Summability of Fourier-Laplace Series with the Method of Lacunary Arithmetical Means at Lebesgue Points

Abstract

Let ∑ _n−1 be the unit sphere in the n-dimensional Euclidean space ℝⁿ. For a funcion ƒ∈L(∑ _n−1 ) denote by $\sigma ^{\delta }_{N} {\left( f \right)}$ the Cesàro means of order δ of the Fourier-Laplace series of ƒ. The special value $\lambda : = \frac{{n - 2}} {2}$ of δ is known as the critical index. In the case when n is even, this paper proves the existence of the ‘rare’ sequence {n _k } such that the summability

$$ \begin{array}{*{20}c} {{\frac{1} {N}{\sum\limits_{k = 1}^N {\sigma ^{\lambda }_{{n_{k} }} {\left( f \right)}{\left( x \right)}} } \to f{\left( x \right)},}} & {{N \to \infty }} \\ \end{array} $$

takes place at each Lebesgue point satisfying some antipole conditions.