Nonuniform average sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces

TL;DR

This paper investigates nonuniform average sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces, proposing two reconstruction algorithms for different types of average sampled values.

Contribution

It introduces two types of average sampling in these subspaces and provides fast algorithms for their reconstruction, advancing sampling theory in mixed Lebesgue spaces.

Findings

Developed reconstruction algorithms for single averaging function sampling.

Developed reconstruction algorithms for multiple averaging functions sampling.

Enhanced understanding of sampling in shift-invariant subspaces of mixed Lebesgue spaces.

Abstract

In this paper, we study nonuniform average sampling problem in multiply generated shift-invariant subspaces of mixed Lebesgue spaces. We discuss two types of average sampled values: average sampled values $\{\left \langle f,\psi_{a}(\cdot-x_{j},\cdot-y_{k}) \right \rangle:j,k\in\mathbb{J}\}$ generated by single averaging function and average sampled values $\left\{\left \langle f,\psi_{x_{j},y_{k}}\right \rangle:j,k\in\mathbb{J}\right\}$ generated by multiple averaging functions. Two fast reconstruction algorithms for this two types of average sampled values are provided.