On the unconditional almost-everywhere convergence of general orthogonal series
Consider an arbitrary measure space \((X,\mathcal{A},\mu)\) and an orthonormal sequence \((\varphi_n)_{n\in \mathbb{N}}\) in the complex Hilbert space \(L^2(\mu)\). Recall that for any sequence \((a_n)_{n\in \mathbb{N}}\) of complex numbers, the series \(\sum_{n=1}^{\infty}a_n\varphi_n\) is said to be unconditionally \(\mu\)-almost everywhere convergent if for any permutation \(\sigma\) of the natural numbers, the series \(\sum_{n=1}^{\infty}a_{\sigma(n)}\varphi_n(x)\) is convergent for \(\mu\)-almost every \(x\in X\). The authors prove the following theorem: if \(\nu_k=2^{2^k}\) and \[ \sum_{k=0}^{\infty}\Bigg(\sum_{n=\nu_k+1}^{\nu_{k+1}}|a_n|^2\log_2^2 n\Bigg)^{1/2}<\infty \] then the series \(\sum_{n=1}^{\infty}a_n\varphi_n\) is unconditionally \(\mu\)-almost everywhere convergent. This result was proved before by \textit{K. Tandori} [Acta Sci. Math. 23, 185--221 (1962; Zbl 0144.32103)] in the special case that \(X\) is a bounded open intervall, \(\mu\) is the Lebesgue measure and the \(\varphi_n\) are real-valued. Using their first result, the authors also show that \(\sum_{n=1}^{\infty}a_n\varphi_n\) is unconditionally \(\mu\)-almost everywhere convergent whenever there exists an increasing sequence \((\omega_n)_{n\in \mathbb{N}}\) of positive numbers such that \[ \sum_{n=2}^{\infty}|a_n|^2\omega_n\log_2^2 n<\infty \;\;\text{and} \;\;\sum_{n=2}^{\infty}\frac{1}{n\omega_n\log_2 n}<\infty, \] which generalises a theorem of \textit{W. Orlicz} [Bulletin Acad. Polonaise (A) 1927, 81--115 (1927; JFM 53.0265.05)] and \textit{P. L. Ul'yanov} [Mat. Sb., N. Ser. 60(102), 39--62 (1963; Zbl 0135.27501), p. 53].
unconditional almost-everywhere convergence
scientific article; zbMATH DE number 6110049
scientific article