Absolute convergence of Fourier series in bounded complete double orthonormal systems (Q2816820)

In this short note, it is proved that for a complete uniformly bounded orthonormal system on \(\mathbb I^2=[0,1]^2\), there exists a function \(f(\mathbf x)=f(x_1,x_2)\in C(\mathbb I^2)\) of bounded Hardy variation such that \(f(x_1,0)=f(x_1,1)=f(0,x_2)=f(1,x_2)=0\) and mixed modulus of continuity NEWLINE\[NEWLINE\omega(f;\delta_1,\delta_2):=\sup_{\mathbf x,\mathbf x+\mathbf h\in \mathbb I^2,0<h_1\leq \delta_1, 0<h_2\leq \delta_2 } \left |\Delta_h(f,\mathbf x)\right | =O \left (\log^{-2}\frac{1}{\max\{\delta_1,\delta_2\}}\right ),NEWLINE\]NEWLINE where \( \mathbf{h}=(h_1,h_2)\) and \(\delta_1,\delta_2\to 0,\) for which the series of moduli of its Fourier coefficients is divergent.