Spherical partial sums of the~double Fourier series of functions of bounded generalized variation

M. I. Dyachenko

Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 29-60

Résumé

The spherical partial sums of the double Fourier series of functions in the Waterman classes are studied. The main result of the paper is as follows. Theorem 1. {\it Let $\Lambda_\varepsilon =\biggl\{\dfrac{n^{3/4}}{(\ln(n+1))^{1/2+\varepsilon}}\biggr\}_{n=1}^\infty$ for $\varepsilon>0$. Let $f(x,y)\in\Lambda_\varepsilon BV(T^2)$ and let \begin{align*} I_r(f)=\sup_{x,y\in T}\sup_{u,v\in[-1,1]}J_r(f) \\ =\sup_{x,y\in T}\sup_{u,v\in[-1,1]}\sum_{r-1|(m,n)|\leqslant r+1}|a_{m,n}(\psi_{x,y,u,v})|\leqslant C \end{align*} for $r\geqslant 1$, where $$ \psi _{x,y,u,v}(s,t)=\psi (s,t)=f(x+t,y+s)w(t)w(s)e^{-i(tu+sv)}, \quad and\quad w(\tau)=\frac\tau{2\sin(\theta/2)}\,. $$ Then $$ \sup_{R\geqslant 1}\sup _{(x,y)\in T^2}|S_R(f,x,y)|\leqslant C(f,\varepsilon). $$ for each $R\geqslant 1$.} Problem of circular convergence of Fourier series of the characteristic function of plane convex sets are also considered.