Pointwise summability of Fourier-Laguerre series of integrable functions

TL;DR

This work analyzes pointwise approximation of functions in the weighted space $L_w$ with $w(t)=e^{-t}t^{\alpha}$, $\alpha>-1$, by the $(C,\gamma)$-means of their Fourier–Laguerre series. The authors derive an explicit $O$-bound on $|S_n^{(\gamma,\alpha)}f(0)-f(0)|$ in terms of a modulus of continuity $\omega$, assuming $\gamma>\alpha+\tfrac{1}{2}$ and certain decay conditions on $\Delta_0 f$, with a tunable rate parameter $\eta$. The proof hinges on a kernel-based decomposition of the error, together with Szegö-type Laguerre polynomial estimates and auxiliary bounds, to control each contribution and produce the stated rate. The results recover Gupta’s earlier findings as a corollary and include concrete examples (e.g., $f(t)=e^{-t/2}$ and $f(t)=t^{\delta}$) that illustrate the decay behavior, thereby advancing the Fourier–Laguerre approximation theory in weighted spaces.

Abstract

We present an approximation version of the results of D. P. Gupta [ J. of Approx. Theory, 7 (1973), 226-238] A. N. S. Singroura [Proc. Japan Acad., 39 (4) (1963), 208-210] and G. Szegö [Math. Z., 25 (1926), 87-115]. Some corollaries and examples will also be given.

Theorems & Definitions (9)

• Theorem 1
• Corollary 1
• Remark 1
• Corollary 2
• Example 1
• Example 2
• Lemma 1
• Lemma 2
• Lemma 3