A Comment on a Controversial Issue: A Generalized Fractional Derivative Cannot Have a Regular Kernel

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a ne...

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Bibliographic Details
Published inFractional calculus & applied analysis Vol. 23; no. 1; pp. 211 - 223
Main Author Hanyga, Andrzej
Format Journal Article
LanguageEnglish
Published Warsaw Versita 01.02.2020
De Gruyter
Nature Publishing Group
Subjects
26A33
34A99
Abstract Harmonic Analysis
Analysis
Bernstein function
completely monotone function
Derivatives
fractional calculus
Functional Analysis
Integral Transforms
Integrals
Kernels
LICM function
Mathematics
Monotone functions
Operational Calculus
Research Paper
Sonine equation
Stieltjes function
Sonine equation
fractional calculus
Bernstein function
26A33
34A99
Stieltjes function
completely monotone function
LICM function
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Summary:The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1311-0454
1314-2224
DOI:10.1515/fca-2020-0008