Lipschitz classes on 0-dimensional groups

1967 ◽  
Vol 63 (4) ◽  
pp. 923-928 ◽  
Author(s):  
P. L. Walker
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Torsion Group ◽  
Character Group ◽  
Lipschitz Classes ◽  
The Fourier Transform

1. Let G be a compact metric 0-dimensional Abelian group. Its dual or character group Γ is a discrete countable torsion group. We denote elements of G by x, of Γ by y, the value of the character y at x by (x, y), and the Fourier transform of f by

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

scholarly journals The conjugation operator onAq(G)

2003 ◽  
Vol 2003 (37) ◽  
pp. 2345-2347
Author(s):  
Sanjiv Kumar Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Bounded Operator ◽  
Conjugation Operator ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
Space Of Integrable Functions

Letq>2. We prove that the conjugation operatorHdoes not extend to a bounded operator on the space of integrable functions defined on any compact abelian group with the Fourier transform inlq.

scholarly journals Multipliers from spaces of test functions to amalgams

1993 ◽  
Vol 54 (1) ◽  
pp. 97-110
Author(s):  
Maria Torres De Squire
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Test Functions ◽  
Space Of Continuous Functions ◽  
Locally Compact ◽  
The Fourier Transform

AbstractIn this paper we study the space of multipliers M(r, s: p, q) from the space of test functions Φrs(G), on a locally compact abelian group G, to amalgams (Lp, lq)(G); the former includes (when r = s = ∞) the space of continuous functions with compact support and the latter are extensions of the Lp(G) spaces. We prove that the space M(∞: p) is equal to the derived space (Lp)0 defined by Figá-Talamanca and give a characterization of the Fourier transform for amalgams in terms of these spaces of multipliers.

scholarly journals The Relationship Between ϵ-Kronecker Sets and Sidon Sets

2016 ◽  
Vol 59 (3) ◽  
pp. 521-527 ◽  
Author(s):  
Kathryn Hare ◽  
L. Thomas Ramsey
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Dual Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Arithmetic Properties ◽  
The Fourier Transform ◽  
The Relationship ◽  
Discrete Abelian Group

AbstractA subset E of a discrete abelian group is called ϵ-Kronecker if all E-functions of modulus one can be approximated to within ϵ by characters. E is called a Sidon set if all bounded E-functions can be interpolated by the Fourier transform of measures on the dual group. As ϵ-Kronecker sets with ϵ < 2 possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

scholarly journals CONVOLUTION EQUATIONS ON THE ABELIAN GROUP $\cA(-1,1)$

2021 ◽  
Author(s):  
Roland Duduchava
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Invariant Measure ◽  
Arbitrary Order ◽  
Convolution Equation ◽  
Bessel Potential ◽  
Convolution Operators ◽  
The Fourier Transform ◽  
Definition Of ◽  
Convolution Equations

The interval $j=[-1,1]$ turns into an Abelian group $\cA(\cJ)$ under the group operation $x+_\cJ y:=(x+y)(1+xy)^{-1},\qquad x,y\in\cJ$. This enables definition of the invariant measure $d_\cJ x=(1-x^2)^{-1}dx$ and the Fourier transform $\cF_\cJ$ on the interval $\cJ$ and, as a consequence, we can consider Fourier convolution operators $W^0_{\cJ,\cA}:=\cF_\cJ^{-1}\cA\cF_\cJ$ on $\cJ$. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative $\fD_\cJ u(x)=-(1-x^2)u’(x)$, $t\in\cJ$. Equations are solved in the scale of Bessel potential $\bH^s_p(\cJ,d_\cJ x)$, $1\leqslant p\leqslant\infty$, and H\”older-Zygmound $\bZ^\nu(\cJ,(1-x^2)^\mu)$, $0<\mu,\nu<\infty$ spaces, adapted to the group $\cA(\cJ)$. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol $\cA(\xi)$, $\xi\in\bR$, of a convolution equation $W^0_{\cJ,\cA}u=f$ defines solvability: the equation is uniquely solvable if and only if the symbol $\cA$ is elliptic. The solution is written explicitely with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Abelian group $\cA(\cJ^n)$.

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.

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