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Mat. Sb. (N.S.), 1977, Volume 103(145), Number 4(8), Pages 563–589 (Mi msb2930)  

This article is cited in 27 scientific papers (total in 27 papers)

Approximation properties of summable functions on sets of full measure

K. I. Oskolkov


Abstract: Estimates are obtained of the rate of approximation almost everywhere as a function of the modulus of continuity of the approximated functions in $L^p$, and of the set from which the approximating functions are chosen. From this point of view the author studies the approximation of functions by Steklov means, partial sums of Fourier–Haar series, arbitrary sequences of polynomials in the Haar and Faber–Schauder systems, and piecewise monotone functions with variable intervals of monotonicity. The estimates of the rate of approximation almost everywhere that are obtained are distinguished from approximation estimates in an integral metric (i.e. from estimates of the type of Jackson's theorem in $L^p$) by unbounded factors depending on the modulus of continuity and the approximating functions. Estimates of the growth of these factors are obtained, and it is established that in a number of cases these estimates are best possible, or almost so.
Bibliography: 17 titles.

Full text: PDF file (2083 kB)
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English version:
Mathematics of the USSR-Sbornik, 1977, 32:4, 489–514

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A25; Secondary 42A56
Received: 03.02.1977

Citation: K. I. Oskolkov, “Approximation properties of summable functions on sets of full measure”, Mat. Sb. (N.S.), 103(145):4(8) (1977), 563–589; Math. USSR-Sb., 32:4 (1977), 489–514

Citation in format AMSBIB
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\by K.~I.~Oskolkov
\paper Approximation properties of summable functions on sets of full measure
\jour Mat. Sb. (N.S.)
\yr 1977
\vol 103(145)
\issue 4(8)
\pages 563--589
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\transl
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 4
\pages 489--514
\crossref{https://doi.org/10.1070/SM1977v032n04ABEH002403}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. K. I. Oskolkov, “Polygonal approximation of functions of two variables”, Math. USSR-Sb., 35:6 (1979), 851–861  mathnet  crossref  mathscinet  zmath  isi
    2. V. I. Kolyada, “On embedding $H_p^{\omega_1,…,\omega_\nu}$ classes”, Math. USSR-Sb., 55:2 (1986), 351–381  mathnet  crossref  mathscinet  zmath
    3. P. L. Ul'yanov, “Luzin's work on the metric theory of functions”, Russian Math. Surveys, 40:3 (1985), 15–77  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. Koliada V., “Estimates of Maximal Functions Connected with Local Smoothness”, 293, no. 3, 1987, 534–537  mathscinet  isi
    5. V. I. Kolyada, “Estimates of rearrangements and imbedding theorems”, Math. USSR-Sb., 64:1 (1989), 1–21  mathnet  crossref  mathscinet  zmath
    6. Kashin B. Kosheleva G., “On an Approach to Correction Theorems”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1988, no. 4, 6–9  mathscinet  isi
    7. V. I. Kolyada, “Rearrangements of functions and embedding theorems”, Russian Math. Surveys, 44:5 (1989), 73–117  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. A. M. Stokolos, “On the strong differentiation of integrals of functions from Hölder classes”, Math. Notes, 55:1 (1994), 57–70  mathnet  crossref  mathscinet  zmath  isi
    9. B. S. Kashin, V. N. Temlyakov, “On best $m$-term approximations and the entropy of sets in the space $L_1$”, Math. Notes, 56:5 (1994), 1137–1157  mathnet  crossref  mathscinet  zmath  isi
    10. A. M. Stokolos, “Rate of strong differentiation of integrals”, Math. Notes, 59:4 (1996), 405–420  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. E. I. Berezhnoi, “Estimates for a uniform modulus of continuity of functions from symmetric spaces”, Izv. Math., 60:2 (1996), 231–248  mathnet  crossref  crossref  mathscinet  zmath
    12. José García-Cuerva, Víctor I. Kolyada, “Rearrangement Estimates for Fourier Transforms inLp andHp in Terms of Moduli of Continuity”, Math Nachr, 228:1 (2001), 123  crossref  mathscinet  zmath
    13. V. I. Ovchinnikov, A. S. Titenkov, “A Criterion for Contiguity of Quasiconcave Functions”, Math. Notes, 70:5 (2001), 708–713  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. Yuri Kryakin, Walter Trebels, “q-Moduli of Continuity in Hp(), p>0, and an Inequality of Hardy and Littlewood”, Journal of Approximation Theory, 115:2 (2002), 238  crossref
    15. A. S. Romanyuk, “Best $M$-term trigonometric approximations of Besov classes of periodic functions of several variables”, Izv. Math., 67:2 (2003), 265–302  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. Gogatishvili A., Pick L., “Discretization and Anti-Discretization of Rearrangement-Invariant Norms”, Publ. Mat., 47:2 (2003), 311–358  mathscinet  zmath  isi
    17. Girela D., Pelaez J., “On the Derivative of Infinite Blaschke Products”, Ill. J. Math., 48:1 (2004), 121–130  mathscinet  zmath  isi
    18. Krotov V., “Weighted l-P-Inequalities for Sharp-Maximal Functions”, Dokl. Math., 72:2 (2005), 684–686  mathscinet  zmath  isi
    19. Girela D., “A Class of Conformal Mappings with Applications to Function Spaces”, Recent Advances in Operator-Related Function Theory, Contemporary Mathematics Series, 393, eds. Matheson A., Stessin M., Timoney R., Amer Mathematical Soc, 2006, 113–121  crossref  mathscinet  zmath  isi
    20. V. G. Krotov, M. A. Prokhorovich, “The Luzin approximation of functions from the classes $W^p_\alpha$ on metric spaces with measure”, Russian Math. (Iz. VUZ), 52:5 (2008), 47–57  mathnet  crossref  mathscinet  zmath  elib
    21. Dai F., Wang K., “Convergence Rate of Spherical Harmonic Expansions of Smooth Functions”, J. Math. Anal. Appl., 348:1 (2008), 28–33  crossref  mathscinet  zmath  isi
    22. I. A. Ivanishko, V. G. Krotov, “Compactness of Embeddings of Sobolev Type on Metric Measure Spaces”, Math. Notes, 86:6 (2009), 775–788  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. Krotov V.G., “Quantitative Form of the Luzin C-Property”, Ukr. Math. J., 62:3 (2010), 441–451  crossref  mathscinet  zmath  isi
    24. V. G. Krotov, “Criteria for compactness in $L^p$-spaces, $p\ge0$”, Sb. Math., 203:7 (2012), 1045–1064  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    25. A. A. Dmitriev, “Ob otsenke konstanty $\mathscr{K}$-delimosti v parakh banakhovykh prostranstv”, Dalnevost. matem. zhurn., 13:2 (2013), 179–191  mathnet
    26. V. I. Ovchinnikov, “Interpolation functions and the Lions–Peetre interpolation construction”, Russian Math. Surveys, 69:4 (2014), 681–741  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    27. V. G. Krotov, A. I. Porabkovich, “Estimates of $L^p$-Oscillations of Functions for $p>0$”, Math. Notes, 97:3 (2015), 384–395  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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