Abstract

In this paper, a set of optimal subspaces is specified for L₂ approximation of three classes of functions in the Sobolev spaces \(W_{2}^{{(r)}}\) defined on a segment and subject to certain boundary conditions. A subspace X of a dimension not exceeding n is called optimal for a function class A if the best approximation of A by X is equal to the Kolmogorov n-width of A. These boundary conditions correspond to subspaces of periodically extended functions with symmetry properties. All approximating subspaces are generated by equidistant shifts of a single function. The conditions of optimality are given in terms of Fourier coefficients of a generating function. In particular, we indicate optimal spline spaces of all degrees d \( \geqslant \)r – 1 with equidistant knots of several different types.

中文翻译：

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段上可微函数类均方逼近的最优子空间

摘要

在本文中，指定了一组最佳子空间，用于在段上定义的Sobolev空间\（W_ {2} ^ {{{（r）}} \）中的三类函数的L ₂逼近，并且要服从某些边界条件。子空间X不超过一个尺寸的Ñ被称为最佳的一个函数类甲如果的最佳近似阿由X等于洛夫Ñ -width的甲。这些边界条件对应于具有对称特性的周期性扩展函数的子空间。所有近似子空间都是由单个函数的等距移位生成的。最优条件是根据生成函数的傅立叶系数给出的。特别是，我们指出了所有度为d \（\ geqslant \）r – 1的最优样条空间，具有几种不同类型的等距结。