Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces |
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Authors: | O L Vinogradov A Yu Ulitskaya |
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Institution: | 1.St. Petersburg State University,St. Petersburg,Russia |
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Abstract: | Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ? and B ∈ L2, let \({{\Bbb S}_{B,n}}\) be the space of functions s of the form \(s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} \). This paper describes all spaces \({{\Bbb S}_{B,n}}\) that satisfy the exact inequality \(E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}\). (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases. |
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