¹-summability and Fourier series of B-splines with respect to their knots

We study the ¹ ℓ 1 -summability of functions in the d -dimensional torus Tᵈ T d and so-called ¹ ℓ 1 -invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the ¹ ℓ 1 -norm of their indices. Such functions are characterized as divided differences that have cos θ ₁,… ,cos θ d cos θ 1 , … , cos θ d as knots for (θ ₁\,… , θ d) ∈ Tᵈ ( θ 1 … , θ d ) ∈ 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.