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- W2089369575 abstract "The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval's equation as (1) ~~~~~(hpf, ( p) = (f , hurt) where (0, hpda E Hy, f E H.. Later, Koh and Zemanian defined the generalized complex Hankel transformation on .J/ = U ?=I Jav ,, where Jav ,. is the testing function space which contains the kernel function, vi~x7J0(xy). A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for f E I (2) (hpf)(y) = (f(x), V/x-yJ1(xy)). In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space Man, which contains the kernel function and show that Hu c Ma it C Jaz ,. We then form the countable union space MI, = U' 1 May,u whose dual M' has J,' as a subspace. Our main result is an inversion theorem stated as follows. Let F(y) = (hf)(y) = (f(x), V/XjJfi(xy)), f E M,#, where y is restricted to the positive real axis. Let p > I . Then, in the sense of convergence in H., r r---=,oo J F(y)uyJ1(xy)dy. This convergence gives a stronger result than the one obtained by Koh and Zemanian (1968). Secondly, we prove that every generalized function belonging to Ma ,A can be represented by a finite sum of derivatives of measurable functions. This proof is analogous to the method employed in structure theorems for Schwartz distributions (Edwards, 1965), and similar to one by Koh (1970). Received by the editors March 10, 1993. 1991 Mathematics Subject Classification. Primary 46F12, 46F10, 44A15." @default.
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- W2089369575 date "1994-04-01" @default.
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- W2089369575 title "The Hankel transformation on $M’sb mu$ and its representation" @default.
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- W2089369575 doi "https://doi.org/10.1090/s0002-9939-1994-1207539-7" @default.
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