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Convolution and Potentials

  • René Erlín Castillo
  • Humberto Rafeiro
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

Abstract

In this chapter we study the convolution which is a very powerful tool and some operators defined using the convolution. We first start with a detailed study about the translation operator and after that we introduce the convolution operator and give some immediate properties of the operator. As an immediate application we show that the convolution with the Gauss-Weierstrass kernel is an approximate identity operator. We also study the Young inequality for the convolution operator. The definition of a support of a convolution is given based upon the definition of the support of a (class of) function which differs from the classical definition of support of a function. Approximate identity operators are studied in a general framework via Dirac sequences and Friedrich mollifiers. We end the chapter with a succinct study of the Riesz potential.

Keywords

Lebesgue Space Convolution Operator Translation Operator Approximate Identity Young Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René Erlín Castillo
    • 1
  • Humberto Rafeiro
    • 2
  1. 1.Universidad Nacional de ColombiaBogotáColombia
  2. 2.Pontificia Universidad JaverianaBogotáColombia

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