On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions

The purpose of this paper is to obtain n-dimensional inversion Laplace transform of retarded, Lorentz invariant functions by means of the passage to the limit of the rth-order derivative of the one-dimensional Laplace transform. This formula (IV.2) can be understood as a generalization of the one-di...

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Publicado: 1993
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v176_n2_p374_Trione
http://hdl.handle.net/20.500.12110/paper_0022247X_v176_n2_p374_Trione
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_0022247X_v176_n2_p374_Trione2025-07-30T17:29:52Z On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions The purpose of this paper is to obtain n-dimensional inversion Laplace transform of retarded, Lorentz invariant functions by means of the passage to the limit of the rth-order derivative of the one-dimensional Laplace transform. This formula (IV.2) can be understood as a generalization of the one-dimensional formula due to Widder [Trans. Amer. Math. Soc.32 (1930)]. This topic is intimately related to the generalized differentiation, the symbolic treatment of the differential equations with constant coefficients and its application to important physical problems (cf. Leibnitz, Pincherle, Liouville, Riemann, Boole, 1-leaviside, and others). Our main theorem (Theorem 15, formula (IV.2)) can be related to a result due to E. Post [6] and we also obtain an equivalent Leray′s formula (cf. (VI.I)) and (VI.2)) which expresses the Laplace transform of retarded, Lorentz invariant functions by means of the mth-order derivative of a K0-transform. Our method consists, essentially, in the following two steps. First: the obtainment of an analog of Bochner′s formula for a Laplace transform of the form (II.1), where φ is a function of the Lorentz distance, whose support is contained in the closure of the domain t0 > 0, t20 - t21 - ··· -t2n - 1 > 0. Formula (II.2) permits us to evaluate n-dimensional integrals by means of a one-dimensional K-transform. This last result was already employed to solve partial differential equations of the hyperbolic type (cf. [A. Gonzalez Dominguez and E. E. Trione, Adv. Math.31 (1979), 51-62]). Second: The passage to the limit of the rth-order derivative of the one-dimensional Laplace transform (via the K-transform). The previous conclusions are related to the classical Functional Analysis and Probability (i.e., the theory of moments, the classical Weierstrass theorem approximation, on compact sets, of continuous functions by polynomials and the inversion of Laplace-Stieltjes integrals). Finally, by appealing to the analytical continuation, we can extend our results to the distributional n-dimensional Laplace integrals. © 1993 Academic Press. Inc. All rights reserved. 1993 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v176_n2_p374_Trione http://hdl.handle.net/20.500.12110/paper_0022247X_v176_n2_p374_Trione
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description The purpose of this paper is to obtain n-dimensional inversion Laplace transform of retarded, Lorentz invariant functions by means of the passage to the limit of the rth-order derivative of the one-dimensional Laplace transform. This formula (IV.2) can be understood as a generalization of the one-dimensional formula due to Widder [Trans. Amer. Math. Soc.32 (1930)]. This topic is intimately related to the generalized differentiation, the symbolic treatment of the differential equations with constant coefficients and its application to important physical problems (cf. Leibnitz, Pincherle, Liouville, Riemann, Boole, 1-leaviside, and others). Our main theorem (Theorem 15, formula (IV.2)) can be related to a result due to E. Post [6] and we also obtain an equivalent Leray′s formula (cf. (VI.I)) and (VI.2)) which expresses the Laplace transform of retarded, Lorentz invariant functions by means of the mth-order derivative of a K0-transform. Our method consists, essentially, in the following two steps. First: the obtainment of an analog of Bochner′s formula for a Laplace transform of the form (II.1), where φ is a function of the Lorentz distance, whose support is contained in the closure of the domain t0 > 0, t20 - t21 - ··· -t2n - 1 > 0. Formula (II.2) permits us to evaluate n-dimensional integrals by means of a one-dimensional K-transform. This last result was already employed to solve partial differential equations of the hyperbolic type (cf. [A. Gonzalez Dominguez and E. E. Trione, Adv. Math.31 (1979), 51-62]). Second: The passage to the limit of the rth-order derivative of the one-dimensional Laplace transform (via the K-transform). The previous conclusions are related to the classical Functional Analysis and Probability (i.e., the theory of moments, the classical Weierstrass theorem approximation, on compact sets, of continuous functions by polynomials and the inversion of Laplace-Stieltjes integrals). Finally, by appealing to the analytical continuation, we can extend our results to the distributional n-dimensional Laplace integrals. © 1993 Academic Press. Inc. All rights reserved.
title On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions
spellingShingle On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions
title_short On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions
title_full On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions
title_fullStr On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions
title_full_unstemmed On the N-Dimensional Inversion Laplace Transform of Retarded, Lorentz Invariant Functions
title_sort on the n-dimensional inversion laplace transform of retarded, lorentz invariant functions
publishDate 1993
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v176_n2_p374_Trione
http://hdl.handle.net/20.500.12110/paper_0022247X_v176_n2_p374_Trione
_version_ 1840320687083880448

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