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Izv. Akad. Nauk SSSR Ser. Mat., 1968, Volume 32, Issue 3, Pages 649–686 (Mi izv2482)  

This article is cited in 50 scientific papers (total in 50 papers)

The imbedding of certain function classes $H_p^\omega$

P. L. Ul'yanov

Abstract: Sufficient conditions on the modulus of continuity $\omega(\delta)$ are found for an imbedding of function classes $H_p^{\omega(\delta)}\subset\varphi(L)$. It is shown that in a number of cases these conditions are also necessary.

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English version:
Mathematics of the USSR-Izvestiya, 1968, 2:3, 601–637

Bibliographic databases:

UDC: 517.5
MSC: 26A15, 26B05, 40A05, 42A20
Received: 13.07.1967

Citation: P. L. Ul'yanov, “The imbedding of certain function classes $H_p^\omega$”, Izv. Akad. Nauk SSSR Ser. Mat., 32:3 (1968), 649–686; Math. USSR-Izv., 2:3 (1968), 601–637

Citation in format AMSBIB
\by P.~L.~Ul'yanov
\paper The imbedding of certain function classes $H_p^\omega$
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1968
\vol 32
\issue 3
\pages 649--686
\jour Math. USSR-Izv.
\yr 1968
\vol 2
\issue 3
\pages 601--637

