The Fourier transform approach to inversion of 𝜆-cosine and funk transforms on the unit sphere

2021 ◽  
Author(s):  
B. Rubin
Keyword(s):  
Fourier Transform ◽  
Unit Sphere ◽  
The Fourier Transform

scholarly journals A UNIQUENESS RESULT FOR THE FOURIER TRANSFORM OF MEASURES ON THE SPHERE

2011 ◽  
Vol 86 (1) ◽  
pp. 78-82 ◽  
Author(s):  
FRANCISCO JAVIER GONZÁLEZ VIELI
Keyword(s):  
Fourier Transform ◽  
Bessel Function ◽  
Unit Sphere ◽  
Nonnegative Integer ◽  
Finite Measure ◽  
Uniqueness Result ◽  
Absolutely Continuous ◽  
Natural Measure ◽  
The Fourier Transform

AbstractA finite measure supported by the unit sphere 𝕊n−1 in ℝn and absolutely continuous with respect to the natural measure on 𝕊n−1 is entirely determined by the restriction of its Fourier transform to a sphere of radius r if and only 2πr is not a zero of any Bessel function Jd+(n−2)/2 with d a nonnegative integer.

The λ-cosine transforms with odd kernel and the hemispherical transform

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Fourier Transform ◽  
Unit Sphere ◽  
Borel Measure ◽  
Fractional Integrals ◽  
Inversion Method ◽  
Spherical Means ◽  
Finite Borel Measure ◽  
Basic Facts ◽  
Hemispherical Transform ◽  
The Fourier Transform

AbstractWe review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in ℝn. These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical Funk- Radon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p-functions.

The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

1986 ◽  
Vol 38 (2) ◽  
pp. 328-359 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Fourier Transforms ◽  
Surface Measure ◽  
Positive Gaussian Curvature ◽  
Smooth Measures ◽  
The Fourier Transform ◽  
Image Position

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

On the Fourier transform of SO(d)-finite measures on the unit sphere

2005 ◽  
Vol 84 (5) ◽  
pp. 470-480 ◽  
Author(s):  
Agata Bezubik ◽  
Agata Dabrowska ◽  
Aleksander Strasburger
Keyword(s):  
Fourier Transform ◽  
Unit Sphere ◽  
The Fourier Transform ◽  
Finite Measures

Efficient variations of the Fourier transform in applications to option pricing

2014 ◽  
Vol 18 (2) ◽  
pp. 57-90 ◽  
Author(s):  
Svetlana Boyarchenko ◽  
Sergei Levendorski˘ı
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
The Fourier Transform

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

scholarly journals Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

2021 ◽  
Vol 166 ◽  
pp. 102941
Author(s):  
Somnath Ghosh ◽  
R.K. Srivastava
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
The Fourier Transform

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lung-Hui Chen
Keyword(s):  
Fourier Transform ◽  
Entire Function ◽  
Convex Hull ◽  
Scattering Theory ◽  
Phase Retrieval ◽  
Indicator Function ◽  
Band Limited ◽  
The Fourier Transform ◽  
Complex Valued ◽  
Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

scholarly journals Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Angela A. Albanese ◽  
Claudio Mele
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Strong Operator ◽  
Dual Spaces ◽  
Structure Theorems ◽  
The Fourier Transform ◽  
Decreasing Functions

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

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