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The Hankel transformation of Banach-space-valued generalized functions

1993; American Mathematical Society; Volume: 119; Issue: 1 Linguagem: Inglês

10.1090/s0002-9939-1993-1149972-7

1088-6826

E. L. Koh, Chunxiu Li,

Mathematical functions and polynomials

The object of this paper is to study Banach-space-valued generalized functions belonging to [ H μ ( A ) ; B ] [{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 ρ \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 ) {D^m}(A) in the testing function spaces of interest. This method is not possible here since the space D m ( A ) {D^m}(A) is not dense in H μ ( A ) {H_\mu }(A) , and the structure of H μ ( A ) {H_\mu }(A) is quite different from that of D m ( A ) {D^m}(A) , which has an inductive-limit topology. Thus, it is necessary to introduce a dense subspace μ D I ( A ) {}_\mu {D_I}(A) of H μ ( A ) {H_\mu }(A) to derive some properties of H μ ( A ) {H_\mu }(A) . We then define the Hankel transformation on [ H μ ( A ) ; B ] [{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.