The continuous -Jacobi transform and its inverse when is a positive integer

Authors:
G. G. Walter and A. I. Zayed

Journal:
Trans. Amer. Math. Soc. **305** (1988), 653-664

MSC:
Primary 44A15; Secondary 33A65

DOI:
https://doi.org/10.1090/S0002-9947-1988-0924774-5

MathSciNet review:
924774

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Abstract | References | Similar Articles | Additional Information

Abstract: The continuous -Jacobi transform is introduced as an extension of the discrete Jacobi transform by replacing the polynomial kernel by a continuous one. An inverse transform is found for both the standard and a modified normalization and applied to a version of the sampling theorem. An orthogonal system forming a basis for the range is shown to have some unusual properties, and is used to obtain the inverse.

**[1]**P. Butzer, R. Stens and M. Wehrens,*The continuous Legendre transform, its inverse transform and applications*, Internat. J. Math. Math. Sci.**3**(1980), 47-67. MR**576629 (81h:44002)****[2]**L. Campbell,*A comparison of the sampling theorems of Kramer and Whittaker*, J. Soc. Indust. Appl. Math.**12**(1964), 117-130. MR**0164173 (29:1472)****[3]**E. Deeba and E. Koh,*The continuous Jacobi transform*, Internat. J. Math. Math. Sci.**6**(1983), 145-160. MR**689452 (84h:44009)****[4]**A. Erdélyi et al.,*Higher transcendental functions*, Vol. 1, McGraw-Hill, New York, 1953.**[5]**A. Jerri,*On the application of some interpolating functions in physics*, J. Res. Nat. Bur. Standards Sect. B Math. Sci.**73B**(1969), 241-245. MR**0256026 (41:686)****[6]**-,*Sampling expansion for the*-*Laguerre integral transform*, J. Res. Nat. Bur. Standards Sect. B Math. Sci.**80B**(1976), 415-418.**[7]**H. Kramer,*A generalized sampling theorem*, J. Math. Phys.**38**(1959), 68-72. MR**0103786 (21:2550)****[8]**T. H. Koornwinder,*Jacobi functions and analysis on noncompact semisimple Lie groups*, Special Functions: Group Theoretical Aspects and Applications (eds., Askey, Koornwinder and Schempp), Reidel, Dordrecht, 1984, pp. 1-85. MR**774055 (86m:33018)****[9]**G. Szegö,*Orthogonal polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1974.**[10]**G. Walter,*A finite continuous Gegenbaur transform and its inverse*(to appear).**[11]**A. Zemanian,*Generalized integral transformations*, Wiley, New York, 1968. MR**0423007 (54:10991)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0924774-5

Keywords:
Jacobi functions,
inverse transform,
Shannon sampling theorem

Article copyright:
© Copyright 1988
American Mathematical Society