Tempered distributions with spectral gaps

1989 ◽  
Vol 106 (1) ◽  
pp. 143-162 ◽  
Author(s):  
Jean-Pierre Gabardo
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
The Other ◽  
Convolution Operators ◽  
Spectral Gaps ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Other Hand

AbstractA tempered distribution on ℝ whose Fourier transform is supported in an interval [−Ω,Ω], where Ω>0, can be characterized by the behaviour of its successive derivatives. On the other hand, a tempered distribution on ℝ whose Fourier transform vanishes in an interval (−Ω,Ω), where Ω>0, can be characterized by the behaviour of a particular sequence of successive antiderivatives. Similar considerations apply to general convolution operators acting on J′(ℝn) and yield characterizations for tempered distributions having their Fourier transforms supported in sets of the form or , where and Ω>0.

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.

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

ORIGINATIONS AND EXTINCTIONS: ANALYSES OF FOSSIL DATA AND SIMULATIONS

2000 ◽  
Vol 11 (02) ◽  
pp. 277-285 ◽  
Author(s):  
TANE RAY ◽  
LEO MOSELEY ◽  
NAEEM JAN
Keyword(s):  
Fourier Transform ◽  
Frequency Dependence ◽  
Frequency Spectrum ◽  
Fossil Record ◽  
Trophic Levels ◽  
The Other ◽  
Simulation Data ◽  
The Past ◽  
Other Hand ◽  
Fossil Data

We analyse the fossil data of Benton1 with and without interpolation schemes. By Fourier transform analysis, we find a frequency dependence of the amplitude of 1/f for the various interpolation schemes used in the past. We illustrate that shuffling the interpolated data changes the spectra only slightly. On the other hand, an identical analysis performed on the raw (uninterpolated) fossil data gives a flat frequency spectrum. We conclude that the 1/f behavior is an artifact of the interpolation schemes. We next introduce a simulation of extinctions driven only by interactions between two trophic levels. Fourier transform analysis of the simulation data shows a frequency dependence of 1/f. When the data are grouped into a form resembling the fossil record the frequency dependence vanishes, giving a flat spectrum. Our simulation produces a frequency spectrum that agrees with the observed fossil record.

The support of tempered distributions

2008 ◽  
Vol 144 (2) ◽  
pp. 495-498 ◽  
Author(s):  
Colin C. Graham
Keyword(s):  
Fourier Transform ◽  
Polynomial Growth ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Schwartz Functions ◽  
Technical Lemma ◽  
Approximate Identities ◽  
The Fourier Transform

AbstractWe identify the support of a tempered distribution by evaluation of a sequence of test functions against the Fourier transform of the distribution. This improves previous results by removing the restriction that the distribution's Fourier transform be in $L^1_{loc}$ and be of polynomial growth. We use an apparently new technical lemma that implies that certain bounded approximate identities for $L^1(\R^n)$ are also topological approximate identities for elements of the space $\Sl$ of Schwartz functions.

Tempered Distributions Supported on a Half-Space of RN and Their Fourier Transforms

1991 ◽  
Vol 43 (1) ◽  
pp. 61-88
Author(s):  
Jean-Pierre Gabardo
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Half Space ◽  
Fourier Transforms ◽  
Fundamental Problem ◽  
Half Line ◽  
Tempered Distributions ◽  
Difficult Question ◽  
Wiener Theorem ◽  
Special Cases

A fundamental problem in Fourier analysis is to characterize the behaviour of a function (or distribution) whose Fourier transform vanishes in some particular set. Of course, this is, in general, a very difficult question and little seems to be known, except in some special cases. For example, a theorem of Paley-Wiener (Theorem XII in [6]) characterizes exactly the behaviour of the modulus of a function in L2(R) whose Fourier transform vanishes on a half-line.

scholarly journals ORBITS OF SEMIGROUPS OF TRUNCATED CONVOLUTION OPERATORS

2012 ◽  
Vol 54 (2) ◽  
pp. 399-414 ◽  
Author(s):  
STANISLAV SHKARIN
Keyword(s):  
Convolution Operator ◽  
The Other ◽  
Convolution Operators ◽  
Other Hand

AbstractWe prove that a semigroup generated by finitely many truncated convolution operators on Lp[0, 1] with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Amplitude and Phase in the MüLler-Lyer Illusion

