Problems of unconditional convergence (Q2678419)

Let \(C_V\) denote the class of functions on \([0,1]\) having derivatives of bounded variation on \([0,1]\) and \(A\) the space of absolutely continuous functions with norm \(\Vert f\Vert_A=\Vert f\Vert_C+\int_0^1 \vert f'(x)\vert \, dx\). Fix an orthonormal system \(\{\varphi_n\}\) for \(L^2[0,1]\) and define \(C_n(f)=\int_0^1 f(x)\varphi_n(x)\, dx\). Let \(A_n(\epsilon)=\log (n+2)(\log (\log(n+2))^{\frac{1+\epsilon}{2}}\), \(Q_n(\epsilon, b,x)=\sum_{k=1}^\infty b_k A_k(\epsilon) \varphi_n(x)\) and \(H_n(\epsilon,b)=\max_{1\leq i\leq n} \bigl\vert \int_0^{i/n} Q_n(\epsilon, b,x)\, dx \bigr\vert \). \par The following problem is considered: Find conditions on ONS \(\{\varphi_n\}\) under which the Fourier series of any \(f\in C_V\) converges unconditionally a.e. on \([0,1]\). This property is known to hold for the classical Fourier basis (\(\{e^{2\pi i nx}\}\)), the Walsh system, and the Haar wavelet system on \([0,1]\). It is proved that if for some \(\epsilon>0\), \(\sum_{n=1}^\infty C_n^2(\text{{1}}) A_n^2(\epsilon)<\infty\) and if for any \((b_n)\in \ell^2\) and some \(\epsilon>0\), \(H_n(\epsilon,b)=O(1)\), then for any \(f\in C_V\), one has \(\sum_{n=1}^\infty C_n^2(f) A_n^2(\epsilon)<\infty\).\par It was shown by \textit{W. Orlicz} [Bulletin Acad. Polonaise (A) 1927, 81--115 (1927; JFM 53.0265.05)] that if for some \(\epsilon>0\), \(\sum_{n=1}^\infty a_n^2 A_n^2(\epsilon)<\infty\) then the series \(\sum_{n=1}^\infty a_n\varphi_n(x)\) converges unconditionally. Consequently, if \(\{\varphi_n\}\) is as in the main result, the series \(\sum_{n=1}^\infty C_n(f)\, \varphi_n(x)\) converges unconditionally a.e. on \([0,1]\) for any \(f\in C_V\). It is also shown that this result is sharp in the sense that if for some \((a_n)\in \ell^2\) and any \(\epsilon>0\), \(\lim\sup H_n(\epsilon, a)=+\infty\) then there is a \(g\in C_V\) such that, for any \(\epsilon>0\), \(\sum_{n=1}^\infty C_n^2(g) A_n^2(\epsilon)=+\infty\). Corresponding results are proved for subsystems. Specifically, it is shown that any ONS in \(L^2[0,1]\) possesses a subsystem whose subsystem expansions of any \(f\in C_V\) converges unconditionally a.e. Finally, it is considered whether the condition \(H_n(\epsilon,b)=O(1)\) can be readily verified. It is proved that the condition holds provided the ONS has the property that \(\int_0^x \varphi_n(x)\, dx=O(1/n)\) uniformly in \([0,1]\). The condition is verified separately for the Haar system on \([0,1]\).