The continuous (𝛼,𝛽)-Jacobi transform and its inverse when 𝛼+𝛽+1 is a positive integer

Walter, G. G.; Zayed, A. I.

doi:10.1090/S0002-9947-1988-0924774-5

The continuous $(\alpha , \beta )$-Jacobi transform and its inverse when $\alpha +\beta +1$ 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: The continuous $(\alpha , \beta )$-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.

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