The Hankel transformation of Banach-space-valued generalized functions
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- by E. L. Koh and C. K. Li PDF
- Proc. Amer. Math. Soc. 119 (1993), 153-163 Request permission
Abstract:
The object of this paper is to study Banach-space-valued generalized functions belonging to $[{H_\mu }(A);B]$ for which the Hankel transformation may be defined. In Realizability theory for continuous linear systems (Academic Press, New York, 1972), Zemanian considered certain $\rho$-type testing function spaces for which the Laplace transformation is defined. Tiwari (Banach space valued distributional Mellin transform and form invariant linear filtering, Indian J. Pure Appl. Math. 20 (1989), 493-504) follows Zemanian in extending the Mellin transform. Their works are based on the denseness of the Schwartz space ${D^m}(A)$ in the testing function spaces of interest. This method is not possible here since the space ${D^m}(A)$ is not dense in ${H_\mu }(A)$, and the structure of ${H_\mu }(A)$ is quite different from that of ${D^m}(A)$, which has an inductive-limit topology. Thus, it is necessary to introduce a dense subspace ${}_\mu {D_I}(A)$ of ${H_\mu }(A)$ to derive some properties of ${H_\mu }(A)$. We then define the Hankel transformation on $[{H_\mu }(A);B]$. We end this paper with some operational formulas, which are analogous with those given by the first author in SIAM J. Math. Anal. 1 (1970), 322-327.References
- A. H. Zemanian, Generalized integral transformations, Pure and Applied Mathematics, Vol. XVIII, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. MR 0423007
- A. H. Zemanian, Realizability theory for continuous linear systems, Mathematics in Science and Engineering, Vol. 97, Academic Press, New York-London, 1972. MR 0449807
- A. K. Tiwari, Banach space valued distributional Mellin transform and form invariant linear filtering, Indian J. Pure Appl. Math. 20 (1989), no. 5, 493–504. MR 1000064
- E. L. Koh, The Hankel transformation of negative order for distributions of rapid growth, SIAM J. Math. Anal. 1 (1970), 322–327. MR 267395, DOI 10.1137/0501028 A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, Vol. II, McGraw-Hill, New York, 1953.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 119 (1993), 153-163
- MSC: Primary 46F12; Secondary 44A15
- DOI: https://doi.org/10.1090/S0002-9939-1993-1149972-7
- MathSciNet review: 1149972