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Mat. Sb. (N.S.), 1970, Volume 81(123), Number 1, Pages 104–131 (Mi msb3364)  

This article is cited in 41 scientific papers (total in 42 papers)

Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics

P. L. Ul'yanov


Abstract: In the first part of the work are established imbedding theorems pertaining to arbitrary classes of functions of a single variable $\varphi(L)$, $L\varphi(L)$, $H_p^{\omega(\delta)}$ and $L^\nu\ln^\beta(1+L)$.
The second part contains estimates for best approximations (moduli of continuity) in different metrics. It is shown that in certain cases these estimates cannot be strengthened.
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 10:1, 103–126

Bibliographic databases:

UDC: 517.51
MSC: 26A15, 26D05, 42A05, 41A10
Received: 08.05.1968

Citation: P. L. Ul'yanov, “Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics”, Mat. Sb. (N.S.), 81(123):1 (1970), 104–131; Math. USSR-Sb., 10:1 (1970), 103–126

Citation in format AMSBIB
\Bibitem{Uly70}
\by P.~L.~Ul'yanov
\paper Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics
\jour Mat. Sb. (N.S.)
\yr 1970
\vol 81(123)
\issue 1
\pages 104--131
\mathnet{http://mi.mathnet.ru/msb3364}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=415303}
\zmath{https://zbmath.org/?q=an:0196.08302|0215.17702}
\transl
\jour Math. USSR-Sb.
\yr 1970
\vol 10
\issue 1
\pages 103--126
\crossref{https://doi.org/10.1070/SM1970v010n01ABEH001589}


