On the Gauss-Weierstrass summability of multiple trigonometric series at \(\mu\)-smoothness points (Q546307)

The notion of \(\mu\)-smoothness points of multi-dimensional periodic functions is introduced. A~point \(x^0\in \mathbb T^n\) is a \(\mu\)-smoothness point of a periodic function \(f\in L_1(\mathbb T^n\)) if \[ \sup_{0<r\leq \rho} \frac{1}{r^n \mu(r)} \int_{|x|\leq r} |f(x^0- x) - f(x^0)| \, dx < \infty, \] where \(0 < \rho < 1\) is a given number and the function \(\mu\) is continuous and increasing on \([0,\rho]\). The rate of convergence of the Gauss-Weierstrass means is investigated at these points.