Geodesic
Criterion of square summability with geometric weight for Jacobi expansions
Dalʹnevostočnyj matematičeskij žurnal, Tome 1 (2000) no. 1, pp. 16-27.

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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, iff $f$ 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 Szegö space $AL_2(\partial{E_{\theta}})$. 
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D. B. Karp. Criterion of square summability with geometric weight for Jacobi expansions. Dalʹnevostočnyj matematičeskij žurnal, Tome 1 (2000) no. 1, pp. 16-27. http://geodesic.mathdoc.fr/item/DVMG_2000_1_1_a2/

[1] N. Aronszajn, “Theory of reprodusing kernels”, Trans. Amer. Math. Soc., 68 (1950), 337–404 | MR | Zbl

[2] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 1, Nauka, M., 1965

[3] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 2, Nauka, M., 1966 | MR

[4] D.-W. Byun, “Inversion of Hermite Semigroup”, Proc. Amer. Math. Soc., 118:2 (1993), 437–445 | MR | Zbl

[5] D. Karp, “Prostranstva analiticheskikh funktsii s gipergeometricheskimi vosproizvodyaschimi yadrami i razlozheniya po ortogonalnym polinomam”, Dalnevostochnaya 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 Räume mit Kernfunktion, Springer-Verlag, Berlin, 1962 | MR | Zbl

[8] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integraly i ryady. Spetsialnye funktsii, Nauka, M., 1983 | MR | Zbl

[9] S. Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, 189, Logman Scientific and Technical, London, 1988 | MR

[10] S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Mathematics Series, 369, Logman, Harlow, 1997 | MR

[11] V. I. Smirnov, N. A. Lebedev, Konstruktivnaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1964 | MR

[12] G. Sege, Ortogonalnye mnogochleny, GIFML, M., 1962