Convolution of Distribution-Valued Functions. Applications.

In this article we examine products and convolutions of vector-valued functions. For nuclear normal spaces of distributions Proposition 25 in \cite[p. 120]{MR0117544} yields a vector-valued product or convolution if there is a continuous product or convolution mapping in the range of the vector-valued functions. For specific spaces, we generalize this result to hypocontinuous bilinear maps at the expense of generality with respect to the function space. We consider holomorphic, meromorphic and differentiable vector-valued functions and state propositions that contain assertions on products and convolutions of distribution-valued functions in literature as particular cases. Moreover we consider the general convolution of analytic distribution-valued functions and give an approach different to \cite{MR2088667}

Convolution of Distribution-Valued Functions. Applications. Convolución de funciones con \emphblackvalores distribuciones. Aplicaciones. CHRISTIAN BARGETZ1

1University of Innsbruck, Innsbruck, Austria. Email:christian.bargetz@uibk.ac.at 

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Abstract

In this article we examine products and convolutions of vector-valued functions. For nuclear normal spaces of distributions Proposition 25 in [31,p. 120] yields a vector-valued product or convolution if there is a continuous product or convolution mapping in the range of the vector-valued functions. For specific spaces, we generalize this result to hypocontinuous bilinear maps at the expense of generality with respect to the function space. We consider holomorphic, meromorphic and differentiable vector-valued functions and state propositions that contain assertions on products and convolutions of distribution-valued functions in literature as particular cases. Moreover we consider the general convolution of analytic distribution-valued functions and give an approach different to [22].

Key words: Distributions, Convolution, Multiplication.

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2000 Mathematics Subject Classification: 46F10, 46E10, 42B20.

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Resumen

En este artículo se investigan los productos y convoluciones de las funciones con valores vectoriales. Para espacios nucleares y normales de distribuciones se obtiene de la Proposition 25 en [31,p. 120] una multiplicación o una convolución con valores vectoriales si existe una multiplicación o una convolución continua en los espacios de las imágenes de las funciones con valores vectoriales. Para espacios particulares se generaliza este resultado a las aplicaciones bilineales hipocontinuas a expensas de la generalidad relativo a los espacios funcionales. Se examinan funciones holomorfas, meromorfas y diferenciables con valores vectoriales y se formulan proposiciones que contienen proposiciones encontradas en la literatura sobre multiplicación y convolución de funciones con \emphblackvalores distribuciones. Además se contempla la convolución general de las funciones analíticas con \emphblackvalores distribuciones y se da un enfoque distinto del presentado en [22].

Palabras clave: Distribuciones, convolución, multiplicación.

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References

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(Recibido en septiembre de 2010. Aceptado en febrero de 2011)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv45n1a05, 
    AUTHOR  = {Bargetz, Christian}, 
    TITLE   = {{Convolution of Distribution-Valued Functions. Applications.}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2011}, 
    volume  = {45}, 
    number  = {1}, 
    pages   = {51-80} 
}