Fourier Transform in Non-Euclidean Space and its Applications to Physics

1990 ◽  
pp. 41-57
Author(s):  
M. Kovalyov ◽  
M. Legare
Keyword(s):  
Fourier Transform ◽  
Euclidean Space

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

Structure Along Arithmetic Patterns in Sequences of Vectors

2009 ◽  
Vol 18 (6) ◽  
pp. 861-870
Author(s):  
P. CANDELA
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Natural Generalization ◽  
Arithmetic Combinatorics ◽  
Vector Valued

We discuss a new direction in which the use of some methods from arithmetic combinatorics can be extended. We consider functions taking values in Euclidean space and supported on subsets of {1, 2, . . ., N}. In this context we present a proof of a natural generalization of Szemerédi's theorem. We also prove a similar generalization of a theorem of Sárkőzy using a vector-valued Fourier transform, adapting an argument of Green and obtaining effective bounds.

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 On Fourier transform multipliers in Lp

1972 ◽  
Vol 13 (2) ◽  
pp. 219-223
Author(s):  
G. O. Okikiolu
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Numbers ◽  
Measurable Functions ◽  
Image Position

We denote by R the set of real numbers, and by Rn, n ≧ 2, the Euclidean space of dimension n. Given any subset E of Rn, n ≧ 1, we denote the characteristic function of E by xE, so that XE(x) = 0 if x ∈ E; and XE(X) = 0 if x ∈ Rn/E.The space L(Rn) Lp consists of those measurable functions f on Rn such that is finite. Also, L∞ represents the space of essentially bounded measurable functions with ║f║>0; m({x: |f(x)| > x}) = O}, where m represents the Lebesgue measure on Rn The numbers p and p′ will be connected by l/p+ l/p′= 1.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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