A note on the Fourier transform of generalized functions

1971 ◽  
Vol 70 (1) ◽  
pp. 49-51
Author(s):  
B. Fisher
Keyword(s):  
Fourier Transform ◽  
Generalized Functions ◽  
Generalized Function ◽  
Convolution Product ◽  
Counter Example ◽  
The Fourier Transform ◽  
Image Position

If F(f) denotes the Fourier transform of a generalized function f and f * g denotes the convolution product of two generalized functions f and g then it is known that under certain conditionsJones (2) states that this is not true in general and gives as a counter-example the case when f = g = H, H denoting Heaviside's function. In this caseand the product (x−1 – iπδ)2 is not defined in his development of the product of generalized functions.

scholarly journals On the Duality of Regular and Local Functions

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
Tempered Distributions ◽  
Regular Functions ◽  
Local Functions ◽  
The Fourier Transform

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

scholarly journals On the Duality of Regular and Local Functions

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
Tempered Distributions ◽  
Regular Functions ◽  
Local Functions ◽  
The Fourier Transform

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

On generalized functions

1982 ◽  
Vol 19 (A) ◽  
pp. 139-156 ◽  
Author(s):  
B. C. Rennie
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Exponential Function ◽  
Generalized Functions ◽  
Simple Theory ◽  
New Space ◽  
The Fourier Transform

There are described in the literature many spaces of what are variously described as generalized functions, distributions, or improper functions. This article introduces another. The new space is like that of M. J. Lighthill in containing the Fourier transform of every element and in having a particularly simple theory of trigonometric and Fourier series; also it is constructed in a somewhat similar way. The new space breaks away from the tradition of every element being, for some n, the nth derivative of an ordinary function, and, for example, the exponential function and its Fourier transform are in the space.

On generalized functions

1982 ◽  
Vol 19 (A) ◽  
pp. 139-156 ◽  
Author(s):  
B. C. Rennie
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Exponential Function ◽  
Generalized Functions ◽  
Simple Theory ◽  
New Space ◽  
The Fourier Transform

There are described in the literature many spaces of what are variously described as generalized functions, distributions, or improper functions. This article introduces another. The new space is like that of M. J. Lighthill in containing the Fourier transform of every element and in having a particularly simple theory of trigonometric and Fourier series; also it is constructed in a somewhat similar way. The new space breaks away from the tradition of every element being, for some n, the nth derivative of an ordinary function, and, for example, the exponential function and its Fourier transform are in the space.

Efficient variations of the Fourier transform in applications to option pricing

2014 ◽  
Vol 18 (2) ◽  
pp. 57-90 ◽  
Author(s):  
Svetlana Boyarchenko ◽  
Sergei Levendorski˘ı
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
The Fourier Transform

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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