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Mat. Sb., 2008, Volume 199, Number 9, Pages 107–148 (Mi msb3941)  

This article is cited in 28 scientific papers (total in 28 papers)

Embedding theorems in constructive approximation

B. V. Simonova, S. Yu. Tikhonovbc

a Volgograd State Technical University
b Scuola Normale Superiore in Pisa
c Institució Catalana de Recerca i Estudis Avancats

Abstract: Necessary and sufficient conditions for the accuracy of embedding theorems of various function classes are obtained. The main result of the paper is a criterion for embeddings between generalized Weyl-Nikol'skiǐ and generalized Lipschitz classes. To define the Weyl-Nikol'skiǐ classes we use the concept of a $(\lambda,\beta)$-derivative, which is a generalization of the derivative in the sense of Weyl. As corollaries, estimates for the norms and moduli of smoothness of transformed Fourier series are obtained.
Bibliography: 59 titles.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm3941

Full text: PDF file (843 kB)
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English version:
Sbornik: Mathematics, 2008, 199:9, 1367–1407

Bibliographic databases:

UDC: 517.518.23+517.518.83
MSC: Primary 46E35, 26A33, 41A17; Secondary 26A16, 42A45
Received: 10.09.2007 and 07.03.2008

Citation: B. V. Simonov, S. Yu. Tikhonov, “Embedding theorems in constructive approximation”, Mat. Sb., 199:9 (2008), 107–148; Sb. Math., 199:9 (2008), 1367–1407

