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.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

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 Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

Singularity of Vector Valued Measures in Terms of Fourier Transform

2006 ◽  
Vol 12 (2) ◽  
pp. 213-223 ◽  
Author(s):  
Maria Roginskaya ◽  
Michal Wojciechowski
Keyword(s):  
Fourier Transform ◽  
Vector Valued

scholarly journals A Parallel Framework with Block Matrices of a Discrete Fourier Transform for Vector-Valued Discrete-Time Signals

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Pablo Soto-Quiros
Keyword(s):  
Fourier Transform ◽  
Parallel Computing ◽  
Discrete Time ◽  
Parallel Implementation ◽  
Multicore Processors ◽  
Block Matrix ◽  
Mathematical Framework ◽  
Matrix Operations ◽  
Time Signals ◽  
Vector Valued

This paper presents a parallel implementation of a kind of discrete Fourier transform (DFT): the vector-valued DFT. The vector-valued DFT is a novel tool to analyze the spectra of vector-valued discrete-time signals. This parallel implementation is developed in terms of a mathematical framework with a set of block matrix operations. These block matrix operations contribute to analysis, design, and implementation of parallel algorithms in multicore processors. In this work, an implementation and experimental investigation of the mathematical framework are performed using MATLAB with the Parallel Computing Toolbox. We found that there is advantage to use multicore processors and a parallel computing environment to minimize the high execution time. Additionally, speedup increases when the number of logical processors and length of the signal increase.

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.

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