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Fractional Operators, Dirichlet Averages, and Splines

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Book cover New Perspectives on Approximation and Sampling Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented.

Dedicated to Paul Leo Butzer on the occasion of his 85th birthday

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Authors and Affiliations

  1. Lehrstuhl Mathematical Modelling (M6), Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, Garching b., München, 85747, Germany

    Peter Massopust

  2. Helmholtz Zentrum München, Ingolstädter Landstr. 1, Neuherberg, 85764, Germany

    Peter Massopust

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  1. Peter Massopust
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Correspondence to Peter Massopust .

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Editors and Affiliations

  1. Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

    Ahmed I. Zayed

  2. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Massopust, P. (2014). Fractional Operators, Dirichlet Averages, and Splines. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_17

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