < \epsilon \leq 1$.  If $A>0$, there exists $t \in [A, A+\epsilon]$ such that $|\sin(\pi t)| \geq |\sin(\pi \epsilon/2)|$.
\end{prop}
\begin{proof}[Proof of Proposition \ref{sinprop}]
The proof is clear from geometric considerations.
\end{proof}

\begin{proof}[Proof of Proposition \ref{linfbound}]
Let $N \geq 0$. Choose $\delta>0$ such that $c := \frac{2N+1}{2N+2}+\delta/2<1$. Define $$A_\ell = (c(N+1)(2N+1)\log(\ell+1/2))^\frac{1}{2(N+1)}(\ell+1/2)^\frac{-1}{2(N+1)}$$ and $$\epsilon_\ell = A_\ell (\ell+1/2)^{1-c}.$$ Note that $\lim_{\ell \rightarrow \infty} \epsilon_\ell = 0$.  Let $t \in [A_\ell (\ell+1/2),A_\ell(\ell+1/2)+\epsilon_\ell].$ Proceeding as before, for sufficiently large $\ell$, we have
\begin{equation}
\frac{1}{2\pi}e^{\Big(\frac{t^{2(N+1)}}{(\ell+1/2)^{2N+1}(N+1)(2N+1))}\Big)} \frac{|\sin(\pi t)|}{t} \leq \Big| Q_{1,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}\Big|.\nonumber
\end{equation}
Now for all $t \in [A_\ell (\ell+1/2),A_\ell(\ell+1/2)+\epsilon_\ell]$,\\
\begin{equation}
\frac{1}{2\pi}\frac{(\ell+1/2)^c}{A_\ell(\ell+1/2)+\epsilon_\ell}|\sin(\pi t)| \leq \Big| Q_{1,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{t^{2k}}{(\ell+1/2)^{2k-1}}}\Big|.\nonumber
\end{equation}
In the multivariate case, if $t \in [A_\ell (\ell+1/2),A_\ell(\ell+1/2)+\epsilon_\ell]^d$, we obtain
\begin{equation}
 \frac{1}{(2\pi)^d}\frac{(\ell+1/2)^{cd}}{(A_\ell(\ell+1/2)+\epsilon_\ell)^d}\prod_{i=1}^d |\sin(\pi t_i)| \leq \Big| Q_{d,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}}\Big|.\nonumber
\end{equation}
For large $\ell$, an application of Proposition \ref{sinprop} yields
\begin{equation}
\frac{1}{(3\pi)^d}\frac{|\sin(\pi \epsilon_\ell/2)|^d}{A_\ell^d(\ell+1/2)^{(1-c)d}} \leq \Big\| Q_{d,\ell} (t) e^{-\sum_{k=1}^N \frac{1}{k(2k-1)}\frac{\|t\|_{2k}^{2k}}{(\ell+1/2)^{2k-1}}} \Big\|_{L_\infty([A_\ell (\ell+1/2),A_\ell(\ell+1/2)+\epsilon_\ell]^d)}.\nonumber
\end{equation}
By the definition of $\epsilon_\ell$, the right hand side of the above equation tends to a positive constant.  The remainder of the proof is almost identical to that of Proposition \ref{l2bound}.
\end{proof}

\noindent The following is trivially deduced from Propositions \ref{l2bound} and \ref{linfbound}:  Fix $N>0$.  If $(E_{\ell,N})_\ell$ is a sequence of intervals such that either equation (\ref{simplecor1}) or equation (\ref{simplecor2}) holds for all $f \in PW_{[-\pi,\pi]^d}$, then $$\max_{x \in (E_{\ell,N})^d} \| x \|_\infty = o\Big(\max_{x \in (E_{\ell,N+1,A})^d} \| x \|_\infty\Big).$$

