Generalization of a theorem of G. Alexits. (English) Zbl 0721.42003
Let \(j_ n(\alpha,\beta,x)\) \((\alpha,\beta >-1)\) be the orthonormal system of Jacobi polynomials and let \(\sum^{\infty}_{k=0}a_ k(f)j_ k(\alpha,\beta,\cos \Theta)\) be the Jacobi-Fourier series of a function \(f(\Theta)\in L^ P(\rho)\), where
\[
\rho (\Theta)=2^{(\alpha +\beta +1)/2}(\sin \Theta /2)^{\alpha +1/2}(\cos \Theta /2)^{\beta +1/2},\quad 0<\Theta <\pi.
\]
The author considers the Fejér means \(\sigma_ n(f,\Theta)\) of the above series and gives a necessary and sufficient condition for the validity of
\[
\| \rho (\Theta)[\sigma_ n(f,\Theta)-f(\Theta)]\|_{L^ p(0,\pi)}=O(1/n),
\]
provided \(1<p<\infty\) and \(\alpha,\beta >-1/2\).
Reviewer: L.Gatteschi (Torino)
MSC:
42A10 | Trigonometric approximation |
42A50 | Conjugate functions, conjugate series, singular integrals |
42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |