Geodesic
Sharp estimates for mean square approximations of classes of periodic convolutions by spaces of shifts
Algebra i analiz, Tome 32 (2020) no. 2, pp. 201-228

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Let $ L_p$ be the classical Lebesgue spaces of $ 2\pi $-periodic functions and $ E(f,X)_2$ the best approximation of  $ f$ by the space  $ X$ in  $ L_2$. For $ n\in \mathbb{N}$, $ B\in L_2$, the symbol $ \mathbb{S}_{B,n}$ stands for the space of functions  $ s$ of the form $\displaystyle s(x)=\sum _{j=0}^{2n-1}\beta _jB\Big (x-\frac {j\pi }{n}\Big ).$     In this paper, all spaces  $ \mathbb{S}_{B,n}$ are described that provide a sharp constant in several inequalities for approximation of classes of convolutions with a kernel  $ G\in L_1$. In particular, necessary and sufficient conditions are obtained under which the inequality $\displaystyle E\bigl (f,\mathbb{S}_{B,n}\bigr )_2\leq \vert c^\ast _{2n+1}(G)\vert\Vert\varphi \Vert _2$     is fulfilled. This inequality is sharp on the class of functions  $ f$ representable in the form $ f=G\ast \varphi $, $ \varphi \in L_2$. The constant $ \vert c^\ast _{2n+1}(G)\vert$ is the $ (2n+1)$th term of the sequence $ \{\vert c_l(G)\vert\}_{l\in \mathbb{Z}}$ of absolute values of the Fourier coefficients of  $ G$ arranged in nonincreasing order. In addition, easily verifiable conditions are indicated that suffice for the above inequality. Examples of kernels and extremal subspaces satisfying these conditions are provided.
Keywords: best approximation, spaces of shifts
Mots-clés : sharp constants, classes of convolutions. 
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     title = {Sharp estimates for mean square approximations of classes of periodic convolutions by spaces of shifts},
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A. Yu. Ulitskaya. Sharp estimates for mean square approximations of classes of periodic convolutions by spaces of shifts. Algebra i analiz, Tome 32 (2020) no. 2, pp. 201-228. http://geodesic.mathdoc.fr/item/AA_2020_32_2_a6/

[1] Korneichuk N. P., Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987 | MR | MR

[2] Sun Yunshen, Li Chun, “Nailuchshee priblizhenie nekotorykh klassov gladkikh funktsii na deistvitelnoi osi splainami vysshego poryadka”, Mat. zametki, 48:4 (1990), 100–109 | Zbl | Zbl

[3] Vinogradov O. L., Ulitskaya A. Yu., “Tochnye otsenki srednekvadratichnykh priblizhenii klassov differentsiruemykh periodicheskikh funktsii prostranstvami sdvigov”, Vestn. Sankt-Peterburg. un-ta. Ser. mat., mekh., astronom., 2018, no. 1, 22–31

[4] A. Pinkus, “$n$-widths in approximation theory”, Ergeb. Math. Grenzgeb. (3), 7 (1985), Springer-Verlag, Berlin | MR | Zbl | MR | Zbl

[5] Schoenberg I. J., Cardinal spline interpolation, Conf. Board Math. Sci. Reg. Conf. Ser. Appl. Math., 12, 2nd ed., Soc. Industr. Appl. Math., Philadelphia, Pa, 1993 | MR | MR

[6] Golomb M., “Approximation by periodic spline interpolants on uniform meshes”, J. Approx. Theory, 1 (1968), 26–65 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Kamada M., Toriachi K., Mori R., “Periodic spline orthonormal bases”, J. Approx. Theory, 55 (1988), 27–34 | DOI | MR | Zbl | DOI | MR | Zbl

[8] Vinogradov O. L., “Analog summ Akhiezera–Kreina–Favara dlya periodicheskikh splainov minimalnogo defekta”, Probl. mat. anal., 25 (2003), 29–56 | Zbl | Zbl

[9] Vinogradov O. L., “Tochnye neravenstva dlya priblizhenii klassov periodicheskikh svertok prostranstvami sdvigov nechetnoi razmernosti”, Mat. zametki, 85:4 (2009), 569–584 | DOI | MR | Zbl | DOI | MR | Zbl

[10] Khardi G. G., Littlvud Dzh. E., Polia G., Neravenstva, IL, M., 1948