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Abstract We study the ¹ ℓ 1 -summability of functions in the d -dimensional torus { {T}}ᵈ 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 ₁, , d cos θ 1, …, cos θ d as knots for (₁\, , d) { {T}}ᵈ (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.
Buhmann et al. (Sat,) studied this question.
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