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Mat. Sb. (N.S.), 1975, Volume 98(140), Number 3(11), Pages 395–415 (Mi msb3717)  

This article is cited in 57 scientific papers (total in 57 papers)

Direct and converse theorems of Jackson type in $L^p$ spaces, $0<p<1$

È. A. Storozhenko, V. G. Krotov, P. Oswald


Abstract: In this paper the connection between properties of functions and best approximations in classical orthogonal systems is studied in the $L^p$-metric, $0<p<1$. Two-sided inequalities are established between moduli of continuity and best approximations in these systems, which are unimprovable in a well-defined sense. Inequalities between best approximations in various metrics are also presented. A number of results are generalized to the classes $\varphi(L)$.
Bibliography: 23 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 27:3, 355–374

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A50; Secondary 42A56
Received: 07.04.1975

Citation: È. A. Storozhenko, V. G. Krotov, P. Oswald, “Direct and converse theorems of Jackson type in $L^p$ spaces, $0<p<1$”, Mat. Sb. (N.S.), 98(140):3(11) (1975), 395–415; Math. USSR-Sb., 27:3 (1975), 355–374

Citation in format AMSBIB
\Bibitem{StoKroOsw75}
\by \`E.~A.~Storozhenko, V.~G.~Krotov, P.~Oswald
\paper Direct and converse theorems of Jackson type in $L^p$ spaces, $0<p<1$
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 98(140)
\issue 3(11)
\pages 395--415
\mathnet{http://mi.mathnet.ru/msb3717}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=402384}
\zmath{https://zbmath.org/?q=an:0314.41004}
\transl
\jour Math. USSR-Sb.
\yr 1975
\vol 27
\issue 3
\pages 355--374
\crossref{https://doi.org/10.1070/SM1975v027n03ABEH002519}


