$L_{p}$ Boundedness of (C, 1) Means of the Generalized Series of the Second Kind for Levin–Lubinsky Weights

Abstract

Let I be a finite or infinite interval containing 0, and let $W:I\rightarrow ( 0,\infty )$. Assume that $W^{2}$ is a weight, so that we may define orthonormal polynomials $\{ p_{j}\} _{j=0}^{\infty }$ corresponding to $W^{2}$. The generalized functions of the second kind are $q_{j}( W^{2},v,x) :=H[ p_{j}Wv] ( x)$, $j\ge 0$, where H denotes the Hilbert transform, and v a bounded function on I. For $f:I\rightarrow \mathbb {R}$, let $s_{m} [ f] :=s_{m}[ W^{2},v,f]$ denote the mth partial sums of the generalized series of the second kind. We investigate boundedness in $L_{p}$ spaces of the (C, 1) means

$$\begin{aligned} \frac{1}{n}\sum _{m=1}^{n}s_{m}[ f]. \end{aligned}$$

The class of weights $W^{2}$ considered includes even and non-even exponential weights.