Category: Functional analysis

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Fractional Calculus Functional analysis

Fractional Calculus for Distributions

R. Hilfer, T. Kleiner

Fractional Calculus and Applied Analysis , (2024)
https://doi.org/10.1007/s13540-024-00306-z

submitted on
Friday, March 29, 2024

Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as D’-convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of D’-convolution.



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Fractional Calculus Functional analysis Mathematics

Convolution on Distribution Spaces Characterized by Regularization

T. Kleiner, R. Hilfer

Mathematische Nachrichten 296, 1938-1963 (2023)
https://doi.org/10.1002/mana.202100330

submitted on
Friday, October 15, 2021

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex latti\-ces of measures.



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Fractional Calculus Functional analysis Mathematics

Sequential generalized Riemann–Liouville derivatives based on distributional convolution

T. Kleiner, R. Hilfer

Fractional Calculus and Applied Analysis 25, 267-298 (2022)
https://doi.org/10.1007/s13540-021-00012-0

submitted on
Friday, October 15, 2021

Sequential generalized fractional Riemann-Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.



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Fractional Calculus Functional analysis Mathematics

On extremal domains and codomains for convolution of distributions and fractional calculus

T. Kleiner, R. Hilfer

Monatshefte für Mathematik 198, 122-152 (2022)
https://doi.org/10.1007/s00605-021-01646-1

submitted on
Wednesday, December 30, 2020

It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz poten- tials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.



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Fractional Calculus Functional analysis Glasses Mathematical Physics Mathematics Special Functions

Fractional glassy relaxation and convolution modules of distributions

T. Kleiner, R. Hilfer

Analysis and Mathematical Physics 11, 130 (2021)
https://doi.org/10.1007/s13324-021-00504-5

submitted on
Wednesday, September 30, 2020

Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.



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Fractional Calculus Functional analysis Uncategorized

Maximal Domains for Fractional Derivatives and Integrals

R. Hilfer, T. Kleiner

Mathematics 8, 1107 (2020)
https://doi.org/10.3390/math8071107

submitted on
Wednesday, March 11, 2020

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. The calculi satisfy desiderata proposed above in Mathematics 7, 149 (2019). For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. The examples suggest to modify desideratum (f) in the original list.



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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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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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