Abstract
Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively.
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References
A. N. Kochubei, “Distributed-order calculus and equations of ultraslow diffusion,” J. Math. Anal. Appl., 340, 252–281 (2008).
N. Yu. Bakaev and R. P. Tarasov, “Semigroups and a method for stably solving the Abel equation,” Sib. Math. J., 19, 1–5 (1978).
I. C. Gohberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, American Mathematical Society, Providence, RI (1970).
N. Jacob and A. M. Krägeloh, “The Caputo derivative, Feller semigroups, and the fractional power of the first order derivative on C ∞(ℝ +0 ),” Fract. Calc. Appl. Anal., 5, 395–410 (2002).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (1993).
R. S. Phillips, “On the generation of semigroups of linear operators,” Pacif. J. Math., 2, 343–369 (1952).
C. Berg, K. Boyadzhiev, R. deLaubenfels, “Generation of generators of holomorphic semigroups,” J. Austral. Math. Soc., Ser. A, 55, 246–269 (1993).
R. L. Schilling, “Subordination in the sense of Bochner and a related functional calculus,” J. Austral. Math. Soc. A, 64, 368–396 (1998).
F. Hirsch, “Intégrales de résolvantes et calcul symbolique,” Ann. Inst. Fourier, 22, No. 4, 239–264 (1972).
F. Hirsch, “Domaines d’opérateurs representés comme intégrales de résolvantes,” J. Funct. Anal., 23, 199–217 (1976).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1966).
E. Hille and R. S. Phillips, Functional Analysis and Semigroups, American Mathematical Society, Providence, RI (1957).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, Gordon and Breach, New York (1986).
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York (1974).
V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, Pergamon Press, Oxford (1965).
V. I. Gorbachuk and A. V. Knyazyuk, “Boundary values of solutions of differential operator equations,” Rus. Math. Surv., 44, No. 3, 67–111 (1989).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 478–486, April, 2008.
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Kochubei, A.N. Distributed-order calculus: An operator-theoretic interpretation. Ukr Math J 60, 551–562 (2008). https://doi.org/10.1007/s11253-008-0076-x
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DOI: https://doi.org/10.1007/s11253-008-0076-x