Über die Konvergenz mehrfacher Orthogonalreihen. (On the convergence of multiple orthogonal series) (Q1187277)

Let \(N_ 2=\{\nu=(k,\ell)\mid\;k,\ell\in\mathbb{N}\}\), and let \(\{\varphi_ \nu\mid\;\nu\in N_ 2\}\) be an orthonormal system (ONS) on the interval \(I=(0,1)\), \(\varphi_ \nu\in L^ 2(I)\), \(\int_ 0^ 1 \varphi_ \nu(x)\varphi_ \mu(x)dx=\begin{cases} 1&\text{ for } \nu=\mu \\ 0 &\text{ otherwise}\end{cases}\). For \(1\leq K\leq\infty\) let \(\Omega(K)\) be those ONSs with \(| \varphi_ \nu(x)|\leq K\), \(x\in I\), \(\nu\in N_ 2\). For a sequence \(a=(a_ \nu)_{n\in N_ 2}\) the series \((1)\;\sum_{\nu\in N_ 2} a_ \nu \varphi_ \nu(x)\) is considered. The series (1) converges regularly in \(x\in I\) iff \(\sum_{\nu=\nu_ 1}^{\nu_ 2} a_ \nu \varphi_ \nu(x)\to 0\), where \(\nu=(k,\ell)\), \(k_ 1\leq k_ 2\), \(\ell_ 1\leq\ell_ 2\), and \(\max(k_ 1,\ell_ 1)\to\infty\). For \(1\leq K\leq\infty\) let \(M(K)\) be the set of sequences of coefficients such that (1) converges regularly almost everywhere in \(I\) for every ONS in \(\Omega(K)\). The main theorem of the paper states: \(M(K)=M(1)\) for \(1<K<\infty\). Whether \(M(\infty)\) equals \(M(1)\) remains an open question.