Waterman classes and triangular sums of double Fourier series (Q5948319)

The classes are families of functions that satisfy a condition of generalized bounded variation, which concept was introduced by \textit{D. Waterman} [Stud. Math. 44, 107-117 (1972; Zbl 0237.42001)]. Definition: Let \(J\) be a bounded interval, let \(f\) be defined on \(J\), and let \(\Omega(J)\) denote the set of all collections of pairwise-disjoint subintervals of \(J\). For \(\Lambda=\{\lambda_n\}_{n=1}^\infty\), a nondecreasing sequence of positive real numbers such that \(\sum_{n=1}^\infty {1\over \lambda_n}\) diverges, the \(\Lambda\)-variation of \(f\) on \(J\) is given by \[ V_\Lambda(f;J)=\sup\left\{\sum_n\;{|f(I_n)|\over \lambda_n} :\{I_n: n=1,2,\dots\}\subset\Omega(J)\right\}, \] where \(f([a,b])=:f(b)-f(a)\), and the range of the summation index is finite. For a function, \(f\), of two real variables the \(\Lambda\)-variation of \(f\) on a rectangle \(D\) is defined, following \textit{A. A. Saakyan} [Izv. Akad. Nauk. Arm. SSR, Mat. 21, No. 6, 517-529 (1986; Zbl 0614.42009)], in the following manner. For a fixed \(y_0\) and interval \(J\), \[ V_x(f(\cdot,y_0);J)=\sup \sum {|f(I_n,y_0)|\over \lambda_n}, \] where \(f([a,b],y_0)=: f(b,y_0)-f(a,y_0)\), and one defines \(V_y(f(x_0,\cdot),J)\) in similar fashion. If \(I=(a,b)\) and \(\Delta=(\alpha,\beta)\), let \(D=I\times\Delta\), let \[ f(I,\Delta)=: f(b,\beta)-f(a,\beta)- f(b,\alpha)+ f(a,\alpha), \] and let \[ V_{xy}(f,D)=\sup\big\{\underset{k,n}\sum {|f(I_n,\Delta_k)|\over \lambda_n\lambda_k}:\{I_n\}\subset I, \{\Delta_n\}\subset \Delta\big\}, \] then \[ V_\Lambda(f,D)=: V_{xy}(f,D)+V_x(f(\cdot,\alpha),I)+V_y(f(a,\cdot),\Delta). \] In case \(V_\Lambda(f,D)<+\infty\), then \(f\) has bounded \(\Lambda\)-variation, \(V_\Lambda(f,D)\) is the \(\Lambda\)-variation of \(f\) on \(D\), and \(\Lambda\text{BV}(D)\) is the set of all such functions. In particular, the author considers the functions of classes \(\Lambda^a \text{BV}(D)\), where \(a\in\mathbb R\) and \(\Lambda^a=: \{\sqrt n \ln^a(n+1)\}_{n=1}^\infty\). Definition: Let \(f\) be defined on the plane, periodic of period \(2\pi\) in each argument, and integrable on \(\mathbf T^2\), where \[ \mathbf T=(-\pi)={1\over 4}(f(x+0,y+0)+ f(x+0,y-0)+f(x-0,y+0)+f(x-0,y-0)) \] is defined and finite, is termed a \textit{regular point} of \(f\). In the present work, the author is concerned with triangular partial sums of the Fourier series \(\underset{m,n\in\mathbb Z}\sum a_{mn}(f)e^{inx}e^{iny}\) of functions, \(f\), belonging to classes of the aforementioned type, \(\Lambda^a\text{BV}(D)\). For integers \(M\) and \(N\), the corresponding triangular partial sum is given by the relation \(S_{MN}^\Delta(f;(x,y))=\underset{{|m|\over M}+{|n|\over N}\leq 1}\sum a_{mn}(f)e^{inx}e^{iny}\). Theorem 1. If \(f\in\Lambda^{-1}\text{BV}(\mathbf T^2)\), then the triangular partial sums \(S_{NN}^\Delta(f;(x,y))\) are uniformly bounded on \(\mathbf T^2\). Theorem 2. If \(f\in\Lambda^{-1-\delta}\text{BV}(\mathbf T^2)\), for some \(\delta>0\), then the sequence of triangular partial sums \(\{S_{NN}(f;(x,y))\}\) converges to \(f^*(x,y)\) at each \((x,y)\in\mathbf T^2\). As for larger values of the exponent \(a\) (in \(\Lambda^a\)) the author offers the following result. Theorem 3. For each \(\epsilon>0\), there exists an \(f\in\Lambda^\epsilon\text{BV}(\mathbf T^2)\) for which the sequence of triangular partial sums \(\{S_{NN}(f;(0,0)\}\) is not bounded.