Multivariate exact and falsified sampling approximation

TL;DR

This paper investigates the approximation capabilities of multivariate sampling expansions, including falsified sampling, providing error estimates based on Fourier analysis and exploring special functions like splines and band-limited functions.

Contribution

It introduces new error bounds for multivariate sampling expansions with both exact and falsified samples, and constructs practical functions for implementation.

Findings

Error estimates depend on signal smoothness and the Strang-Fix condition.

Falsified sampling expansions can be effectively compared with differential expansions.

Constructed functions include splines and band-limited functions suitable for interpolation.

Abstract

Approximation properties of the expansions $\sum_{k\in{\mathbb z}^d}c_k\phi(M^jx+k)$, where $M$ is a matrix dilation, $c_k$ is either the sampled value of a signal $f$ at $M^{-j}k$ or the integral average of $f$ near $M^{-j}k$ (falsified sampled value), are studied. Error estimations in $L_p$-norm, $2\le p\le\infty$, are given in terms of the Fourier transform of $f$. The approximation order depends on how smooth is $f$, on the order of Strang-Fix condition for $\phi$ and on $M$. Some special properties of $\phi$ are required. To estimate the approximation order of falsified sampling expansions we compare them with a differential expansions $\sum_{k\in\,{\mathbb z}^d} Lf(M^{-j}\cdot)(-k)\phi(M^jx+k)$, where $L$ is an appropriate differential operator. Some concrete functions $\phi$ applicable for implementations are constructed. In particular, compactly supported splines and band-limited functions can be taken as $\phi$. Some of these functions provide expansions interpolating a signal at the points $M^{-j}k$.