Far Eastern Mathematical Journal

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Criterion of square summability with geometric weight for Jacobi expansions

D. B. Karp

2000, issue 1, P. 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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References

[1] N. Aronszajn, “Theory of reprodusing kernels”, Trans. Amer. Math. Soc., 68 (1950), 337–404. [2] G. Be'jtmen, A. E'rdeji, Vysshie transcendentnye funkcii, t. 1, Nauka, M., 1965. [3] G. Be'jtmen, A. E'rdeji, Vysshie transcendentnye funkcii, t. 2, Nauka, M., 1966. [4] D.-W. Byun, “Inversion of Hermite Semigroup”, Proc. Amer. Math. Soc., 118:2 (1993), 437–445. [5] D. Karp, “Prostranstva analiticheskix funkcij s gipergeometricheskimi vosproizvodyashhimi yadrami i razlozheniya po ortogonal'nym polinomam”, Dal'nevostochnaya matematicheskaya shkola-seminar imeni akademika E. V. Zolotova, Vladivostok, 26 avgusta – 2 sentyabrya 1999 g., Tezisy dokladov, 1999, 42–44. [6] D. Karp, A class of Holomorphic Pontryagin Spaces and Expansions in Orthogonal polinomials, arXiv: math.CV/9908129. [7] H. Meschkowski, Hilbertsche Ra?ume mit Kernfunktion, Springer-Verlag, Berlin, 1962. [8] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady. Special'nye funkcii, Nauka, M., 1983. [9] S. Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, 189, Logman Scientific and Technical, London, 1988. [10] S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Mathematics Series, 369, Logman, Harlow, 1997. [11] V. I. Smirnov, N. A. Lebedev, Konstruktivnaya teoriya funkcij kompleksnogo peremennogo, Nauka, M., 1964. [12] G. Sege, Ortogonal'nye mnogochleny, GIFML, M., 1962.

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