On linear summation methods of Fourier-Laplace series. II (Q1276301)

Let \(S_n:=\{(x_1,\dots,x_n)\in \mathbb R^n:x^2_1+\dots+x^2_n=1\}\), \(n\in\{3,4,\dots\}\), and \(L(S_n)\) be the space of all functions integrable on \(S_n\). The author proves a theorem on the representation of functions by singular integrals at double Lebesgue points. On the basis of this theorem, the author gives necessary and sufficient conditions for the relation \(\lim_{n\to\infty} U_n(f,x,1)=f(x)\) for an arbitrary integrable function \(f\) at its double Lebesgue point \(x\), where by \(U_n (f,x,1)\) denotes the linear means of the Fourier-Laplace series of \(f\) defined by means of the triangular matrix. [See also the review of Part I, same journal 23, No. 2, 127-148 (1997) in Zbl 0891.42007].