## Abstract and Applied Analysis

### On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions

#### Abstract

We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classical ${L}_{p}$ spaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establish sharp conditions for generalized localization in the class of finitely supported distributions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 649848, 13 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174064

Digital Object Identifier
doi:10.1155/2012/649848

Mathematical Reviews number (MathSciNet)
MR2955035

Zentralblatt MATH identifier
1250.42034

#### Citation

Ashurov, Ravshan; Butaev, Almaz; Pradhan, Biswajeet. On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 649848, 13 pages. doi:10.1155/2012/649848. https://projecteuclid.org/euclid.aaa/1365174064

#### References

• V. Ii'in, “On a generalized interpretation of the principle of localization for Fourier series with respect to fundamental systems of functions,” Sibirskiĭ Matematičeskiĭ Žurnal, vol. 9, no. 5, pp. 1093–1106, 1968.
• P. Sjölin, “Regularity and integrability of spherical means,” Monatshefte für Mathematik, vol. 96, no. 4, pp. 277–291, 1983.
• A. Carbery and F. Soria, “Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an ${L}^{2}$-localisation principle,” Revista Matemática Iberoamericana, vol. 4, no. 2, pp. 319–337, 1988.
• A. Carbery and F. Soria, “Pointwise Fourier inversion and localisation in ${R}^{n}$,” The Journal of Fourier Analysis and Applications, vol. 3, supplement 1, pp. 847–858, 1997.
• A. Bastis, “Generalized localization of Fourier series with respect to the eigenfunctions of the laplace operator in the classes ${L}_{p}$ classes,” Litovskiĭ Matematicheskiĭ Sbornik, vol. 31, no. 3, pp. 387–405, 1991.
• A. Bastis, “The generalized localization principle for an N-fold Fourier integral,” Doklady Akademii Nauk SSSR, vol. 278, no. 4, pp. 777–778, 1984.
• A. Bastis, “On the generalized localization principle for an N-fold Fourier integral in the classes ${L}_{p}$,” Doklady Akademii Nauk SSSR, vol. 304, no. 3, pp. 526–529, 1989.
• R. Ashurov, A. Ahmedov, and A. Rodzi b. Mahmud, “The generalized localization for multiple Fourier integrals,” Journal of Mathematical Analysis and Applications, vol. 371, no. 2, pp. 832–841, 2010.
• F. J. González Vieli and E. Seifert, “Fourier inversion of distributions supported by a hypersurface,” The Journal of Fourier Analysis and Applications, vol. 16, no. 1, pp. 34–51, 2010.
• J. Vindas and R. Estrada, “Distributional point values and convergence of Fourier series and integrals,” The Journal of Fourier Analysis and Applications, vol. 13, no. 5, pp. 551–576, 2007.
• J. Vindas and R. Estrada, “On the order of summability of the Fourier inversion formula,” Analysis in Theory and Applications, vol. 26, no. 1, pp. 13–42, 2010.
• J. Vindas and R. Estrada, “On the support of tempered distributions,” Proceedings of the Edinburgh Mathematical Society 2, vol. 53, no. 1, pp. 255–270, 2010.
• Sh. A. Alimov, “On spectral decompositions of distributions,” Doklady Akademii Nauk, vol. 331, no. 6, pp. 661–662, 1993.
• Sh. A. Alimov and A. Rakhimov, “On the localization of spectral expansions of distributions,” Journal of Differential Equations, vol. 32, no. 6, pp. 792–802, 1996.
• Sh. A. Alimov and A. Rakhimov, “On the localization of spectral expansions of distributions in a closed domain,” Journal of Differential Equations, vol. 33, no. 1, pp. 80–82, 1997.
• Y. Egorov, Linear Differential Equations of Principal Type, Consultants Bureau, New York, NY, USA, 1986.
• E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
• M. V. Fedoryuk, The Saddle-Point Method, Moscow, Russia, 1977.
• A. Garsia, Topics in Almost Everywhere Convergence, vol. 4 of Lectures in Advanced Mathematics, Markham Publishing Corporation, Chicago, Ill, USA, 1970.
• Sh. A. Alimov, R. Ashurov, and A. Pulatov, “Multiple Fourier series and Fourier integrals,” in Commutative Harmonic Analysis IV, vol. 42 of Encyclopaedia of Mathematical Sciences, pp. 1–97, Springer, 1992.