Sampling for approximating $R$-limited functions

Abstract

$R$-limited functions are multivariate generalization of band-limited functions whose Fourier transforms are supported within a compact region $R\subset\mathbb{R}^{n}$. In this work, we generalize sampling and interpolation theorems for band-limited functions to $R$-limited functions. More precisely, we investigated the following question: "For a function compactly supported within a region similar to $R$, does there exist an $R$-limited function that agrees with the function over its support for a desired accuracy?". Starting with the Fourier domain definition of an $R$-limited function, we write the equivalent convolution and a discrete Fourier transform representations for $R$-limited functions through approximation of the convolution kernel using a discrete subset of Fourier basis. The accuracy of the approximation of the convolution kernel determines the accuracy of the discrete Fourier representation. Construction of the discretization can be achieved using the tools from approximation theory as demonstrated in the appendices. The main contribution of this work is proving the equivalence between the discretization of the Fourier and convolution representations of $R$-limited functions. Here discrete convolution representation is restricted to shifts over a compactly supported region similar to $R$. We show that discrete shifts for the convolution representation are equivalent to the spectral parameters used in discretization of the Fourier representation of the convolution kernel. This result is a generalization of the cardinal theorem of interpolation of band-limited functions. The error corresponding to discrete convolution representation is also bounded by the approximation of the convolution kernel using discretized Fourier basis.