\(L_p\) boundedness of \((C,1)\) means of orthonormal expansions for general exponential weights
The paper contains an extension of former investigations of the authors. Namely, for a broad class \({\mathcal F}(C^2)\) of weight functions \(W\) on a finite or infinite interval \(I\) the estimate \[ \|\sigma_n(f)W\Psi_n^{1-1/p}\|_{P_p(I)}\leq C\|fW\Psi_n^{-1/p}\|_{L_p(I)},\quad 1\leq p<\infty, \] is proved, where \(\sigma_n(f)\) denotes the Cesàro \((C,1)\) means of orthonormal expansions for the weight \(W,\) and \(\Psi_n\) is a factor which behaves up to multiplication by positive constants like \[ \left(\left|1-\frac{|x|}{a_n}\right|+n^{-2/3}\right)^{-1/3} \] for a Freud weight \(W\) (with \(a_n\) being the Mhaskar-Rakhmanov-Saff number for \(W).\) For \(p=\infty\) an analogous estimate is proved for strong means, namely means of \(|s_n(f)|,\) instead of ordinary means.
Fourier series with respect to orthogonal polynomials
scientific article; zbMATH DE number 1787698