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A generalization of the Riesz-Fischer theorem and linear summability methods. (English) Zbl 1248.42004

Summary: We extend the classical Riesz-Fischer theorem to biorthogonal systems of functions in Orlicz spaces: from a given double series (not necessarily convergent but satisfying a growth condition) we construct a function (in a given Orlicz space) by a linear summation method, and recover the original double series via the coefficients of the expansion of this function with respect to the biorthogonal system. We give sufficient conditions for the regularity of some linear summation methods for double series. We are inspired by a result of G. A. Fomin [Mat. Zametki 12, 365–372 (1972; Zbl 0247.42012); translation in Math. Notes 12(1972), 651–655 (1973; Zbl 0255.42020)] who extended the Riesz-Fischer theorem to \(L^{p}\) spaces.

MSC:

42A24 Summability and absolute summability of Fourier and trigonometric series
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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[1] Abilov, V. A.; Kerimov, M. K., Sharp estimates for the convergence rate of double Fourier series in terms of orthogonal polynomials in the space \(L_2((a, b) \times(c, d); p(x) q(y))\), Zh. Vychisl. Mat. Mat. Fiz., 49, 8, 1364-1368 (2009) · Zbl 1199.42047
[2] Andrienko, V. A., Summability of double orthogonal series by Riesz methods, Dokl. Akad. Nauk Ukr. SSR Ser. A, 1989, 2, 3-5 (1989) · Zbl 0698.40005
[3] Andrienko, V. A., Rate of approximation by rectangular partial sums of double orthogonal series., J. Anal. Math., 22, 4, 243-266 (1996) · Zbl 0880.41028
[4] Andrienko, V. A.; Kovalenko, L. G., Rate of Cesàro summability of double orthogonal series, J. Anal. Math., 30, 1, 1-31 (2004) · Zbl 1082.42023
[5] Andrienko, V. A.; Kovalenko, L. G., On the rate of \((C, \alpha < 0, 0)\)-almost everywhere summability of double orthogonal series, Math. Student, 23, 1, 68-81 (2005) · Zbl 1061.40007
[6] Chen, Chang-Pao, Weighted integrability and \(L^1\)-convergence of multiple trigonometric series, Studia Math., 108, 2, 177-190 (1994) · Zbl 0821.42007
[7] Cheong, Hyeong-Bin, Application of double Fourier series to the shallow water equations on a sphere, J. Comput. Phys., 165, 1, 261-287 (2000) · Zbl 0986.76064
[8] Cheong, Hyeong-Bin, Double Fourier series on a sphere: applications to elliptic and vorticity equations, J. Comput. Phys., 157, 1, 327-349 (2000) · Zbl 0961.76062
[9] du Bois-Reymond, P., A new theory of convergence and divergence of series with positive terms, Crelle J., 76, 61-91 (1873), (Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern) · JFM 05.0128.01
[10] Fomin, G. A., A generalization of the Riesz-Fischer theorem, Mat. Zametki, 12, 4, 365-372 (1972) · Zbl 0247.42012
[11] Getsadze, Rostom, On the convergence in measure of Nörlund logarithmic means of multiple orthogonal Fourier series, East J. Approx., 11, 3, 237-256 (2005) · Zbl 1143.42012
[12] Getsadze, Rostom, Divergence in measure of rearranged multiple orthogonal Fourier series, Real Anal. Exchange, 34, 2, 501-519 (2009) · Zbl 1183.42007
[13] Hardy, G. H., Divergent Series (1991), Chelsea: Chelsea NY · Zbl 0897.01044
[14] Kantawala, Pragna S., On the summability of double orthonormal series, J. Indian Acad. Math., 24, 2, 273-281 (2002) · Zbl 1046.40008
[15] (Khavin, V. P.; Nikol’skij, N. K.; Gamkrelidze, R. V., Commutative Harmonic Analysis IV: Harmonic Analysis in \(R^n\). Transl from the Russian by J. Peetre. Commutative Harmonic Analysis IV: Harmonic Analysis in \(R^n\). Transl from the Russian by J. Peetre, Encyclopaedia of Mathematical Sciences, vol. 42 (1992), Springer-Verlag), 228 · Zbl 0741.00029
[16] Krasnosel’skiĭ, M. A.; Rutickiĭ, Ya. B., Convex functions and Orlicz spaces (1961), P. Noordhoff LTD. · Zbl 0095.09103
[17] Mazhar, S. M., A generalization of Riesz-Fischer theorem, Arch. Math. (Brno), 14, 51-54 (1978) · Zbl 0403.42022
[18] Móricz, F., On the restricted convergence and (C,1,1)-summability of double orthogonal series, Acta Sci. Math., 49, 221-233 (1985) · Zbl 0598.42026
[19] Móricz, F.; Szalay, I., Absolute summability of double orthogonal series, Acta Sci. Math., 52, 3-4, 349-371 (1988) · Zbl 0668.42013
[20] Móricz, F.; Tandori, K., On the a.e. divergence of the arithmetic means of double orthogonal series, Studia Math., 82, 271-294 (1985) · Zbl 0603.42024
[21] Patel, C. M.; Patel, R. K., Absolute Cesàro summability of double orthogonal series, Nanta Math., 9, 4-9 (1976) · Zbl 0371.40004
[22] Ramis, J.-P., (Séries Divergentes et Théories Asymptotiques. Séries Divergentes et Théories Asymptotiques, Panoramas et Synthèses, vol. 121 (1993), SMF)
[23] Rhoades, B. E., Matrix transformations of some doubly orthogonal series, Indian J. Math., 34, 2, 159-166 (1992) · Zbl 0790.40004
[24] Szalay, I., On the generalized absolute Cesàro summability of double orthogonal series, Acta Sci. Math., 48, 451-458 (1985) · Zbl 0592.40007
[25] Szalay, I., On the strong Cesàro summability of double orthogonal expansions, J. Math. Anal. Appl., 144, 1, 52-74 (1989) · Zbl 0781.42029
[26] Zygmund, A., Trigonometric Series. Volumes I and II Combined (1988), Cambridge University Press · Zbl 0628.42001
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