Illinois Journal of Mathematics

On Abel summability of Jacobi polynomials series, the Watson kernel and applications

Calixto P. Calderón and Wilfredo O. Urbina

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Abstract

In this paper, we return to the study of the Watson kernel for the Abel summability of Jacobi polynomial series. These estimates have been studied for over more than 40 years. The main innovations are in the techniques used to get the estimates that allow us to handle the cases $0<\alpha$ as well as $-1<\alpha<0$, with essentially the same methods. To that effect, we use an integral superposition of Natanson kernels, and the A. P. Calderón-Kurtz, B. Muckenhoupt $A_{p}$-weight theory. We consider also a generalization of a theorem due to Zygmund in the context of Borel measures. The proofs are different from the ones given in (Sobre la conjugación y sumabilidad de series de Jacobi (1971) Universidad de Buenos Aires, Studia Math. 49 (1974) 217–224, Colloq. Math. 30 (1974) 277–288 and Illinois J. Math. 41 (1997) 237–265). We will discuss in detail the Calderón–Zygmund decomposition for nonatomic Borel measures in $\mathbb{R}$. We prove that the Jacobi measure is doubling and following (Studia Math. 57 (1976) 297–306), we study the $A_{p}$ weight theory in the context of Abel summability of Jacobi expansions. We consider power weights of the form $(1-x)^{\overline{\alpha}}$, $(1+x)^{\overline{\beta}}$, $-1<{\overline{\alpha}}<0$, $-1<{\overline{\beta}}<0$. Finally, as an application of the weight theory we obtain $L^{p}$ estimates for the maximal operator of Abel summability of Jacobi function expansions for suitable values of $p$.

Article information

Source
Illinois J. Math., Volume 57, Number 2 (2013), 343-371.

Dates
First available in Project Euclid: 19 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1408453586

Digital Object Identifier
doi:10.1215/ijm/1408453586

Mathematical Reviews number (MathSciNet)
MR3263037

Zentralblatt MATH identifier
06340281

Subjects
Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Secondary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

Citation

Calderón, Calixto P.; Urbina, Wilfredo O. On Abel summability of Jacobi polynomials series, the Watson kernel and applications. Illinois J. Math. 57 (2013), no. 2, 343--371. doi:10.1215/ijm/1408453586. https://projecteuclid.org/euclid.ijm/1408453586


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