Uniform convergence of Fourier--Jacobi series.
Let \(\rho^{(\alpha,\beta)}(x):= (1- x)^\alpha(1+ x)^\beta\) be the Jacobi weight, where \(\alpha,\beta>-1\), and \(\{P^{(\alpha, \beta)}_k(x): k= 0,1,\dots\}\) the corresponding orthonormal system on the interval \([-1,1]\). If a function \(f\) is such that \(f\rho^{(\alpha,\beta)}\in L^1[-1,1]\), then ist Fourier-Jacobi series is defined by \[ \sum^\infty_{k=0} a^{(\alpha,\beta)}_k(f) P^{(\alpha,\beta)}_k(x),\;a^{(\alpha,\beta)}_k(f):= \int^1_{-1} f(x) P^{(\alpha,\beta)}_k(x) \rho^{(\alpha,\beta)}(x)\,dx.\tag{\(*\)} \] Assume that \(f\) is continuous on \([-1,1]\). Denote by \(\omega(\delta)\) ist modulus of continuity and by \(v(n)\), \(n= 1,2,\dots\), its modulus of variation (the latter one was introduced by \textit{Z. A. Chanturiya} [Sov. Math., Dokl. 15, 67--71 (1974); translation from Dokl. Akad. Nauk SSSR 214, 63--66 (1974; Zbl 0295.26008)]). Among others, the following theorem is proved: Assume that the series \((*)\) converges at the endpoints \(x=\pm1\). Then the series \((*)\) converges uniformly on \([-1,1]\) if and only if \[ \lim_{n\to\infty}\, \min_{1\leq\ell\leq n}\, \Biggl\{\omega\Biggl({1\over n}\Biggr) \sum^\ell_{k=1} {1\over k}+ \sum^{n-1}_{k= \ell+1} {v(k)\over k^2}\Biggr\}= 0. \]
scientific article; zbMATH DE number 1891492
scientific article