$\ell^1$-summability and Fourier series of B-splines with respect to their knots

TL;DR

This paper investigates the $ ext{l}^1$-summability of functions on the $d$-dimensional torus, focusing on $ ext{l}^1$-invariant functions related to B-splines and their Fourier series with respect to their knots.

Contribution

It characterizes $ ext{l}^1$-invariant functions as divided differences with specific knots and analyzes the Fourier series of B-splines, revealing a simple bi-orthogonality property.

Findings

Characterization of $ ext{l}^1$-invariant functions as divided differences.

Identification of bi-orthogonality in B-spline Fourier series.

Development of orthogonal series for B-spline functions.

Abstract

We study the $\ell^1$-summability of functions in the $d$-dimensional torus $\mathbb{T}^d$ and so-called $\ell^1$-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the $\ell^1$-norm of their indices. Such functions are characterized as divided differences that have $\cos \theta_1,\ldots,\cos\theta_d$ as knots for $(\theta_1\,\ldots, \theta_d) \in \mathbb{T}^d$. It leads us to consider the $d$-dimensional Fourier series of univariate B-splines with respect to its knots, which turns out to enjoy a simple bi-orthogonality that can be used to obtain an orthogonal series of the B-spline function.