Abstract
Letf∈L p(ℝn),n≥2, be a radial function and letS Rf be the spherical partial sums operator. We prove that if\(\frac{{2n}}{{n + 1}}< p< \frac{{2n}}{{n - 1}}\) thenS Rf(x)→f(x) a.e. asR→∞. The result is false for\(p< \frac{{2n}}{{n + 1}}\) and\(p > \frac{{2n}}{{n + 1}}\).
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Prestini, E. Almost everywhere convergence of the spherical partial sums for radial functions. Monatshefte für Mathematik 105, 207–216 (1988). https://doi.org/10.1007/BF01636929
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DOI: https://doi.org/10.1007/BF01636929