Comments on "Transient analysis of energy equation of dynamical systems"

Paskusz, G. F.

doi:10.1109/13.848080

Cited by 7 publications

(3 citation statements)

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“…Problem 1. Paskusz [20] pointed out that the following proof is problematic, where H (x) is the usual Heaviside function: Since H (x) = H 2 (x), taking the distributional derivative on both sides, we have δ (x) = 2H (x) δ (x) . Hence H (x) δ (x) = 1 2 δ (x) .…”

Section: 42mentioning

confidence: 99%

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Reconstruction of the one-dimensional thick distribution theory

Yang

2020

Journal of Mathematical Analysis and Applications

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The theory of thick distributions (both in dimension 1 and in higher dimensions) was constructed in recent years [7,27]. However this theory of distributions with one thick point in dimension one is very different from that in higher dimensions. In this paper the author uses the language of asymptotic analysis to reconstruct the 1-dimensional thick distribution theory and to incorporate it into the framework of the higher-dimensional thick distribution theory. Some new concepts and interesting results appear in this paper from viewing singular functions in a different way.

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Section: 42mentioning

confidence: 99%

“…Paskusz [20] points out, it is not rigorous to say H (x) δ (x) = 1 2 δ (x) based on H (x) = H 2 (x) . Indeed, taking distributional derivative on both sides of H (x) = H 2 (x) does give us 2H (x) δ (x) = δ (x) .…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the one-dimensional thick distribution theory

Yang

2020

Journal of Mathematical Analysis and Applications

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“…Thick distributions were introduced in one variable in [12] and in several variables in [36,37,38,39]. Thick distributions have found applications in understanding problems in several areas, such as quantum field theory [5], engineering [26,34], the understanding of singularities in mathematical physics as considered in [4] or in [6], or in obtaining formulas for the regularization of multipoles [8,25] that play a fundamental role in the ideas of the late professor Stora on convergent Feyman amplitudes [24,33]. They also appear in other problems, as generalizations of Frahm formulas [16] involving discontinuous test functions [17,37].…”

Section: Introductionmentioning

confidence: 99%

The Fourier transform of thick distributions

2020

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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.

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Distributions in spaces with thick points

Yang

Estrada

2013

Journal of Mathematical Analysis and Applications

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Comments on "Transient analysis of energy equation of dynamical systems"

Cited by 7 publications

References 1 publication

Reconstruction of the one-dimensional thick distribution theory

Reconstruction of the one-dimensional thick distribution theory

The Fourier transform of thick distributions

Distributions in spaces with thick points

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