A Remark on the Convolution with Box Splines

TL;DR

This paper presents a formula involving Bernoulli polynomials to quantify the difference between semi-discrete convolution and continuous convolution with Box Splines, enhancing understanding in approximation theory.

Contribution

It introduces a novel formula connecting semi-discrete and continuous convolutions with Box Splines using Bernoulli polynomials.

Findings

Derived a formula for the convolution difference

Connected Bernoulli polynomials with convolution analysis

Improved theoretical understanding of Box Spline convolutions

Abstract

The semi-discrete convolution with the Box Spline is an important tool in approximation theory. We give a formula for the difference between semi-discrete convolution and convolution with the Box Spline. This formula involves multiple Bernoulli polynomials.