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.

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.

A Central Crack in a Finite Rectangular Plate With Four Hinged Edges Under the Opening Mode

1990 ◽  
Vol 57 (4) ◽  
pp. 1079-1081
Author(s):  
S. W. Ma ◽  
Y. G. Tsuei
Keyword(s):  
Stress Intensity Factor ◽  
Integral Equation ◽  
Fourier Transform ◽  
Fourier Series ◽  
Rectangular Plate ◽  
Uniform Loading ◽  
Central Crack ◽  
Opening Mode ◽  
Loading Case ◽  
The Fourier Transform

By combining linearly the Fourier transform and Fourier series, the stress intensity factor of a central crack in a finite rectangular plate with four hinged edges under the opening mode is expressed as the Fredholm integral equation of the second kind. The uniform loading case is considered in detail. The numerical results include the predictions by Koiter and Fichter as limiting cases.

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