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Functional calculus and a link to fractional calculus. (English) Zbl 1077.47018

The authors give a well readable survey on their functional calculus theory of fractional integration and differentiation. In contrast to the Riemann–Liouville appproach, here the lower limit of integration is not \(0\) (or another finite real number) but \(-\infty\), and the analytical treatment is not via the Laplace but via the Fourier transform. This is particularly appropriate for evolution problems with linear combinations of fractional derivatives of different orders where the starting time is not \(0\) but \(-\infty\). After sketching the \(L_2\) approach and its limitations, the authors extend the theory to distributional spaces, hereby using the Fourier–Laplace transform. Particular attention is paid to the question of causality in evolution problems.

MSC:

47A60 Functional calculus for linear operators
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
35S05 Pseudodifferential operators as generalizations of partial differential operators
26A33 Fractional derivatives and integrals
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