Czechoslovak Mathematical Journal, online first, 12 pp.

Pointwise Fourier inversion of distributions on spheres

Francisco Javier González Vieli

Received July 27, 2016.   First published October 6, 2017.

Francisco Javier González Vieli, Montoie 45, 1007 Lausanne, Switzerland, e-mail:

Abstract: Given a distribution $T$ on the sphere we define, in analogy to the work of Łojasiewicz, the value of $T$ at a point $\xi$ of the sphere and we show that if $T$ has the value $\tau$ at $\xi$, then the Fourier-Laplace series of $T$ at $\xi$ is Abel-summable to $\tau$.

Keywords: distribution; sphere; Fourier-Laplace series; Abel summability

Classification (MSC 2010): 42C10, 46F12

DOI: 10.21136/CMJ.2017.0403-16

Full text available as PDF.

[1] S. Axler, P. Bourdon, W. Ramey: Harmonic Function Theory. Graduate Texts in Mathematics 137, Springer, New York (2001). DOI 10.1007/b97238 | MR 1805196 | Zbl 0959.31001
[2] R. Estrada, R. P. Kanwal: Distributional boundary values of harmonic and analytic functions. J. Math. Anal. Appl. 89 (1982), 262-289. DOI 10.1016/0022-247X(82)90102-0 | MR 0672200 | Zbl 0511.31008
[3] F. J. González Vieli: Fourier inversion of distributions on the sphere. J. Korean Math. Soc. 41 (2004), 755-772. DOI 10.4134/JKMS.2004.41.4.755 | MR 2068151 | Zbl 1066.46031
[4] H. Groemer: Geometric Applications of Fourier Series and Spherical Harmonics. Encyclopedia of Mathematics and Its Applications 61, Cambridge University Press, Cambridge (1996). DOI 10.1017/CBO9780511530005 | MR 1412143 | Zbl 0877.52002
[5] S. Kostadinova, J. Vindas: Multiresolution expansions of distributions: pointwise convergence and quasiasymptotic behavior. Acta Appl. Math. 138 (2015), 115-134. DOI 10.1007/s10440-014-9959-z | MR 3365583 | Zbl 1322.42045
[6] S. Łojasiewicz: Sur la fixation des variables dans une distribution. Stud. Math. 17 (1958), 1-64. (In French.) MR 0107167 | Zbl 0086.09501
[7] E. M. Stein, G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32, Princeton University Press, Princeton (1971). DOI 10.1515/9781400883899 | MR 0304972 | Zbl 0232.42007
[8] J. Vindas, R. Estrada: Distributional point values and convergence of Fourier series and integrals. J. Fourier Anal. Appl. 13 (2007), 551-576. DOI 10.1007/s00041-006-6015-z | MR 2355012 | Zbl 1138.46030
[9] G. Walter: Pointwise convergence of distribution expansions. Stud. Math. 26 (1966), 143-154. MR 0190624 | Zbl 0144.37401
[10] G. G. Walter, X. Shen: Wavelets and Other Orthogonal Systems. Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton (2001). MR 1887929 | Zbl 1005.42018

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