$H^2$ - norms of the partial sums of Fourier series by Laguerre basis for bounded holomorphic functions
Abstract
UDC 517.5
We compute the upper bounds for $H^2$-norms of the partial sums of Fourier series with respect to the Laguerre basis on the unit ball in the space of bounded holomorphic functions on the unit disk.
We give an application of the main result to the solving of some extremal problems of the theory of approximation of holomorphic functions.
References
Y. W. Lee, Synthesis of electric networks by means of the Fourier transforms of Laguerre functions, J. Math. and Phys., 11, № 1 – 4, 83 – 113 (1932).
E. Hille, Bilinear formulas in the theory of the transformation of Laplace, Compos. Math., 6, 93 – 102 (1939).
P. R. Masani, Norbert Wiener, 1894 – 1964, Birkh¨auser, Basel etc. (1990), https://doi.org/10.1007/978-3-0348-9252-0 DOI: https://doi.org/10.1007/978-3-0348-9252-0
B. G. S. Doman, The classical orthogonal polynomials, World Sci., Singapore (2016). DOI: https://doi.org/10.1142/9700
D. J. Newman, H. S. Shapiro, The Taylor coefficients of inner functions, Michigan. Math. J., 9, 249 – 255 (1962). DOI: https://doi.org/10.1307/mmj/1028998724
V. V. Savchuk, Najkrashhi linijni metodi nablizhennya ta optimal`ni ortonormovani sistemi prostoru Gardi, Ukr. mat. zhurn., 60, № 5, 661 – 671 (2008).
V. V. Savchuk, Najkrashhi linijni metodi nablizhennya garmonichnikh funkczij, Nelinijni kolivannya, 11, № 2, 242 – 251 (2008).
V. V. Savchuk, S. O. Chajchenko, Najkrashhi nablizhennya yadra Koshi na dijsnij osi, Ukr. mat. zhurn., 66, № 1, 1540 – 1549 (2014).
V. V. Savchuk, Najkrashhi nablizhennya yadra Koshi – Seg`o v seredn`omu na odinichnomu koli, Ukr. mat. zhurn., 70, № 5, 708 – 714 (2018).
A. Pinkus, $n$--Widths in approximation theory, Springer-Verlag, Berlin (1985), https://doi.org/10.1007/978-3-642-69894-1 DOI: https://doi.org/10.1007/978-3-642-69894-1
S. Ya. Khavinson, Teoriya e`kstremal`ny`kh zadach dlya ogranichenny`kh analiticheskikh funkczij, udovletvoryayushhikh dopolnitel`ny`m usloviyam vnutri oblasti, Uspekhi mat. nauk., 18, № 2, 25 – 98 (1963).
E. Landau, D. Gaier, Darstellung und Bergundung eininger neurer Ergebnisse der Functionentheorie, Springer-Verlag, Berlin (1986), https://doi.org/10.1007/978-3-642-71438-2 DOI: https://doi.org/10.1007/978-3-642-71438-2
S. Ya. Khavinson, Oczenka summ Tejlora ogranichenny`kh analiticheskikh funkczij v kruge, Dokl. AN SSSR, 80, № 3, 333 – 336 (1951).
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