\(l^1\)-SUMMABILITY AND FOURIER SERIES OF B-SPLINES WITH RESPECT TO THEIR KNOTS

Ponente: Martin Buhmann.

Abstract: We study the \(l^1\)-summability of functions in the d-dimensional torus and so-called \(l^1\)-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the -norm of their indices. Such functions are characterized as divided differences that have \(cos \theta_1, . . . , cos \theta_d\) as knots.
It leads us to consider the d-dimensional Fourier series of uni-variate 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. (Joint work with Janin Jäger and Yuan Xu.)