Perception ◽  
10.1068/p2916 ◽  
2000 ◽  
Vol 29 (2) ◽  
pp. 201-209 ◽  
Author(s):  
Bernt C Skottun
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Low Frequency ◽  
Phase Spectrum ◽  
The Other ◽  
Frequency Filtering ◽  
Lyer Illusion ◽  
Amplitude Spectra ◽  
Low Pass

It has previously been claimed that the Müller-Lyer illusion is the result of low-pass spatial filtering. One way to understand this would be that the distribution of amplitudes is what generates this illusion. This possibility was investigated by computing the 2-D Fourier transforms of the two Müller-Lyer stimuli and extracting their phase and amplitude spectra. These spectra were combined to create hybrid spectra having the phase of one Müller-Lyer figure and the amplitudes of the other. Images were then created by computing the inverse Fourier transform of the hybrid spectra. Except in cases where the analysis was performed patchwise on very small patches, the figures generated with the phase spectrum of the stimuli having outward-pointing fins appear the longer. This was also the case when stimuli were generated with flat amplitude spectra. Because they show that the Müller-Lyer illusion does not depend on any particular distribution of amplitudes, these demonstrations do not support the theory that the Müller-Lyer illusion is the result of low-frequency filtering.

scholarly journals Four Particular Cases of the Fourier Transform

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 335 ◽  
Author(s):  
Jens Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform ◽  
Poisson’S Summation Formula

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

UN-Sector and Compositeness Conditions in the Bronzan-Lee Model

1974 ◽  
Vol 29 (11) ◽  
pp. 1531-1542
Author(s):  
A. L. Choudhury
Keyword(s):  
Fourier Transform ◽  
Composite Particle ◽  
Composite Model ◽  
The Other ◽  
Final States ◽  
Lee Model ◽  
Other Hand ◽  
Initial States ◽  
The Fourier Transform ◽  
Limiting Values

The Lehmann-Symanzik-Zimmermann (LSZ) technique has been used to calculate all τ -functions of the UN-sector of the Bronzan-Lee model. Using the prescription of Liossatos, the ZV → 0 limit has been carried out for the fourier transform of the τ-functions in the sector. These limiting functions τ̂α,LUN are then compared with the τ̂C,αUN functions derived from a composite model, proposed by the foregoing author, where V is considered to be a composite particle. It has been found that when the so called composite V-particle does not appear in the initial and the final states, these τ-functions coincide. On the other hand, the limiting values of some τ-functions differ from those of the composite model, when such particles appear in the final or initial states.

scholarly journals On the L1-convergence of Fourier transforms

1996 ◽  
Vol 60 (3) ◽  
pp. 405-420
Author(s):  
Dᾰng Vũ Giang ◽  
Ferenc Móricz
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Inversion Formulas ◽  
Tauberian Conditions ◽  
Tauberian Type ◽  
Necessary And Sufficient

AbstractWe study cosine and sine Fourier transforms defined by F(t):= (2/π) and (t):= (2/π), where f is L1-integrable over[0, ∞]. We also assume than F are locally absolutely continuous over [0, ∞). In particular, this is the case if both f(x) and xf(x) are (L1-integrable over [0, ∞). Motivated by the inversion formulas, we consider the partial integras Sν (f, x):= and ν(f, x):= , the modified partial integrals uν (f, x):= sν(f, x) - F(ν)(sin νx)/x and ũν(f, x):= ν(f, x) + (ν) (cos νx)/x, where ν > 0. We give necessary and sufficient conditions for(L1 [0, ∞)-convergence of uν (f) and ũν (f) as well as for the L1 [0, X]-convergence of sν (f) and ν(f) to f as ν← ∞, where 0 < X < ∞ is fixed. On the other hand, in certain cases we conclude that sν(f) and ν(f) cannot belong to (L1 [0,∞). Conequently, it makes no sense to speak of their (L1 [0, ∞)-convergence as ν ← ∞.As an intermediate tool, we use the Cesàro means of Fourier transforms. Then we prove Tauberian type results and apply Sidon type inequalities in order to obtain Tauberian conditions of Hardy-Karamata kind.We extend these results to the complex Fourier transform defined by G(t):= , where g is L1- integrable over (−∞, ∞).

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