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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. 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
    2. È. A. Storozhenko, “Approximation of functions of class $H^p$, $0<p\leqslant1$”, Math. USSR-Sb., 34:4 (1978), 527–545  mathnet  crossref  mathscinet  zmath
    3. È. A. Storozhenko, “Theorems of Jackson type in $H^p$$0<p<1$”, Math. USSR-Izv., 17:1 (1981), 203–218  mathnet  crossref  mathscinet  zmath  isi
    4. Oswald P., “Spline Approximation in the Lp-Metric, 0 Less-Than-Or-Equal-to P Less-Than-Or-Equal-to 1”, Math. Nachr., 94 (1980), 69–96  crossref  mathscinet  zmath  isi
    5. V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives”, Math. USSR-Izv., 18:1 (1982), 1–17  mathnet  crossref  mathscinet  zmath
    6. Krotov V., “On Differentiability of Functions From Lp and Hp for 0-Less-Than-P-Less-Than-1”, 256, no. 6, 1981, 1311–1314  mathscinet  zmath  isi
    7. Oswald P., “Lp-Approximation Using Series Following the Haar-Orthogonal System and the Faber-Schauder System”, J. Approx. Theory, 33:1 (1981), 1–27  crossref  mathscinet  zmath  isi
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    9. Oswald P., “On Besov-Hardy-Sobolev Spaces of Analytic-Functions in the Unit Disk”, Czech. Math. J., 33:3 (1983), 408–426  mathscinet  zmath  isi
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    11. Vukolova T., “On Some Characteristics of Trigonometric Series with Monotone Coefficients”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1984, no. 6, 18–23  mathscinet  zmath  isi
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    13. 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
    14. V. I. Kolyada, “Estimates of rearrangements and imbedding theorems”, Math. USSR-Sb., 64:1 (1989), 1–21  mathnet  crossref  mathscinet  zmath
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    16. Devore R., Leviatan D., Xiang M., “Polynomial-Approximation in Lp (0-Less-Than-P-Less-Than-1)”, Constr. Approx., 8:2 (1992), 187–201  crossref  mathscinet  zmath  adsnasa  isi
    17. Runovskii K., “On Approximation by Algebraic Polynomials in l(P) Spaces, 0-Less-Than-P-Less-Than-1”, Dokl. Akad. Nauk, 323:2 (1992), 238–240  mathnet  mathscinet  isi
    18. Runovskii K., “On by Angle Approximation in Lp Spaces, O-Less-Than-P-Less-Than-1”, 322, no. 1, 1992, 45–47  mathscinet  isi
    19. K. V. Runovskii, “On families of linear polynomial operators in $L_p$-spaces, $0<p<1$”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 165–173  mathnet  crossref  mathscinet  zmath  isi
    20. Yu. V. Kryakin, “On exact constants in the Whitney theorem”, Math. Notes, 54:1 (1993), 688–700  mathnet  crossref  mathscinet  zmath  isi  elib
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    22. K. V. Runovskii, “On approximation by families of linear polynomial operators in $l_P$-spaces, $0<p<1$”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 441–459  mathnet  crossref  mathscinet  zmath  isi
    23. Tateoka J., “On the Characterization of Hardy-Besov Spaces on the Dyadic Group and its Applications”, Studia Math., 110:2 (1994), 127–148  mathscinet  zmath  isi
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    25. Ditzian Z., Hristov V., Ivanov K., “Moduli of Smoothness and K-Functionals in l(P), 0-Less-Than-P-Less-Than-1”, Constr. Approx., 11:1 (1995), 67–83  crossref  mathscinet  zmath  isi
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    27. È. A. Storozhenko, “A problem of Mahler on the zeros of a polynomial and its derivative”, Sb. Math., 187:5 (1996), 735–744  mathnet  crossref  crossref  mathscinet  zmath  isi
    28. 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
    29. S. G. Pribegin, “A method of approximation in $H^p$, $0<p\leqslant 1$”, Sb. Math., 192:11 (2001), 1705–1719  mathnet  crossref  crossref  mathscinet  zmath  isi
    30. G. A. Akishev, “Obobschennaya sistema Khaara i teoremy vlozheniya v simmetrichnye prostranstva”, Fundament. i prikl. matem., 8:2 (2002), 319–334  mathnet  mathscinet  zmath
    31. S. B. Vakarchuk, “Exact Constants in Jackson-type Inequalities and Exact Values of Widths”, Math. Notes, 78:5 (2005), 735–739  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    32. È. A. Storozhenko, “Nikol'skii-Stechkin inequality for trigonometric polynomials in $L_0$”, Math. Notes, 80:3 (2006), 403–409  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    35. Yu. S. Kolomoitsev, “On approximation of functions by trigonometric polynomials with incomplete spectrum in $L_p$, $0<p<1$”, J. Math. Sci. (N. Y.), 165:4 (2010), 463–472  mathnet  crossref  elib
    36. Shabozov M.Sh., Akobirshoev M.O., “On Exact Values of Quasiwidths for Some Classes of Differentiable Periodic Functions of Two Variables”, Ukr. Math. J., 61:6 (2009), 1013–1024  crossref  mathscinet  zmath  isi
    37. 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
    38. V. V. Arestov, “Sharp inequalities for trigonometric polynomials with respect to integral functionals”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S21–S36  mathnet  crossref  isi  elib
    39. I. N. Katkovskaya, “Riesz' Compactness Criterion for the Space of Measurable Functions”, Math. Notes, 89:1 (2011), 145–149  mathnet  crossref  crossref  mathscinet  isi
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    41. M. Sh. Shabozov, G. A. Yusupov, “Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in $L_2$”, Siberian Math. J., 52:6 (2011), 1124–1136  mathnet  crossref  mathscinet  isi
    42. Matei B., Meignen S., Zakharova A., “Smoothness Characterization and Stability of Nonlinear and Non-Separable Multiscale Representations”, J. Approx. Theory, 163:11 (2011), 1707–1728  crossref  mathscinet  zmath  isi
    43. Vakarchuk S.B. Doronin V.G., “Best Mean Square Approximations by Entire Functions of Finite Degree on a Straight Line and Exact Values of Mean Widths of Functional Classes”, Ukr. Math. J., 62:8 (2011), 1199–1212  crossref  mathscinet  isi
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    45. S. B. Vakarchuk, V. I. Zabutnaya, “Jackson–Stechkin Type Inequalities for Special Moduli of Continuity and Widths of Function Classes in the Space $L_2$”, Math. Notes, 92:4 (2012), 458–472  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    46. A. N. Morozov, “O gladkosti v $L_p$, $0 < p < 1$”, Model. i analiz inform. sistem, 19:3 (2012), 97–104  mathnet
    47. Pichugov S.A., “Inverse Jackson Theorems in Spaces with Integral Metric”, Ukr. Math. J., 64:3 (2012), 394–407  crossref  zmath  isi
    48. Pichugov S.A., “Inequalities for Trigonometric Polynomials in Spaces with Integral Metric”, Ukr. Math. J., 63:12 (2012), 1883–1899  crossref  zmath  isi
    49. Pichugov S.A., “On the Jackson Theorem for Periodic Functions in Metric Spaces with Integral Metric. II”, Ukr. Math. J., 63:11 (2012), 1733–1744  crossref  mathscinet  zmath  isi
    50. 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  isi
    51. Pichugov S.A., “Smoothness of Functions in the Metric Spaces l (Psi)”, Ukr. Math. J., 64:9 (2013), 1382–1402  crossref  zmath  isi
    52. G. A. Yusupov, “Best polynomial approximations and widths of certain classes of functions in the space $L_2$”, Eurasian Math. J., 4:3 (2013), 120–126  mathnet
    53. M. R. Langarshoev, “Tochnye neravenstva tipa Dzheksona–Stechkina i znacheniya poperechnikov nekotorykh klassov funktsii v prostranstve $L_{2}$”, Model. i analiz inform. sistem, 20:5 (2013), 90–105  mathnet
    54. Shabozov M.Sh. Vakarchuk S.B. Zabutnaya V.I., “Sharp Jackson-Stechkin Type Inequalities for Periodic Functions in l-2 and Widths of Function Classes”, Dokl. Math., 88:1 (2013), 478–481  crossref  isi
    55. Yurii Kolomoitsev, Jürgen Prestin, “Sharp estimates of approximation of periodic functions in Hölder spaces”, Journal of Approximation Theory, 2015  crossref
    56. Kolomoitsev Yu., “on Moduli of Smoothness and Averaged Differences of Fractional Order”, Fract. Calc. Appl. Anal., 20:4 (2017), 988–1009  crossref  mathscinet  zmath  isi  scopus
    57. 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  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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