Distributions in spaces with thick points

Yang, Yunyun; Estrada, Ricardo

doi:10.1016/j.jmaa.2012.12.045

Cited by 15 publications

(25 citation statements)

References 21 publications

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“…The definition of D * ,a (R) is very much analogous to the defintion of D * ,a (R n ) when n ≥ 2. [27] On the other hand, the thick test functions defined in [7] is included in the above definition. Recall that a thick test function defined in [7] is a compactly supported function φ with domain R, smooth in R\ {a} , and at x = a all its onesided derivatives,…”

Section: Reconstruction Of the Space Of Test Functions Onmentioning

confidence: 99%

“…Now let us define a topology on D * ,a (R) and make it a topological vector space. Similar to [27], we denote the subspace D…”

Section: Reconstruction Of the Space Of Test Functions Onmentioning

confidence: 99%

“…Furthermore, there is a natural inclusion map i : D (R) ֒→ D * ,a (R). With the topology defined in definition 12, the space of usual test functions D (R) is a closed subspace of D * ,a (R) , similar to the case of higher dimensions [27].…”

Section: Reconstruction Of the Space Of Test Functions Onmentioning

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: Reconstruction Of the Space Of Test Functions Onmentioning

confidence: 99%

“…Now let us define a topology on D * ,a (R) and make it a topological vector space. Similar to [27], we denote the subspace D…”

Section: Reconstruction Of the Space Of Test Functions Onmentioning

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 aim of this article is to construct the Fourier transform of thick tempered distributions in several variables. 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].…”

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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“…lim n→∞ |x|≥1/n G (x) φ (x) dx ,a standard finite part if −λ / ∈ N, and a Hadamard finite part if λ is an integer. Similar ideas are needed to construct thick distributions from locally integrable functions[15].…”

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confidence: 99%

A Generalization of the Banach-Steinhaus Theorem for Finite Part Limits

Estrada

Vindas

2017

Bull. Malays. Math. Sci. Soc.

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Abstract. It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence {y n } ∞ n=1 of linear continuous functionals in a Fréchet space converges pointwise to a linear functional Y, Y (x) = lim n→∞ y n , x for all x, then Y is actually continuous. In this article we prove that in a Fréchet space the continuity of Y still holds if Y is the finite part of the limit of y n , x as n → ∞. We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as LF-spaces, DFS-spaces, and DFS * -spaces, and give examples where it does not hold.

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Distributions in spaces with thick points

Cited by 15 publications

References 21 publications

Reconstruction of the one-dimensional thick distribution theory

Reconstruction of the one-dimensional thick distribution theory

The Fourier transform of thick distributions

A Generalization of the Banach-Steinhaus Theorem for Finite Part Limits

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