The capacity of sets of divergence of certain Taylor series on the unit circle
For an analytic function \(f(z)=\sum_{k=0}^{\infty}a_kz^k\) in the unit disk \(U=\{z\in\mathbb C:|z|<1\}\), \(\sum_{k=0}^{\infty}k^{\alpha}|a_k|^2<\infty\), \(\alpha\in(0,1]\), the author studies the divergence sets of the Taylor series for \(f\) on \(\partial U\). Let \(K\) be a non-negative, even and unbounded integrable function on \(\mathbb R\) that is decreasing on \((0,\infty)\). The convolution \(K*F\) of \(K\) with any \(F\in L^p(\mathbb R)\), \(p>1\), is defined by \[ K*F(x)=\int_{\mathbb R}K(t)F(x-t)dt. \] Define \(T(K,p,E)\), \(E\subset\mathbb R\), to be the set of \(F\in L^p(\mathbb R)\), \(F\geq0\), such that \(K*F(x)\geq1\) for every \(x\in E\). The \(C_{K,p}\) capacity of \(E\) is given by \[ C_{K,p}(E)=\inf\left\{\int_{\mathbb R}F^p(x)dx:F\in T(K,p,E)\right\}. \] Note that \(K*F\) is finite \(C_{K,p}\)-\textit{quasieverywhere}, which means that \(K*F<\infty\) outside a set \(E\) with \(C_{K,p}(E)=0\). The notation \(C_{\alpha,p}\) is used instead of \(C_{K,p}\) when \(K=G_{\alpha}\), \(0<\alpha\leq1\), where the Bessel kernels \(G_{\alpha}\) decay exponentially as \(|x|\to\infty\), and \(G_{\alpha}(x)|x|^{1-\alpha}\) for \(0<\alpha<1\) and \(G_1(x)/\log(1/|x|)\) are separated from 0 and \(\infty\) for \(|x|>0\) small enough. The class \(\mathcal H_{\beta}^p\), \(0<\beta<1\), \(p>1\), consists of the functions \[ f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{F(t)dt}{(1-ze^{-it})^{1-\beta}},\;\;\;F\in L^p(-\pi,\pi),\;\;\;z\in U. \] The author proves the following theorems. \par Theorem 1. Let \(f\in\mathcal H_{\beta}^p\), \(0<\beta<1\), \(p>1\), and \(f(z)=\sum_{k=0}^{\infty}a_kz^k\), \(z\in U\). Then \(\sum_{k=0}^{\infty}a_ke^{ikx}\) converges \(C_{\beta,p}\)-quasieverywhere on \([-\pi,\pi]\) to \(\frac{1}{2\pi}\int_{\pi}^{\pi}\frac{F(t)dt}{(1-e^{-i(t-x)})^{1-\beta}}.\) \par Theorem 2. Let \((\lambda_k)_0^{\infty}\) be a positive convex sequence, \(\lambda_k\to0\), \(\sum\lambda_k=\infty\), and such that the sequence \((\gamma_k)_1^{\infty}\), \(\gamma_k=k(\lambda_k-\lambda_{k+1})\), \(k\geq1\), decreases to 0 as \(k\to\infty\), and \[ K_{\lambda}(x)=\int_1^{\pi/|x|}t(\lambda(t)-\lambda(t+1))dt,\;\;\;0<|x|<\pi, \] where \(\lambda(t)\) is interpolated linearly between \(\lambda_k=\lambda(k)\) and \(\lambda_{k+1}=\lambda(k+1)\). Define \(g(z)=\sum_{k=0}^{\infty}\gamma_kz^k\) for \(z\in U\) and set \[ f(z)=\sum_{k=0}^{\infty}a_kz^k=\int_{-\pi}^{\pi}g(ze^{-it})F(t)dt,\;\;\;z=re^{ix}\in U, \] where \(F\in L^p(-\pi,\pi)\) is extended by periodicity to \([-3\pi,3\pi]\) and \(F\equiv0\) in \(\mathbb R\setminus[-3\pi,3\pi]\). Then \(\sum_{k=0}^{\infty}a_ke^{ikx}\) converges \(C_{K_{\lambda},p}\)-quasieverywhere on \([-\pi,\pi]\) to \(\int_{-\pi}^{\pi}g(e^{-i(t-x)})F(t)dt\).
scientific article; zbMATH DE number 7075282
scientific article