scholarly journals The Fourier Transform on Quantum Euclidean Space

2011 ◽  
Author(s):  
Kevin Coulembier
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
The Fourier Transform

LP -bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with D∞ Type Singularities

2020 ◽  
pp. 350-359
Author(s):  
Nigina A. Soleeva
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
The Fourier Transform

Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erd¨osh and Salmhofer

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

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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