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), 653664
MSC:
Primary 44A15; Secondary 33A65
MathSciNet review:
924774
Fulltext PDF Free Access
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.
L. Butzer, R.
L. Stens, and M.
Wehrens, The continuous Legendre transform, its inverse transform,
and applications, Internat. J. Math. Math. Sci. 3
(1980), no. 1, 47–67. MR 576629
(81h:44002), http://dx.doi.org/10.1155/S016117128000004X
 [2]
L.
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.
Y. Deeba and E.
L. Koh, The continuous Jacobi transform, Internat. J. Math.
Math. Sci. 6 (1983), no. 1, 145–160. MR 689452
(84h:44009), http://dx.doi.org/10.1155/S0161171283000137
 [4]
A. Erdélyi et al., Higher transcendental functions, Vol. 1, McGrawHill, New York, 1953.
 [5]
A.
J. Jerri, On the application of some interpolating functions in
physics, J. Res. Nat. Bur. Standards Sect. B 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), 415418.
 [7]
H.
P. Kramer, A generalized sampling theorem, J. Math. and Phys.
38 (1959/60), 68–72. MR 0103786
(21 #2550)
 [8]
Tom
H. Koornwinder, Jacobi functions and analysis on noncompact
semisimple Lie groups, Special functions: group theoretical aspects
and applications, Math. Appl., 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.
H. Zemanian, Generalized integral transformations,
Interscience Publishers [John Wiley & Sons, Inc.], New
YorkLondonSydney, 1968. Pure and Applied Mathematics, Vol. XVIII. MR 0423007
(54 #10991)
 [1]
 P. Butzer, R. Stens and M. Wehrens, The continuous Legendre transform, its inverse transform and applications, Internat. J. Math. Math. Sci. 3 (1980), 4767. MR 576629 (81h:44002)
 [2]
 L. Campbell, A comparison of the sampling theorems of Kramer and Whittaker, J. Soc. Indust. Appl. Math. 12 (1964), 117130. MR 0164173 (29:1472)
 [3]
 E. Deeba and E. Koh, The continuous Jacobi transform, Internat. J. Math. Math. Sci. 6 (1983), 145160. MR 689452 (84h:44009)
 [4]
 A. Erdélyi et al., Higher transcendental functions, Vol. 1, McGrawHill, 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), 241245. MR 0256026 (41:686)
 [6]
 , Sampling expansion for the Laguerre integral transform, J. Res. Nat. Bur. Standards Sect. B Math. Sci. 80B (1976), 415418.
 [7]
 H. Kramer, A generalized sampling theorem, J. Math. Phys. 38 (1959), 6872. 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. 185. 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)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
44A15,
33A65
Retrieve articles in all journals
with MSC:
44A15,
33A65
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198809247745
PII:
S 00029947(1988)09247745
Keywords:
Jacobi functions,
inverse transform,
Shannon sampling theorem
Article copyright:
© Copyright 1988
American Mathematical Society
