Almost everywhere summability of orthogonal series (Q5959517)

Let \(\{\varphi_k(x): k= 0,1,\dots\}\) be an orthogonal system on some finite interval \((a,b)\), and \(\{\mu_n:n= 1,2,\dots\}\) a sequence of real numbers with the properties \[ 2< \mu_1\leq\mu_2\leq\cdots,\;\mu_n\to\infty\quad\text{and}\quad \mu_n= o(n)\quad\text{as}\quad n\to\infty.\tag{1} \] The means \(T_{\mu_n}(x)\) of the orthogonal series \[ \sum^\infty_{k=0} a_k\varphi_k(x),\quad \sum^\infty_{k=0} a^2_k< \infty,\tag{2} \] are introduced by \[ T_{\mu_n}(x):= \sum^\infty_{k=0} \exp\Biggl[-\Biggl({k\over n}\Biggr)^{\mu_n}\Biggr] a_k\varphi_k(x),\qquad n= 1,2,\dots\;. \] Among others, the following theorem is proved: If the means \(T_{\mu_n}(x)\) of series (2) converge a.e. for every sequence \(\{\mu_n\}\) satisfying (1), then series (2) converges a.e.