A RIEMANN-LEBESGUE LEMMA FOR FOURIER-JACOBI COEFFICIENTS

Abstract

Riemann-Lebesgue Lemma for Fourier-Jacobi coefficients for the class of functions in \(L^p_{(\alpha,\beta)}\), \(1<p\leq\infty\), are studied.

In [3], it is proved that
\[
\lim_{k\to\infty}\widehat{f}_{(\alpha,\beta)}(k)=0
\]
holds for each \(f\in L^1_{(\alpha,\beta)}\) if and only if \(\alpha\geq\beta>-1\), \(\alpha\geq-1/2\).

For \(f\in L^p_{(\alpha,\beta)}\), \(1<p\leq\infty\), and \(\alpha>-1/2\), \(\beta>-1\) Riemann-Lebesgue Lemma is proved.