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arXiv · 2208.02167

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

Martin Buhmann·Janin Jäger·Yuan Xu
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Abstract

We study the $\ell^1$-summability of functions in the $d$-dimensional torus $\mathbb{T}^d$ and so-called $\ell^1$-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the $\ell^1$-norm of their indices. Such functions are characterized as divided differences that have $\cos \theta_1,\ldots,\cos\theta_d$ as knots for $(\theta_1\,\ldots, \theta_d) \in \mathbb{T}^d$. 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.

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Martin Buhmann, Janin Jäger, Yuan Xu. 2022-08-03. $\ell^1$-summability and Fourier series of B-splines with respect to their knots. https://arxiv.org/abs/2208.02167

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