Asymptotic Expansions of Integral Transforms of Functions with Logarithmic Singularities

1977 ◽  
Vol 8 (4) ◽  
pp. 655-672 ◽  
Author(s):  
Norman Bleistein
Keyword(s):  
Asymptotic Expansions ◽  
Integral Transforms ◽  
Logarithmic Singularities

Generalized Mellin Convolutions and their Asymptotic Expansions

1984 ◽  
Vol 36 (5) ◽  
pp. 924-960 ◽  
Author(s):  
R. Wong ◽  
J. P. Mcclure
Keyword(s):  
Asymptotic Expansions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Integral Transforms ◽  
Generalized Function ◽  
Positive Parameter ◽  
Ordinary Sense ◽  
Integrable Functions ◽  
Classical Integral ◽  
Image Position

A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form1.1where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.

Asymptotic expansions for a class of q-integral transforms

2008 ◽  
Vol 198 (1) ◽  
pp. 433-444 ◽  
Author(s):  
Néji Bettaibi ◽  
Jamel El Kamel
Keyword(s):  
Asymptotic Expansions ◽  
Integral Transforms

On integral transforms whose kernels are solutions of singular Sturm–Liouville problems

1988 ◽  
Vol 108 (3-4) ◽  
pp. 201-228 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansions ◽  
Spectral Measure ◽  
Liouville Problem ◽  
Integral Transforms ◽  
Tempered Distributions ◽  
Sturm Liouville ◽  
The Fourier Transform ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known thatwhere the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.

Sign in / Sign up

Export Citation Format

Share Document