Resolvent of harmonic oscillator Hamiltonian and its application to Fourier transform for generalized functions

2016 ◽  
Author(s):  
S. Kuwata
Keyword(s):  
Fourier Transform ◽  
Harmonic Oscillator ◽  
Generalized Functions

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.

Infinite integrals and convolution

1980 ◽  
Vol 371 (1747) ◽  
pp. 479-508 ◽  
Keyword(s):  
Fourier Transform ◽  
Generalized Functions ◽  
Infinite Integral ◽  
Definition Of

A new definition of an infinite integral is discussed. By means of it, a convolution can be defined for generalized functions the behaviour of which at infinity is so singular as to prevent them coming within the scope of customary theories but yet are needed in applications. A Fourier transform gives products such as 8 (p) (α). 8(α) 0 as well as providing multiplication rules for many important 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.

scholarly journals Regular nonlinear generalized functions and applications

2006 ◽  
Vol 133 (31) ◽  
pp. 163-174 ◽  
Author(s):  
A. Delcroix
Keyword(s):  
Fourier Transform ◽  
Integral Operators ◽  
Generalized Functions ◽  
Regular Growth ◽  
Simplified Model ◽  
Local Analysis ◽  
Schwartz Kernel ◽  
Linear Maps ◽  
Nonlinear Generalized Functions ◽  
Kernel Theorem

We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of G8-regularity introduced by M. Oberguggenberger. As a first application we show that these new spaces are useful in a problem of representation of linear maps by integral operators, giving an analogon to Schwartz kernel theorem in the framework of nonlinear generalized functions. Secondly, we remark that these new regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to micro local analysis of singularities of generalized functions, with respect to these regularities. AMS Mathematics Subject Classification (2000): 35A18, 35A27, 42B10, 46E10, 46F30.

scholarly journals A Paradox of Unity

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Summation Formula ◽  
Generalized Functions ◽  
The Other ◽  
Complex Numbers ◽  
Dirac Delta ◽  
Max Born ◽  
Open Question

In previous studies we found that generalized functions can be smooth, discrete, periodic or discrete periodic and they can either be local or global and they are regular or generalized functions. We also saw that these properties were related to Poisson’s summation formula on one hand and to Heisenberg’s uncertainty principle on the other. In this paper, we interlink these studies and show that scalars (real or complex numbers) considered as trivial functions are discrete and periodic, local and global as well as regular and generalized, simultaneously. However, this is also a paradox because it means that Dirac’s δ and 1 (its Fourier transform) coincide. They both are unity. We show that δ and 1 coincide in the sense of scalars (real or complex numbers) but they differ in the sense of (generalized) functions. This result can moreover be related to Max Born’s principle of reciprocity. It also answers an open question in present-day quantum mechanics because it means that the Dirac delta squared is simply delta.

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