Weighted inequalities for the Stieltjes transform and the maximal spherical partial sum operator on radial functions

1995 ◽  
Vol 125 (1) ◽  
pp. 195-204
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Fourier Transform ◽  
Lorentz Space ◽  
Sufficient Conditions ◽  
Weighted Inequalities ◽  
Stieltjes Transform ◽  
Radial Functions ◽  
Partial Sum ◽  
Power Weights ◽  
The Fourier Transform ◽  
Spherical Partial Sum

If TRf(x) is the spherical partial sum of the Fourier transform of f and T*f(x) = SUPR > 0 | TRf(x)|, sufficient conditions are given on the non-negative weight function ω(x) which ensure that T* restricted to radial functionsis bounded on the Lorentz space Lp,s(Rn,ω) into Lp,q(Rn, ω) For power weights, these conditions are also necessary. The weight pairs (u,v) for which the generalised Stieltjes transform Sλ is bounded from LP,S(R+, v)into Lp,q(R+, u)are also characterised. These are an essential ingredient for the study of T*.

scholarly journals The class Bp for weighted generalized Fourier transform inequalities

2015 ◽  
Vol 14 (1) ◽  
pp. 121-133
Author(s):  
Chokri Abdelkefi ◽  
Mongi Rachdi
Keyword(s):  
Fourier Transform ◽  
Dunkl Transform ◽  
Weighted Inequalities ◽  
Generalized Fourier Transform ◽  
Class B ◽  
Power Weights ◽  
Pitt’S Inequality ◽  
The Fourier Transform

Abstract In the present paper, we prove weighted inequalities for the Dunkl transform (which generalizes the Fourier transform) when the weights belong to the well-known class Bp. As application, we obtain the Pitt’s inequality for power weights.

scholarly journals On fourier transforms of radial functions

1965 ◽  
Vol 5 (3) ◽  
pp. 289-298 ◽  
Author(s):  
James L. Griffith
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Radial Functions ◽  
Cartesian Space ◽  
The Fourier Transform ◽  
Image Position

The Fourier transformF(y) of a functionf(t) inL1(Ek) whereEkis thek-dimensional cartesian space will be defined by.

Fourier Transforms of Unbounded Measures

1979 ◽  
Vol 31 (6) ◽  
pp. 1281-1292 ◽  
Author(s):  
James Stewart
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Stieltjes Transform ◽  
The Real ◽  
Bounded Complex ◽  
The Fourier Transform ◽  
Image Position

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:(1.1)More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function(1.2)where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that(1.3)for all ϕ ∈ L1(G) where

About the convergence of the Fourier transform

2021 ◽  
pp. 1-21
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Fourier Transform ◽  
Sufficient Conditions ◽  
Integral Form ◽  
Weak Form ◽  
Central Difference ◽  
Similar Theorem ◽  
Principal Objective ◽  
The Fourier Transform

The main result is the proof of the theorems, the results of which one can characterize as a weak form of the formula for the inversion of the bi-dimmensional Fourier transform. Sufficient conditions on a function are obtained for a weak (of degree $r$) convergence of bi-dimmensional Fourier transform for a function $f(x;y)$. These conditions have an integral form and describe the behavior of the function near the border of a rectangle. A similar theorem is proved, in which the Fourier transform of a function $f$ is replaced by the Fourier transform of another function $g$, the norm of the central difference of which does not exceed the norm of the central difference of $f$. The principal objective is to study the behavior of the Fourier transform of $g$ and $f$.

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.

Determination of starch crystallinity with the Fourier-transform terahertz spectrometer

2021 ◽  
Vol 262 ◽  
pp. 117928
Author(s):  
Shusaku Nakajima ◽  
Shuhei Horiuchi ◽  
Akifumi Ikehata ◽  
Yuichi Ogawa
Keyword(s):  
Fourier Transform ◽  
The Fourier Transform ◽  
Starch Crystallinity
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