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A general approach to approximation theory of operator semigroups. (English. French summary) Zbl 1440.47011

Approximation theory of operator semigroups has important applications in the study of partial differential equations, probability theory, approximation of functions, and the study of semigroups themselves. Despite this fact, it seems that a comprehensive account on approximation of \(C_0\)-semigroups is still missing. Several important formulas require separate approaches, and it is natural to try to unify the separate considerations. One of the aims of this paper is to develop a general functional calculus approach to the approximation of \(C_0\)-semigroups on Banach spaces by bounded completely monotone functions of their generators. In this direction, the authors extend ideas from [A. Gomilko and Y. Tomilov, J. Funct. Anal. 266, No. 5, 3040–3082 (2014; Zbl 1317.47042)].
It was remarked in [loc.cit.]that many approximation formulas for \(C_0\)-semigroups follow from relations of the form \[ e^{-tz}-e^{-n\varphi\left(tz/n\right)}\to 0,\ n\to\infty \] or \[ e^{-tz}-e^{-nt\varphi\left(z/n\right)}\to 0,\ n\to\infty \] for all \(t\geq 0\), \(\operatorname{Re}(z)\geq 0\), and \(\varphi\) a suitable Bernstein function. Extending this observation, the authors consider the asymptotic relation \[ g_t^n\left(tz/n\right)-e^{-tz}\to 0,\ n\to\infty,\tag{1} \] where \((g_t)_{t>0}\) is a family of suitable bounded completely monotone functions. Using the functional calculi ideas from [loc. cit.] and the approximation formula (1), the authors unify most of the known approximation formulas for bounded \(C_0\)-semigroups and equip them with optimal convergence rates. Second order approximation formulas with optimal rates and approximation results for holomorphic semigroups with sharp constants are also presented. This approach based on functional calculi is compared with a probabilistic approach existing in the literature. The conclusion is that, from a practical point of view, the two approaches can probably be considered as complementary rather than covering each other. Some aspects of applications of semigroup approximations, although different from the setting of this paper, are presented in [F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications. Berlin: Walter de Gruyter (1994; Zbl 0924.41001)] and [F. Altomare et al., Markov operators, positive semigroups and approximation processes. Berlin: De Gruyter (2014; Zbl 1352.47001)].

MSC:

47A60 Functional calculus for linear operators
65J08 Numerical solutions to abstract evolution equations
47D03 Groups and semigroups of linear operators
46N40 Applications of functional analysis in numerical analysis
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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