In this expository article we survey some properties of completely monotonic functions and give various examples, including some famous special functions.Such function are useful, for example, in probability theory. It is known, [3], p.450, for example, that a function w is the Laplace transform of an infinitely divisible probability distribution on (0, ∞), if and only if w = e −h where the derivative of h is completely monotonic and h(0+) = 0.
This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4-9, 2003, Minsk, Belarus. We give a survey of investigations on 'the variable exponent business', concentrating mainly on recent advances in the operator theory and harmonic analysis in the generalized Lebesgue and Sobolev spaces L p (·) and W m,p(·) .
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