Identities for the Hankel transform and their applications (original) (raw)
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It is well known that the Hankel transform possesses neither a shift-modulation nor a convolution-multiplication rule, both of which have found many uses when used with other integral transforms. In this paper, the generalized shift operator, as defined by Levitan, is applied to the Hankel transform. It is shown that under this generalized definition of shift, both convolution and shift theorems now apply to the Hankel transform. The operation of a generalized shift is compared to that of a simple shift via example.
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In the present paper the authors consider several new integral transforms including the F S,2 -transform, the F C,2 -transform and the P 4 -transform as generalizations of the classical Fourier sine transform, the classical Fourier cosine transform, and the classical Stieltjes transform, respectively. Many identities involving these transforms are given. By making use of these identities, a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some example are also given as illustrations of the results presented here.
Applied Mathematics and Computation, 2008
In the present paper the authors consider several new integral transforms including the F S,2-transform, the F C,2-transform and the P 4-transform as generalizations of the classical Fourier sine transform, the classical Fourier cosine transform, and the classical Stieltjes transform, respectively. Many identities involving these transforms are given. By making use of these identities, a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some example are also given as illustrations of the results presented here.
Numerical evaluation of Hankel transforms via Gaussian-Laguerre polynomial expansions
IEEE Transactions on Acoustics, Speech, and Signal Processing
Hankel transforms of individual members of the orthonormal set of Gaussian-Laguerre (G-L) functions yield the same functional form as the original members, Thus, the Hankel transform of arbitrary functions can be accomplihed in principle by expanding the function into G-L functions and using the same expansion coefficients for calld a t i n g the Hankel transformation. Numerical evaluation for the &culm function illustrates the convergence features of this type of expansion.