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.

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

Characterization of Dini–Lipschitz functions for the Helgason Fourier transform on rank one symmetric spaces

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Salah El Ouadih ◽  
Radouan Daher
Keyword(s):  
Fourier Transform ◽  
Symmetric Spaces ◽  
Translation Operator ◽  
Lipschitz Functions ◽  
Generalized Translation Operator ◽  
Generalized Translation ◽  
Rank One

AbstractIn this paper, using a generalized translation operator, we obtain an analog of Younis’ theorem, [

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.

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.

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