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.
\Bibitem{Osk77}
\by K.~I.~Oskolkov
\paper Approximation properties of summable functions on sets of full measure
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 4
\pages 489--514
\mathnet{http://mi.mathnet.ru/eng/sm2930}
\crossref{https://doi.org/10.1070/SM1977v032n04ABEH002403}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=473679}
\zmath{https://zbmath.org/?q=an:0371.41008}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1977GL81400007}
Linking options:
https://www.mathnet.ru/eng/sm2930
https://doi.org/10.1070/SM1977v032n04ABEH002403
https://www.mathnet.ru/eng/sm/v145/i4/p563
This publication is cited in the following 32 articles:
Per G. Nilsson, “Representation of real interpolation spaces using weighted couples of radon measures”, Anal.Math.Phys., 15:3 (2025)
V. G. Krotov, A. I. Porabkovich, “Estimates of $L^p$-Oscillations of Functions for $p>0$”, Math. Notes, 97:3 (2015), 384–395
V. I. Ovchinnikov, “Interpolation functions and the Lions–Peetre interpolation construction”, Russian Math. Surveys, 69:4 (2014), 681–741
A. A. Dmitriev, “Ob otsenke konstanty $\mathscr{K}$-delimosti v parakh banakhovykh prostranstv”, Dalnevost. matem. zhurn., 13:2 (2013), 179–191
V. G. Krotov, “Criteria for compactness in $L^p$-spaces, $p\geqslant0$”, Sb. Math., 203:7 (2012), 1045–1064
Veniamin G. Krotov, Springer Proceedings in Mathematics & Statistics, 25, Recent Advances in Harmonic Analysis and Applications, 2012, 197
Alexander M. Stokolos, Walter Trebels, Springer Proceedings in Mathematics & Statistics, 25, Recent Advances in Harmonic Analysis and Applications, 2012, 339
Krotov V.G., “Quantitative Form of the Luzin C-Property”, Ukr. Math. J., 62:3 (2010), 441–451
Luboš Pick, International Mathematical Series, 13, Around the Research of Vladimir Maz'ya III, 2010, 279
I. A. Ivanishko, V. G. Krotov, “Compactness of Embeddings of Sobolev Type on Metric Measure Spaces”, Math. Notes, 86:6 (2009), 775–788
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
Dai F., Wang K., “Convergence Rate of Spherical Harmonic Expansions of Smooth Functions”, J. Math. Anal. Appl., 348:1 (2008), 28–33
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
Girela D., Pelaez J., “On the Derivative of Infinite Blaschke Products”, Ill. J. Math., 48:1 (2004), 121–130
A. S. Romanyuk, “Best $M$-term trigonometric approximations of Besov classes of periodic functions of several variables”, Izv. Math., 67:2 (2003), 265–302
Gogatishvili A., Pick L., “Discretization and Anti-Discretization of Rearrangement-Invariant Norms”, Publ. Mat., 47:2 (2003), 311–358
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
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
V. I. Ovchinnikov, A. S. Titenkov, “A Criterion for Contiguity of Quasiconcave Functions”, Math. Notes, 70:5 (2001), 708–713
Citation in format AMSBIB
Contact us: