Convergence of termwise derived Fourier-Jacobi series and expansions in the Faber-Schauder system (Q1326058)

A classical result of Sidon for cosine series \(a_ 0/2+ \sum_{k=1}^ \infty a_ k \cos kt\) asserts that if \(a_ k\to 0\), \(k\to\infty\), and \(\sum_{k=1}^ \infty | a_ k- a_{k-1}| \log(k+ 1)<\infty\) then the cosine series is the Fourier series of some \(L_ 1\) function. In this interesting paper, the author considers extensions and generalizations of this for expansions in Jacobi polynomials. As a corollary, the author deduces results for Fourier series, for example, if \(a_ k\to 0\), \(k\to \infty\), and \(\sum_{k=1}^ \infty | a_ k- a_{k-1}| k<\infty\), then the cosine series is the Fourier series of some function \(f\) differentiable in \((0, \pi]\), and moreover, the term by term differentiated cosine series converges to \(f'(x)\) uniformly on \([\delta, \pi]\), for every \(\delta>0\). Let \(P_ n^{( \alpha, \beta)} (x)\) denote the orthonormal polynomial for the Jacobi weight \((1-x)^ \alpha (1+ x)^ \beta\) and let \(\lambda_ n:=a_ n/ P_ n^{(\alpha, \beta)} (1)\). Let \(\alpha>-1\) and \(\sum_{k=1}^ \infty |\lambda_ k- \lambda_{k-1}| k^{\alpha+ 3/2} <\infty\). The author shows that \(\sum_{k=0}^ \infty a_ k P_ k^{(\alpha, \beta)} (x)\) is the Jacobi expansion of some function continuously differentiable on \((-1,1)\). The author also presents related results for Faber-Schauder expansions.