On convergence of Riesz spherical means of multiple Fourier series

An $N$-dimensional analog is proved of a theorem of Plessner and Ul'yanov on equivalent conditions for convergence of certain series and integrals. There is obtained from it a sufficient condition on the quadratic modulus of continuity of a periodic function of $N\geqslant2$ variables ensuring the a.e. convergence of the spherical sums of its Fourier series. A two-dimensional analog of a theorem of Luzin and Denjoy and an $N$-dimensional analog of the Dini–Lipschitz criterion are proved. A necessary and sufficient condition on a function $\Phi(u)$ is derived ensuring the pointwise convergence of the Riesz spherical means of critical order of multiple Fourier series of functions of bounded $\Phi$-variation.
Bibliography: 33 titles.