Discretization of Continuous Frames by Quasi-Monte Carlo Methods

TL;DR

This paper presents a novel discretization method for continuous localized frames using quasi-Monte Carlo techniques, providing a universal sampling scheme with controlled density based on discrepancy measures.

Contribution

It generalizes classical discrepancy concepts to the entire phase space and establishes a Koksma-Hlawka inequality for frame discretization.

Findings

Discrepancy measure defined on phase space $\\mathbb{R}^2$

A Koksma-Hlawka inequality for frame discretization

Universal sampling set controlled by discrepancy and Sobolev seminorm

Abstract

We introduce a discretization scheme for continuous localized frames using quasi-Monte Carlo integration and discrepancy theory. By generalizing classical concepts, we define a discrepancy measure on the entire phase space $\mathbb{R}^2$ and establish a corresponding Koksma-Hlawka inequality. This approach enables control over the density of the discretized frame and ensures the universality of the sampling set, relying only on the discrepancy of the sampling set and on the Sobolev-type seminorm of an iterated kernel rather than on specific frame properties.