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 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.

scholarly journals Four Particular Cases of the Fourier Transform

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 335 ◽  
Author(s):  
Jens Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform ◽  
Poisson’S Summation Formula

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals Four Particular Cases of the Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals Four Particular Cases of the Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

scholarly journals Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Gaussian Functions ◽  
The Fourier Transform ◽  
Definition Of ◽  
Quaternion Linear Canonical Transform

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

scholarly journals Strong boundedness of analytic functions in tubes

1979 ◽  
Vol 2 (1) ◽  
pp. 15-28
Author(s):  
Richard D. Carmichael
Keyword(s):  
Fourier Transform ◽  
Analytic Function ◽  
Analytic Functions ◽  
Weak Topology ◽  
Complex Space ◽  
Direct Proof ◽  
Tempered Distributions ◽  
Boundedness Property ◽  
Tube Domains ◽  
The Fourier Transform

Certain classes of analytic functions in tube domainsTC=ℝn+iCinn-dimensional complex space, whereCis an open connected cone inℝn, are studied. We show that the functions have a boundedness property in the strong topology of the space of tempered distributionsg′. We further give a direct proof that each analytic function attains the Fourier transform of its spectral function as distributional boundary value in the strong (and weak) topology ofg′.

scholarly journals Fourier transforms in generalized Fock spaces

1990 ◽  
Vol 13 (3) ◽  
pp. 431-441
Author(s):  
John Schmeelk
Keyword(s):  
Fourier Transform ◽  
Taylor Series ◽  
Fourier Transforms ◽  
Fock Space ◽  
Natural Generalization ◽  
Fock Spaces ◽  
Test Functions ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

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