On the Equiconvergence of the Fourier Series and Integral of Distributions


We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.

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Rakhimov, A. (2015) On the Equiconvergence of the Fourier Series and Integral of Distributions. Journal of Applied Mathematics and Physics, 3, 1361-1366. doi: 10.4236/jamp.2015.311163.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Alimov, Sh.A., Il’in, V.A. and Nikishin, E.M. (1977) Problems of Convergence of Multiple Trigonometric Series and Spectral Decompositions. Russian Mathematical Surveys, 32, 115-139.
[2] Bloshanskii, I.L. and Grafov, D.A. (2014) Equiconvergence of Expansions in Multiple Trigonometric Fourier Series and Fourier Integral with Jk-lacunary Sequences of Rectangular Partial Sums. Acta et commentationes Universittatis Tartuensis de Mathematica, 18, 69-80.
[3] Grafov, D.A. (2015) Equiconvergence of Expansions into Triple Trigonometric Series and Fourier Integral for Continuous Functions with a Certain Modulus of Continuity. Moscow University Mathematics Bulletin, 70, 24-32.
[4] Denisov, S.A. (1998) Equiconvergence of a Spectral Expansion Corresponding to a Schrodinger Operator with Summable Potential, with Fourier Integral. Differential Equations, 34, 1043-1048.
[5] Sadovnichaya, V. (2010) Equiconvergence Theorems for Sturm-Lioville Operators with Singular Potentials (Rate of Equiconvergence in -Norm). Eurasian Mathematical Journal, 1, 137-146.
[6] Sadovnichaya, V. (2010) Equiconvergence of Eigenfunction Expansions for Sturm-Liouville Operators with a Distributional Potential. Sbornik: Mathematics, 201, 61-76.
[7] Marcokova, M. (1995) Equiconvergence of Two Fourier Series. Journal of Approximation Theory, 80, 151-163.
[8] Alimov, Sh.A. (1993) On the Spectral Decompositions of Distributions. Doklady Mathematics, 331, 661-662.
[9] Alimov, Sh.A. and Rakhimov, A.A. (1996) Localization of Spectral Expansions of Distributions. Difference Equations, 32, 798-802.
[10] Alimov, Sh.A. and Rakhimov, A.A. (1997) Localization of Spectral Expansions of Distributions in a Closed Domain. Difference Equations, 33, 80-82.
[11] Rakhimov, A.A. (2000) On the Localization of Multiple Trigonometric Series of Distributions. Dokl. Math., 62, 163-165. (translation from Dokl. Akad. Nauk, Ross. Akad. Nauk, 374, 20-22).
[12] Rakhimov, A.A. (1996) Localization Conditions for Spectral Decompositions Related to Elliptic Operators from Class Ar. Mathematical Notes, 59, 298-302. (translation from Mat. Zametki, 59, 421-427).
[13] Rakhimov, A., Ahmedov, A. and Zainuddin, H. (2012) On the Spectral Expansions of Distributions Connected with Schrodinger Operator. Applied Mathematics Letters, 25, 921-924.
[14] Rakhimov, A.A. (1996) Spectral Decompositions of Distributions from Negative Sobolev Classes. Difference Equations, 32, 1011-1013.
[15] Bochner, S. (1936) Summation of Multiple Fourier Series by Spherical Means. Transactions of the American Mathematical Society, 40, 175-207.
[16] Stein, E.M. (1958) Localization and Summability of Multiple Fourier Series. Acta Mathematica, 100, 93-147.
[17] Levitan, B.M. (1954) On the Eigenfunction Expansions of the Laplace Operator. Matematicheskii Sbornik, 35, 267-316.
[18] Il’in, V.A. (1995) Spectral Theory of Differential Operators: Self-Adjoint Differential Operators. Consultants Bureau, New York.
[19] Bastis, A.Y. (1983) On the Asymptotic of the Riesz-Bochner Kernel. Analysis Mathematica, 9, 247-258.

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