Ukrainian Mathematical Bulletin
Volume 3, Issue 1, 2006 pp. 41-62.
Strong Spherical Means and Convergence of Multiple Trigonometric Series in $L$Authors: Olga I. Kuznetsova
Author institution: Institute of Applied Mathematics and Mechanics of NAS of Ukraine 74 R. Luxemburg Str., 83114, Donetsk, Ukraine kuznets@iamm.ac.donetsk.ua
Summary: For any $p \ge 1$, we find an exact order of growth for norms of $p$-strong means of spherical partial Fourier sums in the space of measurable a.e. bounded functions on the $m$-dimensional torus $T^m = [- \pi, \pi)^m$ for $m \ge 3$. We obtain estimates, which can not be improved, for integral norms of linear means of spherical Dirichlet kernels in terms of the coefficients of these means (Sidon's inequality type). For multiple trigonometric series with coefficients that have a radial symmetry, we obtain conditions for the considered series to be Fourier series, and also conditions that are necessary and sufficient for convergence of these series on spheres in $L (T^m)$. In particular, we prove a criterion for the series $$ \sum_{k \in Z^m} \frac {b ( |k| )}{|k|^\alpha} e^{i k x} $$ to converge on spheres in $L (T^m)$, where $\alpha > 0, b (t)$ is a function that is slowly changing in Zigmund's sense.
Contents Full-Text PDF