On the singularities of the continuous Jacobi transform when

Author:
Ahmed I. Zayed

Journal:
Proc. Amer. Math. Soc. **101** (1987), 67-75

MSC:
Primary 44A20; Secondary 33A70

DOI:
https://doi.org/10.1090/S0002-9939-1987-0897072-0

MathSciNet review:
897072

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let and , where is the Jacobi function of the first kind, , and . Let

In this paper, we devise a technique to continue analytically to the complex -plane and locate the singularities of by relating them to the singularities of

However, this will be done in the more general case where the limit in (*) exists in the sense of Schwartz distributions and defines a generalized function . In this case, we pass from to its analytic representation

**[1]**Hans Bremermann,*Distributions, complex variables, and Fourier transforms*, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. MR**0208364****[2]**E. Y. Deeba and E. L. Koh,*The continuous Jacobi transform*, Internat. J. Math. Math. Sci.**6**(1983), no. 1, 145–160. MR**689452**, https://doi.org/10.1155/S0161171283000137**[3]**R. P. Gilbert,*Integral operator methods in bi-axially symmetric potential theory*, Contributions to Differential Equations**2**(1963), 441–456 (1963). MR**0156998****[4]**J. Levin,*Distribution of zeros of entire functions*, Amer. Math. Soc. Transl.**5**(1972).**[5]**Zeev Nehari,*On the singularities of Legendre expansions*, J. Rational Mech. Anal.**5**(1956), 987–992. MR**0080747****[6]**G. Szegö,*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R.I., 1978.**[7]**Gilbert Walter,*On real singularities of Legendre expansions*, Proc. Amer. Math. Soc.**19**(1968), 1407–1412. MR**0257635**, https://doi.org/10.1090/S0002-9939-1968-0257635-0**[8]**Gilbert G. Walter and Ahmed I. Zayed,*On the singularities of continuous Legendre transforms*, Proc. Amer. Math. Soc.**97**(1986), no. 4, 673–681. MR**845986**, https://doi.org/10.1090/S0002-9939-1986-0845986-9**[9]**Ahmed I. Zayed,*A generalized inversion formula for the continuous Jacobi transform*, Internat. J. Math. Math. Sci.**10**(1987), no. 4, 671–692. MR**907785**, https://doi.org/10.1155/S0161171287000796**[10]**Ahmed I. Zayed,*A generalized inversion formula for the continuous Jacobi transform*, C. R. Math. Rep. Acad. Sci. Canada**8**(1986), no. 4, 271–276. MR**850114**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
44A20,
33A70

Retrieve articles in all journals with MSC: 44A20, 33A70

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0897072-0

Keywords:
Continuous Jacobi transform,
analytic continuation,
singular points

Article copyright:
© Copyright 1987
American Mathematical Society