Approximation properties of generalized Bochner-Riesz means in the Hardy spaces $H_p$, $0$

Yu. S. Kolomoitsev

Sbornik. Mathematics, Tome 203 (2012) no. 8, pp. 1151-1168

Résumé

A test for the convergence of the generalized spherical and $\ell_1$ Bochner-Riesz means in the Hardy spaces $H_p(D^n)$, $0<p\le 1$, is obtained, where $D^n$ is the unit polydisc. Precise orders of the approximation of functions by the generalized $\ell_q$ Bochner-Riesz means in terms of the $K$-functional and special moduli of smoothness are found. Bibliography: 31 titles.

Détail

Keywords: Hardy spaces in a polydisc, generalized Bochner-Riesz means, $K$-functional, moduli of smoothness, Bernstein-type inequalities.

Bibliographie

[1] W. Rudin, Function theory in polydiscs, Benjamin, New York–Amsterdam, 1969 | MR | MR | Zbl | Zbl

[2] È. A. Storoženko, “Approximation of functions of class $H^p$, $0

\le 1$”, Math. USSR-Sb., 34:4 (1978), 527–545 | DOI | MR | Zbl | Zbl

[3] A. A. Soljanik, “On the order of approximation to functions of $H^p(\mathbb R)$ $(0

\le 1)$ by certain means of Fourier integrals”, Anal. Math., 12:1 (1986), 59–75 | DOI | MR

[4] Eh. S. Belinskij, “Strong summability of periodic functions and imbedding theorems”, Russian Acad. Sci. Dokl. Math., 48:2 (1994), 255–258 | MR | Zbl

[5] R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus”, Math. USSR-Izv., 17:3 (1981), 567–593 | DOI | MR | Zbl | Zbl

[6] O. I. Kuznetsova, R. M. Trigub, “Two-sided estimates of the approximation of functions by Riesz and Marcinkiewicz means”, Soviet Math. Dokl., 21:2 (1980), 374–377 | MR | Zbl

[7] A. N. Podkorytov, “Summation of multiple Fourier series over polyhedra”, Vestnik Leningrad Univ. Math., 13 (1981), 69–77 | MR | Zbl | Zbl

[8] R. M. Trigub, “Multipliers in the Hardy spaces $H_p(D^m)$ with $p\in (0,1]$ and approximation properties of summability methods for power series”, Sb. Math., 188:4 (1997), 621–638 | DOI | MR | Zbl

[9] Yu. S. Kolomoitsev, “O dvustoronnikh otsenkakh priblizheniya funktsii obobschennymi srednimi Bokhnera–Rissa v $H_p$, $0

\le 1$”, Tr. IPMM NAN Ukrainy, 20 (2010), 116–123

[10] S. G. Pribegin, “Approximation of functions in $H^p$, $0

\le 1$, by generalized Riesz means with fractional exponents”, Sb. Math., 197:7 (2006), 1025–1035 | DOI | MR | Zbl

[11] S. G. Pribegin, “Some summability methods for power series of functions in $H^p(D^n)$, $0

\infty$”, Sb. Math., 200:2 (2009), 243–260 | DOI | MR | Zbl

[12] S. G. Pribegin, “A method of approximation in $H^p$, $0

\le 1$”, Sb. Math., 192:11 (2001), 1705–1719 | DOI | MR | Zbl

[13] St. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, New York, NY, 1993 | MR | MR | Zbl | Zbl

[14] Ya. Valashek, “O priblizheniyakh v mnogomernykh prostrasntvakh Khardi $H^p$, $0

\le\nobreak 1$”, Soobsch. AN GSSR, 105:1 (1982), 21–24 | MR | Zbl

[15] R. M. Trigub, E. S. Bellinsky, Fourier analysis and approximation of functions, Kluwer Acad. Publ., Dordrecht, 2004 | MR | Zbl

[16] V. V. Savchuk, M. V. Savchuk, “Norms of multipliers and best approximations of holomorphic multivariable functions”, Ukrainian Math. J., 54:12 (2002), 2025–2037 | DOI | MR | Zbl

[17] C. Fefferman, E. M. Stein, “$H^p$ spaces of several variables”, Acta Math., 129:1 (1972), 137–193 | DOI | MR | Zbl

[18] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., 43, Princenton Univ. Press, Princenon, NJ, 1993 | MR | Zbl

[19] A. B. Aleksandrov, “Essays on nonlocally convex Hardy classes”, Complex analysis and spectral theory (Leningrad, 1979/1980), Lecture Notes in Math., 864, Springer-Verlag, Berlin–New York, 1981, 1–89 | DOI | MR | Zbl

[20] R. E. Edwards, Fourier series. A modern introduction, v. 2, Grad. Texts in Math., 85, Springer-Verlag, New York–Heidelberg–Berlin, 1982 | MR | MR | Zbl | Zbl

[21] S. Bochner, “Classes of holomorphic functions of several variables in circular domains”, Proc. Nat. Acad. Sci. U.S.A., 46:5 (1960), 721–723 | DOI | MR | Zbl

[22] Z. Liu, “Multipliers on real Hardy spaces”, Sci. China Ser. A, 35:1 (1992), 55–69 | MR | Zbl

[23] D. Chen, D. Fan, “Multiplier transformations on $H^p$ spaces”, Studia Math., 131:2 (1998), 189–204 | MR | Zbl

[24] J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud., 116, North Holland, Amsterdam, 1985 | MR | Zbl

[25] È. A. Storoženko, V. G. Krotov, P. Osval'd, “Direct and converse theorems of Jackson type in $L^p$ spaces, $0

1$”, Math. USSR-Sb., 27:3 (1975), 355–374 | DOI | MR | Zbl | Zbl

[26] V. I. Ivanov, “Direct and converse theorems of the theory of approximation in the metric of $L_p$ for $0

1$”, Math. Notes, 18:5 (1975), 972–982 | DOI | MR | Zbl | Zbl

[27] E. Belinskii, E. Liflyand, “Approximation properties in $L_p$, $0

1$”, Funct. Approx. Comment. Math., 22 (1993), 189–199 | MR | Zbl

[28] K. Runovski, H.-J. Schmeisser, “On some extensions of Bernstein's inequality for trigonometric polynomials”, Funct. Approx. Comment. Math., 29 (2001), 125–142 | MR

[29] E. S. Belinsky, “Strong summability for the Marcinkiewicz means in the integral metric and related questions”, J. Austral. Math. Soc. Ser. A, 65:3 (1998), 303–312 | DOI | MR | Zbl

[30] Vit. V. Volchkov, “Neravenstvo Bernshteina v prostranstvakh Khardi $H^p$, $0

1$”, Ryady Fure: teoriya i prilozheniya (Kamenets-Podolskii, 1997), In-t matem. NAN Ukrainy, Kiev, 1998, 77–84 | MR | Zbl

[31] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., 32, Princeton Univ. Press, Princeton, NJ, 1971 | MR | Zbl | Zbl