The Complex Hankel and<i>I</i>-Transformations of Generalized Functions

Koh, E. L.; Zemanian, A. H.

doi:10.1137/0116076

Cited by 55 publications

(27 citation statements)

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“…In this paper we define the distributional F transform by using the kernel method inspired in the ideas of Dube and Pandey [9] and Koh and Zemanian [11]. We obtain a representation of the elements of the corresponding dual spaces that allows us to get an explicit form for the distributional F transform.…”

Section: Introductionmentioning

confidence: 99%

Distributional Chébli–Trimèche transforms

Méndez

2006

Journal of Mathematical Analysis and Applications

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In this paper we investigate the distributional Chébli-Trimèche transforms. We use the so-called kernel method and we are inspired by the papers of Dube and Pandey [L.S. Dube, J.N. Pandey, On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975) 337-354] and Koh and Zemanian [E.L. Koh, A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968) 945-957] about distributional Hankel transforms.We note that our procedure, supported in a representation of the elements in the corresponding dual spaces, is simpler than the methods described in the above mentioned papers. Some applications of our distributional theory are presented.  2005 Elsevier Inc. All rights reserved.

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Section: Introductionmentioning

confidence: 99%

Distributional Chébli–Trimèche transforms

Méndez

2006

Journal of Mathematical Analysis and Applications

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“…[7] and [21]) neither one seems very suitable for our use. His definition in [7] does not admit a generalized Hankel transform for the function t n (n > 0) and even though his definition in [21] admits such a transform, it does not lead directly to the results obtained by Wong [16]. Therefore, we shall construct a new testingfunction space that will enable us to define a generalized Hankel transform for We list some properties of the space S,.…”

Section: Then For Sufficiently Large S We Havementioning

confidence: 99%

Asymptotic Expansions of Some Integral Transforms by Using Generalized Functions

Zayed¹

1982

Transactions of the American Mathematical Society

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ABSTRACT. The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong's technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the K and the Hankel transforms.

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“…It is proved in [1] that any generalized function that has a Hankel transform according to (2) will also have a Hankel transform according to (1), and the two transforms will agree.…”

Section: Introductionmentioning

confidence: 99%

“…The generalized functions in the dual H^ of Hß act like distributions of slow growth as x -> oo . Moreover, H^ is the domain of the generalized Hankel transformation hß , which is defined via (1). It follows that hß is an automorphism on H'ß .…”

Section: Introductionmentioning

confidence: 99%

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The Hankel transformation on 𝑀’_{𝜇} and its representation

Koh¹,

Li²

1994

Proc. Amer. Math. Soc.

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Abstract. The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval's equation as (1) (h"f, 9) = (/, K where Jov.it is the testing function space which contains the kernel function, yf^yJit(xy) ■ A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for feJp,(hitf)(y) = (f(x),Vxyjfl(xy)).In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space Ma,ii which contains the kernel function and show that Hf, c Ma,p¡ C Ja,it ■ We then form the countable union space M^ = \J%X May,n whose dual M^ has J^ as a subspace. Our main result is an inversion theorem stated as follows.

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The Complex Hankel andI-Transformations of Generalized Functions

Cited by 55 publications

References 2 publications

Distributional Chébli–Trimèche transforms

Distributional Chébli–Trimèche transforms

Asymptotic Expansions of Some Integral Transforms by Using Generalized Functions

The Hankel transformation on 𝑀’_{𝜇} and its representation

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