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    This publication is cited in the following articles:
    1. V. A. Andrienko, “Necessary conditions for imbedding the function classes $H^\omega_p$”, Math. USSR-Sb., 7:2 (1969), 273–292  mathnet  crossref  mathscinet  zmath
    2. P. L. Ul'yanov, “Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics”, Math. USSR-Sb., 10:1 (1970), 103–126  mathnet  crossref  mathscinet  zmath
    3. V. P. Motornyi, “Approximation of functions by algebraic polynomials in the $L_p$ metric”, Math. USSR-Izv., 5:4 (1971), 889–914  mathnet  crossref  mathscinet  zmath
    4. 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
    5. 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
    6. L. V. Zhizhiashvili, “Some problems in the theory of simple and multiple trigonometric and orthogonal series”, Russian Math. Surveys, 28:2 (1973), 65–127  mathnet  crossref  mathscinet  zmath
    7. V. I. Kolyada, “On imbedding in classes $\varphi(L)$”, Math. USSR-Izv., 9:2 (1975), 395–413  mathnet  crossref  mathscinet  zmath
    8. È. A. Storozhenko, “Embedding theorems and best approximations”, Math. USSR-Sb., 26:2 (1975), 213–224  mathnet  crossref  mathscinet  zmath
    9. 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
    10. 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
    11. P. L. Ul'yanov, “On Fourier coefficients”, Math. USSR-Sb., 38:1 (1981), 11–29  mathnet  crossref  mathscinet  zmath  isi
    12. È. A. Storozhenko, “On a problem of Hardy-Littlewood”, Math. USSR-Sb., 47:2 (1984), 557–577  mathnet  crossref  mathscinet  zmath
    13. S. M. Nikol'skii, “Aleksandrov and Kolmogorov in Dnepropetrovsk”, Russian Math. Surveys, 38:4 (1983), 41–55  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    14. P. L. Ul'yanov, “Kolmogorov and divergent Fourier series”, Russian Math. Surveys, 38:4 (1983), 57–100  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. V. I. Kolyada, “On embedding $H_p^{\omega_1,…,\omega_\nu}$ classes”, Math. USSR-Sb., 55:2 (1986), 351–381  mathnet  crossref  mathscinet  zmath
    16. V. I. Kolyada, “Estimates of rearrangements and imbedding theorems”, Math. USSR-Sb., 64:1 (1989), 1–21  mathnet  crossref  mathscinet  zmath
    17. V. I. Kolyada, “Rearrangements of functions and embedding theorems”, Russian Math. Surveys, 44:5 (1989), 73–117  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    18. A. M. Stokolos, “Rate of strong differentiation of integrals”, Math. Notes, 59:4 (1996), 405–420  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. S. A. Anikin, “Error estimate for a regularization method in problems of the reconstruction of inputs of dynamic systems”, Comput. Math. Math. Phys., 37:9 (1997), 1020–1031  mathnet  mathscinet  zmath
    20. M. V. Medvedeva, “Embeddings of the classes $H^\omega$”, Math. Notes, 64:5 (1998), 616–621  mathnet  crossref  crossref  mathscinet  zmath  isi
    21. 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
    22. 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
    23. N. Yu. Antonov, “Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity”, Math. Notes, 76:5 (2004), 606–619  mathnet  crossref  crossref  mathscinet  zmath  isi
    24. Z. Ditzian, S. Tikhonov, “Ul’yanov and Nikol'skii-type inequalities”, Journal of Approximation Theory, 133:1 (2005), 100  crossref
    25. Sh. U. Azhgaliev, N. Temirgaliev, “Informativeness of all the linear functionals in the recovery of functions in the classes $H_p^\omega$”, Sb. Math., 198:11 (2007), 1535–1551  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    26. S. S. Volosivets, “Hardy and Bellman transformations of series with respect to multiplicative systems”, Sb. Math., 199:8 (2008), 1111–1137  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    27. M. B. Sikhov, “The embedding and approximation of classes of functions with a dominant mixed difference”, Russian Math. (Iz. VUZ), 53:8 (2009), 69–71  mathnet  crossref  mathscinet  zmath
    28. 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
    29. M. K. Potapov, B. V. Simonov, S. Yu. Tikhonov, “Relations between mixed moduli of smoothness, and embedding theorems for Nikol'skii classes”, Proc. Steklov Inst. Math., 269 (2010), 197–207  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    30. B. V. Simonov, “Embedding Nikol'skiĭclasses into Lorentz spaces”, Siberian Math. J., 51:4 (2010), 728–744  mathnet  crossref  mathscinet  isi  elib  elib
    31. Amiran Gogatishvili, Luboš Pick, Jan Schneider, “Characterization of a rearrangement-invariant hull of a Besov space via interpolation”, Rev Mat Complut, 2011  crossref
    32. I. Notarangelo, “Polynomial inequalities and embedding theorems with exponential weights on (−1,1)”, Acta Math Hung, 2011  crossref
    33. 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
    34. Shai Dekel, “On the analysis of anisotropic smoothness”, Journal of Approximation Theory, 2012  crossref
    35. 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
    36. Amiran Gogatishvili, Bohumír Opic, Sergey Tikhonov, Walter Trebels, “Ulyanov-type Inequalities Between Lorentz–Zygmund Spaces”, J Fourier Anal Appl, 2014  crossref
    37. S. S. Volosivets, “Modified Bessel ${\mathbf P}$-integrals and $\mathbf P$-derivatives and their properties”, Izv. Math., 78:5 (2014), 877–901  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    39. N. Temirgaliev, M. A. Zhainibekova, G. T. Dzhumakaeva, “Criteria of embedding of classes of Morrey type”, Russian Math. (Iz. VUZ), 59:5 (2015), 69–73  mathnet  crossref
    40. 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
    41. 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  isi
    42. 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
    43. N. Temirgaliev, A. Zhubanysheva, “Order estimates of the norms of derivatives of functions with zero values on linear functionals and their applications”, Russian Math. (Iz. VUZ), 61:3 (2017), 77–82  mathnet  crossref  isi
    44. N. A. Il'yasov, “The direct theorem of the theory of approximation of periodic functions with monotone Fourier coefficients in different metrics”, Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 100–114  mathnet  crossref  crossref  isi  elib
    45. 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
    46. 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
    47. 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  mathscinet
    48. E. D. Kosov, “Besov classes on finite and infinite dimensional spaces”, Sb. Math., 210:5 (2019), 663–692  mathnet  crossref  crossref  adsnasa  isi  elib
    49. Kolyada V.I., “Embedding Theorems For Sobolev and Hardy-Sobolev Spaces and Estimates of Fourier Transforms”, Ann. Mat. Pura Appl., 198:2 (2019), 615–637  crossref  mathscinet  zmath  isi  scopus
    50. Gabdolla Akishev, “Estimates of best approximations of functions with logarithmic smoothness in the Lorentz space with anisotropic norm”, Ural Math. J., 6:1 (2020), 16–29  mathnet  crossref  mathscinet
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