Geodesic
On the convergence of Fourier–Laplace series
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2009), pp. 3-7.

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In the present paper we prove the following theorem. For any $\varepsilon>0$ there exists a measurable set $G\subset S^3$ with measure $mes G>4\pi-\varepsilon$, such that for each $f(x)\in L^1(S^3)$ there is a function $g(x)\in L^1(S^3)$, coinciding with $f(x)$ on $G$ with the following properties. Its Fourier–Laplace series converges to $g(x)$ in metrics $L^1(S^3)$ and the inequality holds $\displaystyle\sup_N||\sum_{n=1}^N Y_n[g,(\theta, \varphi)]||_{L^1(S^3)}\ll 3||g||_{L^1(S^3)}\leq12||f||$.
Keywords: spherical harmonics
Mots-clés : Legendre polynomials, convergence of Fourier series. 
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A. A. Sargsyan. On the convergence of Fourier–Laplace series. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2009), pp. 3-7. http://geodesic.mathdoc.fr/item/UZERU_2009_1_a0/

[1] E. Stein, G. Weiss, Introduction to Fourier Analiysis in Euclidian spaces, Mir, M., 1974 (in Russian)

[2] Proc. Steklov Inst. Math., 166 (1986), 207–222 | MR | Zbl

[3] Proc. Steklov Inst. Math., 172 (1987), 295–302 | MR | Zbl

[4] M.G. Grigorian, DAN SSSR, 315:3 (1990), 265–266 (in Russian)

[5] A. Bonami, J. Clerc, “Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques”, Trans. Amer. Math. Soc., 183 (1973), 223–263 | MR | Zbl

[6] Russian Math. (Iz. VUZ), 36:2 (1992), 17–23 | MR | MR | Zbl

[7] M.G. Grigorian, S.A. Episkoposian, “On universal trigonometric series in weighted spaces $L_{\mu} ^1$”, East journal on approximations, 5:4 (1999), 483–492 | MR | Zbl

[8] D.E. Menshoff, “Sur la convergence uniforme des séries de Fourier”, Mat. Sbornik, 53:2 (1942), 67–96 (in Russian)

[9] D. E. Men'shov, “On Fourier series of summable functions”, Tr. Mosk. Mat. Obs., 1, GITTL, Moscow–Leningrad, 1952, 5–38 (in Russian) | MR | Zbl

[10] Math. Notes, 33:5 (1983), 368–372 | DOI | MR | Zbl

[11] F.G. Arutunian, DAN Arm SSR, 64:4 (1976), 205–209 (in Russian)

[12] M.G. Grigorian, “On the representation of functions by orthogonal series in weighted spaces”, Studia. Math., 134:3 (1999), 207–216 | MR | Zbl