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/