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

We study the ${\mathrm{\ell }}^1$-summability of functions in the $d$-dimensional torus ${\mathbb{T}}^d$ and so-called ${\mathrm{\ell }}^1$-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the ${\mathrm{\ell }}^1$-norm of their indices. Such functions are characterized as divided differences that have $\cos {\theta }_1,\dots ,\cos {\theta }_d$ as knots for $({\theta }_1\dots ,{\theta }_d)\in {\mathbb{T}}^d$. 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.