Taylor formula for distributions

Bogdan Ziemian

Displaying similar documents to “Taylor formula for distributions”

A characterization of the space $D_{F}^{'}$

S. Pilipović (1982)

Matematički Vesnik

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Results on Colombeau product of distributions

Blagovest Damyanov (1997)

Commentationes Mathematicae Universitatis Carolinae

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The differential $ℂ$ -algebra $𝒢 (ℝ^{m})$ of generalized functions of J.-F. Colombeau contains the space $𝒟^{'} (ℝ^{m})$ of Schwartz distributions as a $ℂ$ -vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in $𝒟^{'} (ℝ^{m})$ . This is particularly useful for evaluation of certain products of distributions, as they are embedded in $𝒢 (ℝ^{m})$ , in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions $x_{\pm}^{a}$ and $δ^{(p)} (x)$ ,...

Topological imbedding of Laplace distributions in Laplace hyperfunctions

Zofia Szmydt, Bogdan Ziemian

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CONTENTS Foreword..............................................................................................................................5 Introduction..........................................................................................................................6 1. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7...

On regular $K^{'} M_{P}$ -distributions

Stevan Pilipović (1988)

Annales Polonici Mathematici

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Local analysis of nonstandard $C^{\infty}$ functions of pre-distributional type

V. Komkov, T. G. McLaughlin (1984)

Annales Polonici Mathematici

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Premium evaluation for different loss distributions using utility theory

Harman Preet Singh Kapoor, Kanchan Jain (2011)

Discussiones Mathematicae Probability and Statistics

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For any insurance contract to be mutually advantageous to the insurer and the insured, premium setting is an important task for an actuary. The maximum premium ( $P_{m a x})$ that an insured is willing to pay can be determined using utility theory. The main focus of this paper is to determine $P_{m a x}$ by considering different forms of the utility function. The loss random variable is assumed to follow different Statistical distributions viz Gamma, Beta, Exponential, Pareto, Weibull, Lognormal and Burr....

Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker)

Christian Brouder, Nguyen Viet Dang, Frédéric Hélein (2016)

Studia Mathematica

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The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front sets satisfy some conditions. Thus, it is natural to investigate the topological properties of these operations between spaces $_{Γ}^{'}$ of distributions having a wave front set included in a given closed cone Γ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on $_{Γ}^{'}$ , and the tensor product is not separately continuous....

Lifting distributions to the cotangent bundle

Włodzimierz M. Mikulski (2008)

Annales Polonici Mathematici

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A classification of all $ℳ f_{m}$ -natural operators $A : G r_{p} ⟿ G r_{q} T *$ lifting p-dimensional distributions D ⊂ TM on m-manifolds M to q-dimensional distributions A(D) ⊂ TT*M on the cotangent bundle T*M is given.

One-parameter semigroups in the convolution algebra of rapidly decreasing distributions

(2012)

Colloquium Mathematicae

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The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra $_{C}^{'} (ℝ ⁿ; M_{m \times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G \in_{C}^{'} (ℝ ⁿ; M_{m \times m})$ is the generating distribution of an i.d.c.s. if and only if the operator $\partial_{t} \otimes_{m \times m} - G *$ on $ℝ^{1 + n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

An inversion formula and a note on the Riesz kernel

Andrejs Dunkels (1976)

Annales de l'institut Fourier

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For potentials $U_{K}^{T} = K * T$ , where $K$ and $T$ are certain Schwartz distributions, an inversion formula for $T$ is derived. Convolutions and Fourier transforms of distributions in $(D_{L}^{'} p)$ -spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order $α$ , $0 < α < m$ , of a compact subset $E$ of $R^{m}$ has the following property: its restriction to the interior of $E$ is an absolutely continuous measure with analytic density which is expressed by an explicit formula.

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.

Global approximations for the γ-order Lognormal distribution

Thomas L. Toulias (2013)

Discussiones Mathematicae Probability and Statistics

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A generalized form of the usual Lognormal distribution, denoted with $_{γ}$ , is introduced through the γ-order Normal distribution $_{γ}$ , with its p.d.f. defined into (0,+∞). The study of the c.d.f. of $_{γ}$ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.