scholarly journals The Meijer transformation of generalized functions

1987 ◽  
Vol 10 (2) ◽  
pp. 267-286 ◽  
Author(s):  
E. L. Koh ◽  
E. Y. Deeba ◽  
M. A. Ali
Keyword(s):  
Bessel Function ◽  
Function Space ◽  
Kernel Function ◽  
Dual Space ◽  
Generalized Functions ◽  
Modified Bessel Function ◽  
Inversion Theorem ◽  
Distributional Sense

This paper extends the Meijer transformation,Mμ, given by(Mμf)(p)=2pΓ(1+μ)∫0∞f(t)(pt)μ/2Kμ(2pt)dt, wherefbelongs to an appropriate function space,μ ϵ (−1,∞)andKμis the modified Bessel function of third kind of orderμ, to certain generalized functions. A testing space is constructed so as to contain the Kernel,(pt)μ/2Kμ(2pt), of the transformation. Some properties of the kernel, function space and its dual are derived. The generalized Meijer transform,M¯μf, is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established.

scholarly journals A distributional Hardy transformation

1979 ◽  
Vol 2 (4) ◽  
pp. 693-701 ◽  
Author(s):  
R. S. Pathak ◽  
J. N. Pandey
Keyword(s):  
Function Space ◽  
Inversion Formula ◽  
Generalized Functions ◽  
Distributional Sense

The Hardy'sF-transformF(t)=∫0∞Fv(ty)yf(y)dyis extended to distributions. The corresponding inversion formulaf(x)=∫0∞Cv(tx)tF(t)dtis shown to be valid in the weak distributional sense. This is accomplished by transferring the inversion formula onto the testing function space for the generalized functions under consideration and then showing that the limiting process in the resulting formula converges with respect to the topology of the testing function space.

scholarly journals Iterated Stieltjes transform of generalized functions

1985 ◽  
Vol 8 (3) ◽  
pp. 425-432 ◽  
Author(s):  
L. S. Dube
Keyword(s):  
Generalized Functions ◽  
Stieltjes Transform ◽  
Inversion Theorem ◽  
Distributional Sense ◽  
Space Of Generalized Functions

The generalizedS2-transform of a member offof a certain space of generalized functions is defined asF(x)=〈f(t),k(x,t;ρ)〉, wherek(x,t;ρ)=∫0∞1(x+y)ρ(y+t)ρ  dy,  ρ>12,0<x<∞  and  0<t<∞.An inversion theorem for the transform is established interpreting the oonvergene in the weak distributional sense.

scholarly journals Analytic representation of the distributional finite Hankel transform

1985 ◽  
Vol 8 (2) ◽  
pp. 325-344
Author(s):  
O. P. Singh ◽  
Ram S. Pathak
Keyword(s):  
Series Representation ◽  
Hankel Transform ◽  
Generalized Functions ◽  
Analytic Representation ◽  
Inversion Theorem ◽  
Distributional Sense ◽  
Hankel Transforms ◽  
Finite Hankel Transform

Various representations of finite Hankel transforms of generalized functions are obtained. One of the representations is shown to be the limit of a certain family of regular generalized functions and this limit is interpreted as a process of truncation for the generalized functions (distributions). An inversion theorem for the gereralized finite Hankel transform is established (in the distributional sense) which gives a Fourier-Bessel series representation of generalized functions.

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

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