Back
 JAMP  Vol.3 No.11 , November 2015
On the Equiconvergence of the Fourier Series and Integral of Distributions
Abstract: We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.
Cite this paper: 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.
References

[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.
http://dx.doi.org/10.3103/S0027132215010052

[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.
http://dx.doi.org/10.1006/jath.1995.1012

[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.
https://zbmath.org/?q=an:0890.47031

[15]   Bochner, S. (1936) Summation of Multiple Fourier Series by Spherical Means. Transactions of the American Mathematical Society, 40, 175-207.
http://dx.doi.org/10.1090/S0002-9947-1936-1501870-1

[16]   Stein, E.M. (1958) Localization and Summability of Multiple Fourier Series. Acta Mathematica, 100, 93-147.
http://dx.doi.org/10.1007/BF02559603

[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.
http://dx.doi.org/10.1007/BF01910305

 
 
Top