On the rate of convergence of integrals of Gauss–Weierstrass type for functions of several variables

A one-parameter class of summability methods for multiple Fourier series and Fourier integrals is considered. This class includes the Abel–Poisson method and the Gauss–Weierstrass method. These methods are used to investigate the rate of summability of Fourier series and integrals of differentiable functions. As corollaries, criteria are obtained for harmonicity and polyharmonicity of functions in given domains of a multidimensional Euclidean space. For example, a criterion is obtained for harmonicity and polyharmonicity of a polynomial in $N$ variables. Moreover, the rate of convergence in the $L_p$-metric is studied for singular integrals of the class under discussion for functions in the Nikol'skii class $H_p^\alpha$ ($\alpha>0$, $1\leqslant p\leqslant\infty$).
Bibliography: 14 titles.