l1-summability and Fourier series of B-splines with respect to their knots

Abstract

We study the l1-summability of functions in the d-dimensional torus Td and so-called l1-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the l1-norm of their indices. Such functions are characterized as divided differences that have cos θ1, . . . , cos θd as knots for (θ1 . . . , θd ) ∈ Td . It leads us to consider the ddimensional 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.