论文标题
$ \ ell^1 $ -Summability and Fourier系列B-Splines相对于他们的结
$\ell^1$-summability and Fourier series of B-splines with respect to their knots
论文作者
论文摘要
我们研究$ d $ -D $ - 二维圆环$ \ mathbb {t}^d $和所谓的$ \ ell^1 $ -Invariant函数中的$ \ ell^1 $ - 单位。这些是在圆环上的功能，其傅立叶系数仅取决于$ \ ell^1 $ norm的索引。此类函数的特征是具有$ \cosθ_1，\ ldots，\cosθ_d$的划分差异为$（θ_1\，\ ldots，θ_d）\ in \ mathbb {t}^d $。它导致我们考虑了相对于其结的$ D $维傅立叶系列B-Splines系列，事实证明，它享受了一种简单的双轴性，可用于获得B-Spline功能的正交系列。
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 $\cos θ_1,\ldots,\cosθ_d$ as knots for $(θ_1\,\ldots, θ_d) \in \mathbb{T}^d$. 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.
