An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces

1966 ◽  
Vol 165 (4) ◽  
pp. 297-308 ◽  
Author(s):  
Sigurdur Helgason
Keyword(s):  
Fourier Transform ◽  
Symmetric Spaces ◽  
Wiener Theorem ◽  
The Fourier Transform

A weighted version of the Paley–Wiener theorem

1989 ◽  
Vol 105 (2) ◽  
pp. 389-395 ◽  
Author(s):  
T. G. Genchev
Keyword(s):  
Fourier Transform ◽  
Exponential Type ◽  
Entire Functions ◽  
Weighted Version ◽  
Cauchy Theorem ◽  
Wiener Theorem ◽  
Functions Of Exponential Type ◽  
The Fourier Transform

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

scholarly journals Growth properties of the Fourier transform

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 755-760 ◽  
Author(s):  
William Bray ◽  
Mark Pinsky
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Symmetric Spaces ◽  
Bessel Functions ◽  
Spherical Means ◽  
Compact Type ◽  
Growth Property ◽  
Growth Properties ◽  
Rank One ◽  
The Fourier Transform

In a recent paper by the authors, growth properties of the Fourier transform on Euclidean space and the Helgason Fourier transform on rank one symmetric spaces of non-compact type were proved and expressed in terms of a modulus of continuity based on spherical means. The methodology employed first proved the result on Euclidean space and then, via a comparison estimate for spherical functions on rank one symmetric spaces to those on Euclidean space, we obtained the results on symmetric spaces. In this note, an analytically simple, yet overlooked refinement of our estimates for spherical Bessel functions is presented which provides significant improvement in the growth property estimates.

scholarly journals Fourier transforms of spherical distributions on compact symmetric spaces

2011 ◽  
Vol 109 (1) ◽  
pp. 93 ◽  
Author(s):  
Gestur Ólafsson ◽  
Henrik Schlichtkrull
Keyword(s):  
Fourier Transform ◽  
Symmetric Spaces ◽  
Fourier Transforms ◽  
Type Theorem ◽  
Compact Type ◽  
Smooth Functions ◽  
Invariant Distributions ◽  
Compact Symmetric Spaces ◽  
The Fourier Transform ◽  
Small Support

In our previous articles [27] and [28] we studied Fourier series on a symmetric space $M=U/K$ of the compact type. In particular, we proved a Paley-Wiener type theorem for the smooth functions on $M$, which have sufficiently small support and are $K$-invariant, respectively $K$-finite. In this article we extend these results to $K$-invariant distributions on $M$. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. Finally, we characterize the singular support of a distribution in terms of its Fourier transform, and we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator. The extension from symmetric spaces of compact type to all compact symmetric spaces is sketched in an appendix.

scholarly journals Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group: مبرهنات بالي _ وينر لتحويل فورييه اعتماداً على زمرة هايزنبرغ

2020 ◽  
Vol 4 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Fourier Transforms ◽  
Mathematical Concepts ◽  
Weyl Transform ◽  
Wiener Theorem ◽  
Group Fourier Transform ◽  
The Fourier Transform ◽  
Compactly Supported Functions

  In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we discuss the representation theory of this group and the relationship between the representation theory of the Heisenberg group and the position and momentum operators and momentum operators relationship between the representation theory of the Heisenberg group and the position and momentum that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schrodinger picture. That is, all the representations we considered are realized in the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator-valued function, and other facts and properties. In our research, we depended on new formulas for some mathematical concepts such as Fourier Transform and Weyl transform. The main aim of our research is to introduce the Paley_ Wiener theorem for the Fourier transform on the Heisenberg group. We will show that the classical Paley_ Wiener theorem for the Euclidean Fourier transform characterizes compactly supported functions in terms of the behaviour of their Fourier transforms and Weyl transform. And we are interested in establishing results for the group Fourier transform and the Weyl transform.

scholarly journals Complete transformation of Radon-Kipriyanov. Some properties

2019 ◽  
Vol 489 (2) ◽  
pp. 125-130
Author(s):  
L. N. Lyakhov ◽  
M. G. Lapshina ◽  
S. A. Roshchupkin
Keyword(s):  
Fourier Transform ◽  
Differential Operator ◽  
Differential Operators ◽  
Support Theorem ◽  
Singular Differential Operator ◽  
Wiener Theorem ◽  
Bessel Transform ◽  
Bessel Transforms ◽  
The Fourier Transform ◽  
Complete Transformation

The even Radon-Kipriyanov transform (Kg-transform) is suitable for investigating problems with the Bessel singular differential operator Bi = 2i2+iii,i 0. In this paper, we introduce the odd Radon-Kipriyanov transform and complete Radon-Kipriyanov transform to investigation more general equations containing odd B‑derivativesiBik, k = 0, 1, 2, ... (in particular, gradients of functions). Formulas of K-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B. M. Levitan and the odd Bessel transform introduced by I. A. Kipriyanov and V. V. Katrakhov, a connection was obtained between the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform. An analogue of Helgasons support theorem and an analogue of the Paley-Wiener theorem are presented.

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

Determination of starch crystallinity with the Fourier-transform terahertz spectrometer

2021 ◽  
Vol 262 ◽  
pp. 117928
Author(s):  
Shusaku Nakajima ◽  
Shuhei Horiuchi ◽  
Akifumi Ikehata ◽  
Yuichi Ogawa
Keyword(s):  
Fourier Transform ◽  
The Fourier Transform ◽  
Starch Crystallinity
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