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Mat. Sb., 2004, Volume 195, Number 8, Pages 3–46 (Mi msb838)  

This article is cited in 13 scientific papers (total in 13 papers)

On Jackson's inequality for a generalized modulus of continuity in $L_2$

A. I. Kozko, A. V. Rozhdestvenskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The value of the sharp constant $\varkappa$ in the Jackson type inequality in the space $L_2(\mathbb T^d)$
\begin{equation} E_{n-1}(f)\leqslant\varkappa\overline\omega_\psi(f,T) \end{equation}
is studied for the generalized modulus of continuity
$$ \overline\omega_\psi(f,T)=\max_{t\in T}(\sum_{s}\psi(st)|\widehat f_s|^2)^{1/2}. $$
The value $\overset{*}{\varkappa}$ of the minimum sharp constant in inequality (1) is found.
A class of generalized moduli of continuity is introduced which contains the moduli $\widetilde\omega_{a,r}(f,\delta):=\sup_{0\leqslant t\leqslant\delta}\|\Delta_{a^{r-1}t}\dotsb \Delta_{at}\Delta_{t}f\|_2$, with even $a$. The relation $\varkappa=\overset{*}\varkappa$ is proved in this class for all $\delta\geqslant\pi/n$.

DOI: https://doi.org/10.4213/sm838

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English version:
Sbornik: Mathematics, 2004, 195:8, 1073–1115

Bibliographic databases:

UDC: 517.518.8
MSC: 41A17
Received: 14.06.2002 and 10.11.2003

Citation: A. I. Kozko, A. V. Rozhdestvenskii, “On Jackson's inequality for a generalized modulus of continuity in $L_2$”, Mat. Sb., 195:8 (2004), 3–46; Sb. Math., 195:8 (2004), 1073–1115

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. S. Balaganskii, “Exact constant in the Jackson–Stechkin inequality in the space $L^2$ on the period”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S78–S102  mathnet  crossref  isi  elib
    2. S. N. Vasil'ev, “Jackson inequality in $L_2(\mathbb T^N)$ with generalized modulus of continuity”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S218–S226  mathnet  crossref  isi  elib
    3. V. I. Ivanov, “Direct and inverse theorems in approximation theory for periodic functions in S. B. Stechkins papers and the development of these theorems”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S1–S13  mathnet  crossref  elib
    4. S. N. Vasil'ev, “Jackson inequality in $L_2(\mathbb R^N)$ with generalized modulus of continuity”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S163–S170  mathnet  crossref  isi  elib
    5. V. S. Balaganskii, “On the Continuity of the Sharp Constant in the Jackson–Stechkin Inequality in the Space $L^2$”, Math. Notes, 93:1 (2013), 12–28  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Vakarchuk S.B., Zabutnaya V.I., “On the Best Polynomial Approximation in the Space l (2) and Widths of Some Classes of Functions”, Ukr. Math. J., 64:8 (2013), 1168–1176  crossref  mathscinet  zmath  isi
    7. D. V. Gorbachev, “An estimate of an optimal argument in the sharp multidimensional Jackson–Stechkin $L_2$-inequality”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 70–78  mathnet  crossref  mathscinet  isi  elib
    8. S. Yu. Artamonov, “Quality of Approximation by Fourier Means in Terms of General Moduli of Smoothness”, Math. Notes, 98:1 (2015), 3–10  mathnet  crossref  crossref  mathscinet  isi  elib
    9. S. B. Vakarchuk, V. I. Zabutnaya, “Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space $L_2$ and Widths of Classes of Functions”, Math. Notes, 99:2 (2016), 222–242  mathnet  crossref  crossref  mathscinet  isi  elib
    10. K. V. Runovskii, “Approximation by Fourier Means and Generalized Moduli of Smoothness”, Math. Notes, 99:4 (2016), 564–575  mathnet  crossref  crossref  mathscinet  isi  elib
    11. M. Sh. Shabozov, A. D. Farozova, “Tochnoe neravenstvo Dzheksona–Stechkina s neklassicheskim modulem nepreryvnosti”, Tr. IMM UrO RAN, 22, no. 4, 2016, 311–319  mathnet  crossref  mathscinet  elib
    12. Ivanov V., Ivanov A., “Generalized Logan'S Problem For Entire Functions of Exponential Type and Optimal Argument in Jackson'S Inequality in l-2((3))”, Acta. Math. Sin.-English Ser., 34:10 (2018), 1563–1577  crossref  mathscinet  isi  scopus
    13. Babenko V.F., Konareva S.V., “Jackson-Stechkin-Type Inequalities For the Approximation of Elements of Hilbert Spaces”, Ukr. Math. J., 70:9 (2019), 1331–1344  crossref  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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