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\mytitle{Cardinal and semi-cardinal interpolation with Mat\'{e}rn kernels}
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Aurelian Bejancu \\ %
Kuwait University \\
\url{aurelianbejancu@gmail.com}
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Let $\phi:{\mathbb R}^d\to{\mathbb R}$ be a continuous and symmetric function (i.e.\ $\phi(-x) = \phi(x)$,
$x\in{\mathbb R}^d$), satisfying a suitable decay condition for large $\|x\|$, and define the shift-invariant
space
\[
S(\phi) = \{ \sum_{k\in \mathbb{Z}^d} c_k \phi(\cdot-k) : (c_k) \in \ell^\infty (\mathbb{Z}^d) \}.
\]
The problem of \emph{cardinal interpolation} with the kernel $\phi$ is to find, for some data
$(y_j)_{j\in\mathbb{Z}^d}$, a function $s\in S(\phi)$, such that $s(j) = y_j$, for all $j\in\mathbb{Z}^d$.
If this problem admits a unique solution for any bounded data, it is known that specific
algebraic or exponential decay of the kernel $\phi$ is transferred to the corresponding \emph{Lagrange
function} for cardinal interpolation on $\mathbb{Z}^d$, leading to a well-localized Lagrange representation
of the solution $s$.
The first part of the talk presents a similar result obtained for the related problem of
\emph{semi-cardinal interpolation}, in which the multi-integer grid $\mathbb{Z}^d$ is replaced by a half-space
lattice $H\subset\mathbb{Z}^d$. Despite the loss of shift-invariance in this case, we prove that the
algebraic or exponential decay still carries over from $\phi$ to the corresponding $H$-indexed family of
semi-cardinal Lagrange functions, with constants that are independent of the index $j\in H$ (see \cite{ab20a}).
In the second part, we discuss two recent applications and refinements of the above results for the
Mat\'{e}rn kernel $\phi := \phi_{m,d}$, defined, for a positive integer $m>d/2$, as the exponentially decaying
fundamental solution of the elliptic operator $(1-\Delta)^m$ in $\mathbb{R}^d$, where $\Delta$ is the Laplace
operator. Namely, for a scaling parameter $h>0$, we consider non-stationary interpolation to data prescribed
on $h\mathbb{Z}^d$ from the \emph{flat ladder} collection $\{S_h(\phi)\}_h$ generated by $\phi$ via
\[
S_h (\phi) = \{ \sum_{k\in \mathbb{Z}^d} c_k \phi(\cdot-hk) : (c_k) \in \ell^\infty (\mathbb{Z}^d) \}.
\]
For this problem, we prove that the Lebesgue constant of the associated interpolation operator is
uniformly bounded as $h\to 0$, which allows us to deduce the maximal $L^\infty$-convergence rate
$O(h^{2m})$ for the Mat\'{e}rn flat ladder interpolation scheme (see \cite{ab20b}). On the other hand, if $d=1$, then
the translates of the Mat\'{e}rn kernel $\phi_{m,1}$ span a linear space of exponential splines, for which
we show that non-stationary semi-cardinal interpolation on the scaled grid $h\mathbb{Z}_+$ achieves the
convergence rate $O(h^{m})$ in $L^\infty(\mathbb{R}_+)$, amounting to half of the approximation order of
the corresponding cardinal scaled scheme.
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\paragraph{Acknowledgements:} This work was supported by Kuwait University, Research Grant No.\ SM01/18.
%%%% BIBLIOGRAPHY -- MODIFY THIS %%%%%
\begin{thebibliography}{1}
\bibitem{ab20a}
A.~Bejancu.
\newblock Wiener-Hopf difference equations and semi-cardinal interpolation with integrable convolution kernels.
\newblock {\em Preprint}, arXiv:2006.05282, 2020.
\bibitem{ab20b}
A.~Bejancu.
\newblock Uniformly bounded Lebesgue constants for scaled cardinal interpolation with Mat\'{e}rn kernels.
\newblock {\em Preprint}, arXiv:2009.00711, 2020.
\end{thebibliography}
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