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

We study the ℓ ^1 -summability of functions in the d -dimensional torus 𝕋^d and so-called ℓ ^1 -invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the ℓ ^1 -norm of their indices. Such functions are characterized as divided differences that have cosθ_1,… ,cosθ_d as knots for (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.