$$L_{p}$$ 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.