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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. P. L. Ul'yanov, “Representation of functions by series and classes $\varphi(L)$”, Russian Math. Surveys, 27:2 (1972), 1–54  mathnet  crossref  mathscinet  zmath
    2. B. I. Golubov, “Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials”, Math. USSR-Sb., 16:2 (1972), 265–285  mathnet  crossref  mathscinet  zmath
    3. È. A. Storozhenko, “Necessary and sufficient conditions for the imbedding of some classes of functions”, Math. USSR-Izv., 7:2 (1973), 388–400  mathnet  crossref  mathscinet  zmath
    4. V. I. Kolyada, “On imbedding in classes $\varphi(L)$”, Math. USSR-Izv., 9:2 (1975), 395–413  mathnet  crossref  mathscinet  zmath
    5. È. A. Storozhenko, “Embedding theorems and best approximations”, Math. USSR-Sb., 26:2 (1975), 213–224  mathnet  crossref  mathscinet  zmath
    6. È. A. Storozhenko, V. G. Krotov, P. Oswald, “Direct and converse theorems of Jackson type in $L^p$ spaces, $0<p<1$”, Math. USSR-Sb., 27:3 (1975), 355–374  mathnet  crossref  mathscinet  zmath
    7. V. I. Kolyada, “Imbedding theorems and inequalities in various metrics for best approximations”, Math. USSR-Sb., 31:2 (1977), 171–189  mathnet  crossref  mathscinet  zmath  isi
    8. È. A. Storozhenko, “On the approximation by algebraic polynomials of functions of class $L^p$, $0<p<1$”, Math. USSR-Izv., 11:3 (1977), 613–623  mathnet  crossref  mathscinet  zmath
    9. È. A. Storozhenko, “On a problem of Hardy-Littlewood”, Math. USSR-Sb., 47:2 (1984), 557–577  mathnet  crossref  mathscinet  zmath
    10. V. N. Temlyakov, “Approximation of periodic functions of several variables by trigonometric polynomials, and widths of some classes of functions”, Math. USSR-Izv., 27:2 (1986), 285–322  mathnet  crossref  mathscinet  zmath
    11. Koliada V., “Embedding Into the Bmo and Exp l-Alpha Spaces”, 287, no. 2, 1986, 277–280  isi
    12. V. I. Ivanov, “Approximation in $L_p$ by polynomials in the Walsh system”, Math. USSR-Sb., 62:2 (1989), 385–402  mathnet  crossref  mathscinet  zmath
    13. Kaljabin G. Lizorkin P., “Spaces of Functions of Generalized Smoothness”, Math. Nachr., 133 (1987), 7–32  crossref  isi
    14. V. I. Kolyada, “Estimates of rearrangements and imbedding theorems”, Math. USSR-Sb., 64:1 (1989), 1–21  mathnet  crossref  mathscinet  zmath
    15. V. I. Kolyada, “Rearrangements of functions and embedding theorems”, Russian Math. Surveys, 44:5 (1989), 73–117  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. N. N. Pustovoitov, “Estimates of the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson's theorem”, Sb. Math., 188:10 (1997), 1507–1520  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. G. V. Radzievskii, “Direct and converse theorems in problems of approximation by vectors of finite degree”, Sb. Math., 189:4 (1998), 561–601  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. V. I. Kolyada, “Embeddings of fractional Sobolev spaces and estimates of Fourier transforms”, Sb. Math., 192:7 (2001), 979–1000  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. M. B. Sikhov, “Inequalities of Bernstein and Jackson–Nikol'skii types and their applications”, Russian Math. (Iz. VUZ), 46:8 (2002), 54–61  mathnet  mathscinet  zmath  elib
    20. V. I. Kolyada, “Inequalities of Gagliardo–Nirenberg type and estimates for the moduli of continuity”, Russian Math. Surveys, 60:6 (2005), 1147–1164  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Kolyada V. Marcellan F., “Kernels and Best Approximations Related to the System of Ultraspherical Polynomials”, J. Approx. Theory, 133:2 (2005), 173–194  crossref  isi
    22. S. I. Dudov, A. M. Zakharov, D. V. Prokhorov, A. P. Khromov, “Petr Lavrentevich Ulyanov”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 7:1 (2007), 89–93  mathnet  elib
    23. V. A. Andrienko, “Embedding of $H_p^\omega$ in the class $e^L$”, Russian Math. (Iz. VUZ), 54:3 (2010), 1–6  mathnet  crossref  mathscinet  elib
    24. N. A. Ilyasov, “Skorostnaya $L_p$-versiya kriteriya M. Rissa absolyutnoi skhodimosti trigonometricheskikh ryadov Fure”, Tr. IMM UrO RAN, 16, no. 4, 2010, 193–202  mathnet  elib
    25. Tikhonov S., “Weak Type Inequalities for Moduli of Smoothness: The Case of Limit Value Parameters”, Journal of Fourier Analysis and Applications, 16:4 (2010), 590–608  crossref  isi  elib
    26. Simonov B., Tikhonov S., “Sharp Ul'yanov-type inequalities using fractional smoothness”, Journal of Approximation Theory, 162:9 (2010), 1654–1684  crossref  isi  elib
    27. Tikhonov S., Trebels W., “Ulyanov-type inequalities and generalized Liouville derivatives”, Proc Roy Soc Edinburgh Sect A, 141:1 (2011), 205–224  crossref  isi
    28. E. S. Smailov, A. I. Takuadina, “O neuluchshaemosti predelnoi teoremy vlozheniya raznykh metrik v prostranstvakh Lorentsa s vesom Ermitta”, Ufimsk. matem. zhurn., 3:3 (2011), 140–151  mathnet  zmath
    29. M. K. Potapov, B. V. Simonov, “Improvement of the Ul'yanov inequality for mixed moduli of smoothness for functions with lacunary Fourier coefficients”, Moscow University Mathematics Bulletin, 67:1 (2012), 16–22  mathnet  crossref  mathscinet
    30. I. P. Irodova, “Piecewise polynomial approximation methods in the theory of Nikol'skiĭ–Besov spaces”, Journal of Mathematical Sciences, 209:3 (2015), 319–480  mathnet  crossref
    31. T. F. Ismagilov, “Embedding theorems for classes of functions with a dominating mixed modulus of smoothness”, Moscow University Mathematics Bulletin, 68:5 (2013), 215–220  mathnet  crossref  mathscinet
    32. M. K. Potapov, B. V. Simonov, “Connection between moduli of smoothness in the metrics of $L_p$ and $C$”, Moscow University Mathematics Bulletin, 70:1 (2015), 6–13  mathnet  crossref  mathscinet
    33. A. U. Bimendina, E. S. Smailov, “Fourier–Price coefficients of class GM and best approximations of functions in the Lorentz space $L_{p\theta}[0,1)$, $1<p<+\infty$, $1<\theta<+\infty$”, Proc. Steklov Inst. Math., 293 (2016), 77–98  mathnet  crossref  crossref  mathscinet  isi  elib
    34. N. A. Ilyasov, “Obratnaya teorema v raznykh metrikakh teorii priblizhenii periodicheskikh funktsii s monotonnymi koeffitsientami Fure”, Tr. IMM UrO RAN, 22, no. 4, 2016, 153–162  mathnet  crossref  mathscinet  elib
    35. Ullrich M. Ullrich T., “The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative”, SIAM J. Numer. Anal., 54:2 (2016), 969–993  crossref  mathscinet  zmath  isi  scopus
    36. 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
    37. N. A. Ilyasov, “Pryamaya teorema v raznykh metrikakh teorii priblizhenii periodicheskikh funktsii s monotonnymi koeffitsientami Fure”, Tr. IMM UrO RAN, 23, no. 3, 2017, 144–158  mathnet  crossref  elib
    38. 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
    39. 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  isi
    40. N. A. Ilyasov, “O ravnosilnosti nekotorykh sootnoshenii v raznykh metrikakh mezhdu normami, nailuchshimi priblizheniyami i modulyami gladkosti periodicheskikh funktsii i ikh proizvodnykh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 176–188  mathnet  crossref  elib
    41. Niyazi A. Il'yasov, “Order equalities in different metrics for moduli of smoothness of various orders”, Ural Math. J., 4:2 (2018), 24–32  mathnet  crossref  elib
    42. E. D. Kosov, “Klassy Besova na konechnomernykh i beskonechnomernykh prostranstvakh”, Matem. sb., 210:5 (2019), 41–71  mathnet  crossref  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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