On fourier transform, parseval equality, and the inversion formula in idempotent analysis

2013 ◽  
Author(s):  
Antonio Avantaggiati ◽  
Paola Loreti
Keyword(s):  
Fourier Transform ◽  
Inversion Formula ◽  
Parseval Equality ◽  
Idempotent Analysis

Product of continuous fractional wave packet transforms

2019 ◽  
Vol 17 (04) ◽  
pp. 1950021
Author(s):  
Akhilesh Prasad ◽  
Praveen Kumar ◽  
Tanuj Kumar
Keyword(s):  
Fourier Transform ◽  
Wave Packet ◽  
Inversion Formula ◽  
Fractional Fourier Transform

In this paper, we investigated the fractional Fourier transform (FrFT) of the continuous fractional wave packet transform and studied some properties of continuous fractional wave packet transform. The product of continuous fractional wave packet transforms (CFrWPTs) is defined. Parseval’s relation and an inversion formula for product of CFrWPT are obtained. An example is also given.

scholarly journals On Katugampola Fourier Transform

2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Tariq O. Salim ◽  
Atta A. K. Abu Hany ◽  
Mohammed S. El-Khatib
Keyword(s):  
Fourier Transform ◽  
Inversion Formula ◽  
Convolution Theorem ◽  
The Fourier Transform

The aim of this article is to introduce a new definition for the Fourier transform. This new definition will be considered as one of the generalizations of the usual (classical) Fourier transform. We employ the new Katugampola derivative to obtain some properties of the Katugampola Fourier transform and find the relation between the Katugampola Fourier transform and the usual Fourier transform. The inversion formula and the convolution theorem for the Katugampola Fourier transform are considered.

scholarly journals From dS to AdS and back

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Charlotte Sleight ◽  
Massimo Taronna
Keyword(s):  
Perturbation Theory ◽  
Fourier Transform ◽  
Inversion Formula ◽  
Initial Conditions ◽  
General Relation ◽  
De Sitter ◽  
Partial Wave Expansion ◽  
The Fourier Transform ◽  
The Many ◽  
Dilatation Group

Abstract We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

scholarly journals The Radon Transforms on the Generalized Heisenberg Group

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Tianwu Liu ◽  
Jianxun He
Keyword(s):  
Fourier Transform ◽  
Differential Operator ◽  
Heisenberg Group ◽  
Radon Transform ◽  
Inversion Formula ◽  
The Other ◽  
Radon Transforms

Let Hna be the generalized Heisenberg group. In this paper, we study the inversion of the Radon transforms on Hna. Several kinds of inversion Radon transform formulas are established. One is obtained from the Euclidean Fourier transform; the other is derived from the differential operator with respect to the center variable t. Also by using sub-Laplacian and generalized sub-Laplacian we deduce an inversion formula of the Radon transform on Hna.

Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two

2014 ◽  
Vol 66 (3) ◽  
pp. 700-720 ◽  
Author(s):  
Jianxun He ◽  
Jinsen Xiao
Keyword(s):  
Fourier Transform ◽  
Lie Group ◽  
Radon Transform ◽  
Wavelet Transforms ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Nilpotent Lie Group ◽  
Differentiability Property ◽  
3 Dimensional ◽  
Group Fourier Transform

AbstractLet F2n;2 be the free nilpotent Lie group of step two on 2n generators, and let P denote the affine automorphism group of F2n;2. In this article the theory of continuous wavelet transformon F2n;2 associated with P is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on F2n;2 is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if n = 1, F2;2 is the 3-dimensional Heisenberg group H1, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on F2;2. This result cannot be extended to the case n ≥ 2.

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