Tag: fractional integral

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Functional analysis Mathematics

Weyl Integrals on Weighted Spaces

T. Kleiner, R. Hilfer

Fractional Calculus and Applied Analysis 22, 1225-1248 (2019)
DOI: 10.1515/fca-2019-0065

submitted on
Thursday, January 31, 2019

Weighted spaces of continuous functions are introduced such that Weyl fractional integrals with orders from any finite nonnegative interval define equicontinuous sets of continuous linear endomorphisms for which the semigroup law of fractional orders is valid. The result is obtained from studying continuity and boundedness of convolution as a bilinear operation on general weighted spaces of continuous functions and measures.



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Fractional Calculus Mathematics

Desiderata for Fractional Derivatives and Integrals

R. Hilfer, Yu. Luchko

Mathematics 7, 149 (2019)
https://doi.org/10.3390/math7020149

submitted on
Friday, January 11, 2019

The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.



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Functional analysis Mathematics

Convolution Operators on Weighted Spaces of Continuous Functions and Supremal Convolution

T. Kleiner, R. Hilfer

Annali di Matematica Pura ed Applicata 199, 1547-1569 (2020)
https://doi.org/10.1007/s10231-019-00931-z

submitted on
Tuesday, September 25, 2018

The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.



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Fractional Calculus Functional analysis Mathematical Physics Mathematics Stochastic Processes

Mathematical and physical interpretations of fractional derivatives and integrals

R. Hilfer

in: Handbook of Fractional Calculus with Applications: Basic Theory, Vol. 1
edited by: A. Kochubei and Y. Luchko
Walter de Gruyter GmbH, Berlin, 47-86 (2019)
https://doi.org/10.1515/9783110571622
ISBN: 9783110571622

submitted on
Saturday, June 2, 2018

Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpretation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief description of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point.



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