Trigonometric analogs of the Szeg\H o equiconvergence theorem for Fourier--Jacobi series

V. M. Badkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 68-79

Résumé

Let $\{\Phi^{\alpha,\beta}_k(\tau)\}_{k=0}^\infty$ be an orthonormal system of trigonometric Jacobi polynomials obtained by orthogonalizing the sequence $1,\sin\tau,\cos\tau,\sin2\tau,\cos2\tau,\dots$ by Schmidt method on $[0,2\pi]$ with a weight $\varphi^{\alpha,\beta}(\tau):=(1-\cos\tau)^{\alpha+1/2}(1+\cos\tau)^{\beta+1/2}$; $s_n^{\alpha,\beta}(F;\theta):=\sum_{k=0}^nc_k(\varphi^{\alpha,\beta};F)\Phi^{\alpha,\beta}_k(\theta)$ is $n$-th Fourier sum of function $F$ in system $\Phi^{\alpha,\beta}_k(\tau)\}_{k=0}^\infty$; $s_n(F;\theta)=s_{2n}^{-1/2,-1/2}(F;\theta)$ is usual Fourier sum. It is proved that if $\alpha,\beta>-1$, $A:=\min\{\alpha+1/2,\alpha/2+1/4\}$, $B:=\min\{\beta+1/2,\beta/2+1/4\}$, $\varepsilon\in(0,\pi/2)$, $F$ is measurable, $F(\tau)(1-\cos\tau)^A(1+\cos\tau)^B\in L^1$ and $\varepsilon\in(0,\pi/2)$ $F\varphi^{\alpha,\beta}\in L^1$ and the sum $s_{2n}^{\alpha,\beta}(F;\theta)$ equiconverges with each of sequences $s_n(F\sqrt{\varphi^{\alpha,\beta}};\theta)/\sqrt{\varphi^{\alpha,\beta}(\theta)}$ and $s_n(F\varphi^{\alpha,\beta};\theta)/\varphi^{\alpha,\beta}(\theta)$ uniformly on intervals $[-\pi+\varepsilon,-\varepsilon]$ and $[\varepsilon,\pi-\varepsilon]$. For even function $F$ similar results were obtained by G. Szegő and Ye.  A. Pleshchyova.

Détail

Keywords: trigonometric Jacobi polynomials, Fourier sums
Mots-clés : equiconvergens.