Far Eastern Mathematical Journal

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Criterion of square summability with geometric weight for Jacobi expansions


D. B. Karp

2000, issue 1, Ń. 16–27


Abstract
In this paper we prove that $\sum_{k=0}^{\infty}|f_k|^2\theta^k<\infty$, where $\theta>1$ and $f_k$ is the k-th Fourier coefficient of a function $f\in{L_1(-1,1;(1-x)^{\lambda}(1+x)^{\mu})}$ in orthonormal Jacobi polynomials, if $f$ can be analytically continued to the ellipse $E_{\theta}=\{z:~|z-1|+|z+1|<\theta^{\frac{1}{2}}+ \theta^{-\frac{1}{2}}\}$ and its analytic continuation belongs to the Szego? space $AL_2(\partial{E_{\theta}})$.

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