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      Power Series.Orthogonal Operational Expansions

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            Summary

            The power series are argued by the Cauchy formula interpreted as an orthogonal concept through the Fourier series, there is no direct expansion theorem similar to the one known for orthogonal functions. In this work we argue the power series expansons with a new method; from the position of orthogonal differential operators, which, unlike the classic one that requires knowledge of the analytical character at a point, uses said behavior in the positive semiaxis  and derive every series as a differential transformation of a single function, giving meaning to the idea of ​​origin of the series. In short, every function is the result of the action of an operator expressed as an orthogonal series of the differentiation operator acting on a basic function, the coefficients of development are expressed as integrals similar to the classic orthogonal ones. These facts allow us to formulate the concept of orthogonal operational space in analogy with the Hilbert space. In addition, we deduce the Laplace inverse in two versions for analytical functions, one already known integral and the other in terms of differential operators.

            Abstract

            Formulation of the classic Taylor series as an orthogonal concept based on identifying the expansions coefficients as differential transformation applied to a unique function; definition of operational orthogonality by analogy with the Hilbert space and identification of the nth derivative at a point based on the improper integral on the positive semiaxis ; deduction of inversion integrals for Laplacen transforms for analytical functions.

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            Author and article information

            Journal
            ScienceOpen Preprints
            ScienceOpen
            11 October 2020
            Affiliations
            [1 ] Independent researcher , retired
            Article
            10.14293/S2199-1006.1.SOR-.PP5HHW9.v1
            cf496dae-159b-408c-875d-3798482168e1

            This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .


            All data generated or analysed during this study are included in this published article (and its supplementary information files).
            Analysis,Applied mathematics,Functional analysis
            Taylor serie origin,operational expansions,operational coefficient,Laguerre differential operator,operational orthogonality,Laplace inversion,orthogonal operators

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