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The Foundations of Fractional Calculus in the Mellin Transform Setting with Applications

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Abstract

In this article we study the basic theoretical properties of Mellin-type fractional integrals, known as generalizations of the Hadamard-type fractional integrals. We give a new approach and version, specifying their semigroup property, their domain and range. Moreover we introduce a notion of strong fractional Mellin derivatives and we study the connections with the pointwise fractional Mellin derivative, which is defined by means of Hadamard-type fractional integrals. One of the main results is a fractional version of the fundamental theorem of differential and integral calculus in the Mellin frame. In fact, in this article it will be shown that the very foundations of Mellin transform theory and the corresponding analysis are quite different to those of the Fourier transform, alone since even in the simplest non-fractional case the integral operator (i.e. the anti-differentiation operator) applied to a function \(f\) will turn out to be the \(\int _0^x f(u)du/u\) with derivative \((xd/dx)f(x).\) Thus the fundamental theorem in the Mellin sense is valid in this form, one which stands apart from the classical Newtonian integral and derivative. Among the applications two fractional order partial differential equations are studied.

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Notes

  1. An important question is whether there exists a fractional integral operator of order \(\alpha > 0\) which is the counterpart of the fractional Marchaud derivative in the sense that the fundamental theorem of the calculus is also valid in the Fourier transform setting, thus that each operator is the inverse operator of the other, under certain conditions apart from \(f\) belonging to \(L^2 ( \mathbb {R} ).\) This problem is treated in the book of Samko ([54], pp. 14–24, 75–76, 85, 212). The proofs of his results are modeled upon the corresponding ones for Fourier series developed in [33], which is based upon a fundamental theorem of Westphal [60] and is cited regularly in the relevant literature. However, the fully complete fundamental theorem is unfortunately not there. His very useful volume, listing 318 books and papers in the field, with an elaborate author index, and excellent biographical notes after the eleven chapters, is carried out in a multidimensional setting.

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Acknowledgments

The authors would like to thank Boris Ivanovich Golubov (Moscow) for his great help in regard to the short biography of R.G. Mamedov. He contacted his colleague in Baku, who in turn received a four pages biography of Mamedov together with a list of his publications kindly sent by his son Aykhan Mammadov, and made a first translation of this biography. The present biography is the extended and polished version kindly carried out by Peter Oswald (Bremen). The authors are grateful to Aykhan Mammadov for his extraordinary help in regard to various aspects of his father’s life, Azerbaijan, and the paper itself. The translation of the preface of [45] is due to Andi Kivinukk (Tallinn). Also, the authors wish to thank Anna Rita Sambucini for her technical support in including photos in the text. The first and third authors have been partially supported by the Gruppo Nazionale Analisi Matematica, Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), through the INdAM—GNAMPA Project 2014, and by the Department of Mathematics and Computer Sciences of University of Perugia.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, University of Perugia, via Vanvitelli 1, 06123, Perugia, Italy

    Carlo Bardaro & Ilaria Mantellini

  2. Lehrstuhl A fur Mathematik RWTH-Aachen, Templergraben 55, 52062, Aachen, Germany

    Paul L. Butzer

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  1. Carlo Bardaro
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  2. Paul L. Butzer
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  3. Ilaria Mantellini
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Correspondence to Carlo Bardaro.

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Communicated by Hans G. Feichtinger.

This article is a far-reaching extension of the paper “Mellin analysis and exponential sampling. Part I: Mellin fractional integrals” ( jointly with C. Bardaro and P.L. Butzer, see [17] ), presented by Ilaria Mantellini at “SAmpta2013”, held at Jacobs University, Bremen, on 1–5 July, 2013, and was conducted by Goetz Pfander, Peter Oswald, Peter Massopust and Holger Rauhut. The meeting marked two decades of SAmpta Workshops [held at Riga (1995), Aveiro (1997), Loen (1999), Tampa (2001), Strobl (2003), Samsun (2005), Marseille (2007), Singapore (2009) ] and the Commemoration of the 85th Birthday of P.L. Butzer. In Memory of Rashid Gamid-oglu Mamedov, a pioneer in Mellin Analysis.

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Bardaro, C., Butzer, P.L. & Mantellini, I. The Foundations of Fractional Calculus in the Mellin Transform Setting with Applications. J Fourier Anal Appl 21, 961–1017 (2015). https://doi.org/10.1007/s00041-015-9392-3

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