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. I. Dyachenko, E. D. Nursultanov, “Hardy-Littlewood theorem for trigonometric series with $\alpha$-monotone coefficients”, Sb. Math., 200:11 (2009), 1617–1631  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Tikhonov S., “Weak type inequalities for moduli of smoothness: the case of limit value parameters”, J. Fourier Anal. Appl., 16:4 (2010), 590–608  crossref  mathscinet  zmath  isi  elib  scopus
    3. Tikhonov S., Trebels W., “Ulyanov-type inequalities and generalized Liouville derivatives”, Proc. Roy. Soc. Edinburgh Sect. A, 141:1 (2011), 205–224  crossref  mathscinet  zmath  isi  elib  scopus
    4. Akgün R., Kokilashvili V., “The refined direct and converse inequalities of trigonometric approximation in weighted variable exponent Lebesgue spaces”, Georgian Math. J., 18:3 (2011), 399–423  mathscinet  zmath  isi  elib
    5. Akgün R., “Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth”, Georgian Math. J., 18:2 (2011), 203–235  mathscinet  zmath  isi
    6. Akgun R., “Improved Converse Theorems and Fractional Moduli of Smoothness in Orlicz Spaces”, Bull. Malays. Math. Sci. Soc., 36:1 (2013), 49–62  mathscinet  zmath  isi
    7. St. Petersburg Math. J., 26:5 (2015), 741–756  mathnet  crossref  mathscinet  isi  elib  elib
    8. M. K. Potapov, B. V. Simonov, “Properties of the mixed modulus of smoothness of positive order in a mixed metric”, Moscow University Mathematics Bulletin, 69:6 (2014), 258–266  mathnet  crossref  mathscinet
    9. A. Jumabayeva, “Liouville–Weyl derivatives, best approximations, and moduli of smoothness”, Acta Math. Hungar, 2015  crossref  mathscinet  zmath  elib  scopus
    10. N. A. Ilyasov, “O poryadke ravnomernoi skhodimosti chastnykh kubicheskikh summ kratnykh trigonometricheskikh ryadov Fure na klassakh funktsii $H_{1,m}^{l}[\omega]$”, Tr. IMM UrO RAN, 21, no. 4, 2015, 161–177  mathnet  mathscinet  elib
    11. Akgun R., “Some inequalities of trigonometric approximation in weighted Orlicz spaces”, Math. Slovaca, 66:1 (2016), 217–234  crossref  mathscinet  zmath  isi  scopus
    12. Jafarov S.Z., “Approximation in weighted Lorentz spaces On moduli of smoothness and approximation by trigonometric polynomials in weighted Lorentz spaces”, Hacet. J. Math. Stat., 45:6 (2016), 1675–1684  crossref  mathscinet  zmath  isi  elib  scopus
    13. M. K. Potapov, B. V. Simonov, “Interrelations between full moduli of smoothness in the metrics of $L_1$ и $L_\infty$”, Moscow University Mathematics Bulletin, 71:1 (2016), 15–22  mathnet  crossref  mathscinet  isi
    14. M. K. Potapov, B. V. Simonov, “Interrelations between mixed moduli of smoothness in metrics of $L_p$ and $L_\infty$”, Moscow University Mathematics Bulletin, 72:3 (2017), 107–120  mathnet  crossref  mathscinet  isi
    15. Jumabayeva A., “Sharp Ul'Yanov Inequalities For Generalized Liouville-Weyl Derivatives”, Anal. Math., 43:2 (2017), 279–302  crossref  mathscinet  zmath  isi  scopus
    16. Akgun R. Yildirir Yu.E., “Generalized Derivatives and Approximation in Weighted Lorentz Spaces”, Bull. Belg. Math. Soc.-Simon Steven, 24:3 (2017), 355–366  mathscinet  isi
    17. Jafarov S.Z., “Derivatives of a Polynomial of Best Approximation and Modulus of Smoothness in Generalized Lebesgue Spaces With Variable Exponent”, Demonstr. Math., 50:1, SI (2017), 245–251  crossref  mathscinet  zmath  isi  scopus
    18. M. K. Potapov, B. V. Simonov, “Estimates for mixed moduli of smoothness in $L_q$ metric via mixed moduli of smoothness in $L_1$ metric”, Moscow University Mathematics Bulletin, 73:2 (2018), 50–63  mathnet  crossref  mathscinet  zmath  isi
    19. Kolomoitsev Yu., “Best Approximations and Moduli of Smoothness of Functions and Their Derivatives in l-P, 0 < P < 1”, J. Approx. Theory, 232 (2018), 12–42  crossref  mathscinet  zmath  isi  scopus
    20. Jumabayeva A., Simonov B., “Inequalities For Moduli of Smoothness of Functions and Their Liouville-Weyl Derivatives”, Acta Math. Hung., 156:1 (2018), 1–17  crossref  mathscinet  zmath  isi  scopus
    21. Danelia A., “Conjugate Functions and the Modulus of Smoothness of Fractional Order”, J. Contemp. Math. Anal.-Armen. Aca., 53:5 (2018), 288–293  crossref  mathscinet  zmath  isi  scopus
    22. Artamonov S. Runovski K. Schmeisser H.-J., “Approximation By Bandlimited Functions, Generalized K-Functionals and Generalized Moduli of Smoothness”, Anal. Math., 45:1 (2019), 1–24  crossref  mathscinet  zmath  isi  scopus
    23. K. V. Runovskii, “Generalized Smoothness and Approximation of Periodic Functions in the Spaces $L_p$, $1<p<+\infty$”, Math. Notes, 106:3 (2019), 412–422  mathnet  crossref  crossref  isi  elib
    24. M. I. Dyachenko, A. B. Mukanov, S. Yu. Tikhonov, “Smoothness of functions and Fourier coefficients”, Sb. Math., 210:7 (2019), 994–1018  mathnet  crossref  crossref  adsnasa  isi  elib
    25. Jafarov S.Z., “On Moduli of Smoothness of Functions in Orlicz Spaces”, Tbil. Math. J., 12:3 (2019), 121–129  crossref  mathscinet  zmath  isi
    26. Jafarov S.Z., “Best Trigonometric Approximation and Modulus of Smoothness of Functions in Weighted Grand Lebesgue Spaces”, Bull. Karaganda Univ-Math., 94:2 (2019), 26–32  crossref  isi
    27. Testici A., “On Derivative of Trigonometric Polynomials and Characterizations of Modulus of Smoothness in Weighted Lebesgue Space With Variable Exponent”, Period. Math. Hung., 80:1 (2020), 59–73  crossref  mathscinet  isi
    28. A. A. Jumabayeva, B. V. Simonov, “Transformation of Fourier Series by Means of General Monotone Sequences”, Math. Notes, 107:5 (2020), 740–758  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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