Summability of Fourier-Laplace series with the method of lacunary arithmetical means at Lebesgue points

Let Sigma (n-1) be the unit sphere in the n-dimensional Euclidean space R-n . For a function f is an element of L(Sigma (n-1)) denote by sigma (delta)( N)(f) the Cesaro means of order delta of the Fourier-Laplace series of f. T he special value lambda := n-2/2 of delta is known as the critical index. I n the case when n is even, this paper proves the existence of the `rare' se quence {nk} such that the summability 1\N Sigma (N)(K=1) sigma (lambda)(nk) (f)(x) --> f(x), N --> infinity takes place at each Lebesgue point satisfying some antipole conditions.