Abstract:
In this paper we formulate operators which constitute an essential generalization of the Riemann–Liouville fractional integro-differentiation operator. With the aid of these operators fundamentally new analogues of the classical formulas of Cauchy, Schwarz, and Poisson for the representation of analytic and harmonic functions in the interior of a circle are established. These formulas enable us to give a complete structural representation for the broad classes of harmonic and analytic functions associated with the generalized operators. Finally, in this paper we also establish sufficient conditions for the solvability of the Hausdorff and Stieltjes moment problems for some general families of sequences of positive numbers.
\Bibitem{Dzh68}
\by M.~M.~Dzhrbashyan
\paper A generalized Riemann--Liouville operator and some of its
applications
\jour Math. USSR-Izv.
\yr 1968
\vol 2
\issue 5
\pages 1027--1063
\mathnet{http://mi.mathnet.ru/eng/im2506}
\crossref{https://doi.org/10.1070/IM1968v002n05ABEH000699}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=233992}
\zmath{https://zbmath.org/?q=an:0174.11701}
Linking options:
https://www.mathnet.ru/eng/im2506
https://doi.org/10.1070/IM1968v002n05ABEH000699
https://www.mathnet.ru/eng/im/v32/i5/p1075
This publication is cited in the following 13 articles:
Armen M. Jerbashian, Joel E. Restrepo, Frontiers in Mathematics, Functions of Omega-Bounded Type, 2024, 163
Armen M. Jerbashian, Joel E. Restrepo, Frontiers in Mathematics, Functions of Omega-Bounded Type, 2024, 3
S. M. Sitnik, E. L. Shishkina, “O dvukh klassakh operatorov obobschennogo drobnogo integro-differentsirovaniya”, Differentsialnye uravneniya i matematicheskaya fizika, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 198, VINITI RAN, M., 2021, 109–122
I. I. Bavrin, O. E. Iaremko, “Inverting of generalized Riemann–Liouville operator by means of integral Laplace transform”, Ufa Math. J., 8:3 (2016), 41–48
A. M. Jerbashian, “On boundary properties and biorthogonal systems in the spaces A ω 2 ⊂ H 2”, J. Contemp. Mathemat. Anal, 49:1 (2014), 17
A. G. Butkovskii, S. S. Postnov, E. A. Postnova, “Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation”, Autom. Remote Control, 74:4 (2013), 543–574
Armen M. Jerbashian, Modern Analysis and Applications, 2009, 335
A.M. Jerbashian *, “On the theory of weighted classes of area integrable regular functions”, Complex Variables, Theory and Application: An International Journal, 50:3 (2005), 155
A. M. Dzhrbashyan, “Embedding the classes N{ω} of Nevanlinna type”, Izv. Math., 63:4 (1999), 687–705
N. U. Arakelian, A. G. Vitushkin, V. S. Vladimirov, A. A. Gonchar, “Mkhitar Mkrtichevich Dzhrbashyan (on his sixtieth birthday)”, Russian Math. Surveys, 34:2 (1979), 269–275
M. M. Dzhrbashyan, “The theory of factorization and boundary properties of functins meromorphic in a disc”, Russian Math. Surveys, 28:4 (1973), 1–12
M. M. Dzhrbashyan, V. S. Zakharyan, “Boundary properties of subclasses of meromorphic functions of
bounded form”, Math. USSR-Izv., 4:6 (1970), 1273–1354
M. M. Dzhrbashyan, “Theory of factorization of functions meromorphic in the disk”, Math. USSR-Sb., 8:4 (1969), 493–592
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