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The strong converse inequality for de la Vallée Poussin means on the sphere. (English) Zbl 1334.41016

Summary: This paper discusses the approximation by de la Vallée Poussin means \(V_nf\) on the unit sphere. Especially, the lower bound of approximation is studied. As a main result, the strong converse inequality for the means is established. Namely, it is proved that there are constants \(C_1\) and \(C_2\) such that \(C_1\omega(f,\frac1{\sqrt{n}})_p\leq\|V_nf-f\|_p\leq C_2\omega(f,\frac1{\sqrt{n}})_p\) for any \(p\)th Lebesgue integrable or continuous function \(f\) defined on the sphere, where \(\omega(f,t)_p\) is the modulus of smoothness of \(f\).

MSC:

41A25 Rate of convergence, degree of approximation
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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