Uniform convergence of double Fourier series for classes of functions with anisotropic smoothness (Q1360832)

The author considers the double Fourier series of functions \(f\in C(\mathbb{T}^2)\) endowed with the norm \(|f|:= \sup\{|f(x,y)|:(x, y)\in\mathbb{T}^2\}\). We recall that \[ \omega(f,\delta):= \sup\{|f(x+h, y+k)- f(x,y)|: |(h, k)|\leq\delta\} \] is called the joint modulus of continuity of \(f\), where \(|(h, k)|:=(h^2+ k^2)^{1/2}\); while the partial moduli of continuity are defined by \[ \begin{aligned}\omega_1(f,\delta) &:=\sup\{|f(x+h,y)- f(x,y)|:|h|\leq\delta\},\\ \omega_2(f,\delta) &:= \sup\{|f(x,y+k)- f(x,y)|:|k|\leq\delta\}.\end{aligned} \] The author proves the uniform \(u\)-convergence of the double Fourier series of \(f\) under certain conditions expressed in terms of \(\omega\), \(\omega_1\), and \(\omega_2\). We note that a numerical double series \(\sum a_{mn}\) is called \(u\)-convergent to \(\alpha\) if for every \(\varepsilon>0\) there exists \(N\) such that for any \(U\subset A_3\) satisfying the condition \(\{(m,n)\in Z^2:|(m, n)|\leq N\}\subseteq U\) we have \[ \Biggl|\sum_{(m,n)\in U} a_{mn}-\alpha\Biggr|< \varepsilon. \] Here \(U\subset A_3\) means that if \((k_1,k_2)\in U\), then all the integers of the parallelepiped \([-|k_1|,|k_1|]\times [-|k_2|,|k_2|]\) are also contained in \(U\).