scholarly journals On the Fourier transform of Bessel functions over complex numbers—II: The general case

2019 ◽  
Vol 372 (4) ◽  
pp. 2829-2854 ◽  
Author(s):  
Zhi Qi
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Complex Numbers ◽  
The Fourier Transform

scholarly journals On the Fourier Transform of Regularized Bessel Functions on Complex Numbers and Beyond Endoscopy Over Number Fields

2019 ◽  
Author(s):  
Zhi Qi
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Number Fields ◽  
Complex Numbers ◽  
Type Integral ◽  
The Fourier Transform ◽  
Totally Real ◽  
Beyond Endoscopy ◽  
Special Case ◽  
Integral Formulae

AbstractIn this article, we prove certain Weber–Schafheitlin-type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel functions on complex numbers. This is applied to extend the work of A. Venkatesh on Beyond Endoscopy for $\textrm{Sym}^2$ on $\textrm{GL}_2$ from totally real to arbitrary number fields.

The Fourier transform of 1/r

Geophysics ◽  
1986 ◽  
Vol 51 (2) ◽  
pp. 417-418
Author(s):  
Maurice Craig
Keyword(s):  
Fourier Transform ◽  
Potential Theory ◽  
Bessel Functions ◽  
Integral Formula ◽  
Fourier Integral ◽  
Elementary Functions ◽  
The Fourier Transform

Potential theory employs the Fourier integral [Formula: see text] where [Formula: see text]. Bhattacharyya (1966) obtained the value [Formula: see text] with the use of Bessel functions; the same argument has been repeated often in the geophysical literature (see Courant, 1961; Fuller, 1971; Lourenço, 1972; Gunn, 1975; Weil, 1976; Nabighian, 1984). The following alternative, short derivation involves only elementary functions and may, therefore, be of interest.

Comment on “Fourier transform of hydrogen-type atomic orbitals”

2019 ◽  
Vol 97 (12) ◽  
pp. 1349-1360 ◽  
Author(s):  
Ernst Joachim Weniger
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Fourier Transforms ◽  
Laguerre Polynomials ◽  
Bound State ◽  
Atomic Orbitals ◽  
Generalized Laguerre Polynomials ◽  
Classic Result ◽  
The Fourier Transform ◽  
Finite Sum

Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi: 10.1103/PhysRev.34.109 ) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü (Can. J. Phys. 96, 724 (2018) doi: 10.1139/cjp-2017-0728 ), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of Slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling’s formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov’s functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions.

scholarly journals On the spectrum of the distributional kernel related to the residue

2001 ◽  
Vol 27 (12) ◽  
pp. 715-723
Author(s):  
Amnuay Kananthai
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Nonnegative Integer ◽  
Dimensional Euclidean Space ◽  
Complex Numbers ◽  
Alpha Beta ◽  
The Fourier Transform

We study the spectrum of the distributional kernelKα,β(x), whereαandβare complex numbers andxis a point in the spaceℝnof then-dimensional Euclidean space. We found that for any nonzero pointξthat belongs to such a spectrum, there exists the residue of the Fourier transform(−1)kK2k,2k(ξ)ˆ, whereα=β=2k,kis a nonnegative integer andξ∈ℝn.

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 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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