By Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber, Frank Neubrander
This monograph offers a scientific account of the speculation of vector-valued Laplace transforms, starting from illustration conception to Tauberian theorems. In parallel, the idea of linear Cauchy difficulties and semigroups of operators is built thoroughly within the spirit of Laplace transforms. lifestyles and strong point, regularity, approximation and certainly asymptotic behaviour of strategies are studied. diversified functions to partial differential equations are given. The ebook includes an advent to the Bochner crucial and a number of other appendices on history fabric. it truly is addressed to scholars and researchers drawn to evolution equations, Laplace and Fourier transforms, and sensible analysis.
The authors have succeeded admirably in bringing jointly a wealth of contemporary fabric, a lot of which appears to be like in e-book shape for the 1st time. This authoritative paintings is probably going to develop into a customary reference on either the Laplace rework and its purposes to the summary Cauchy challenge. …
The publication is a superb textbook to boot. Proofs are regularly obvious and whole, and lots of themes which may were regarded as heritage fabric are coated to boot. All this makes the textual content very obtainable and self-contained. purposes to concrete differential operators are given in the course of the textual content. every one bankruptcy ends with historic and bibliographical reviews. … In precis, this ebook could be of curiosity to a large viewers of (functional) analysts and it may have a spot in each arithmetic library. Warmly recommended!
Jan van Neerven, Nieuw Archief voor Wiskunde, No. three, 2003
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Additional info for Vector-valued Laplace Transforms and Cauchy Problems: Second Edition
THE RADON-NIKODYM PROPERTY 19 for all t ≥ s ≥ 0. Hence, F ◦h−1 : R+ → X is Lipschitz continuous. e. Since |h(t) − h(s)| ≥ |t − s|, h−1 maps null sets to null sets. , since G is absolutely continuous. e. 5. 4 are satisﬁed. By the remarks above, the space X has the Radon-Nikodym property if and only if every Lipschitz continuous function F : [0, 1] → X is diﬀerentiable almost everywhere. It is clear that a closed subspace of a space with the Radon-Nikodym property has the Radon-Nikodym property. Next we exhibit a class of spaces having the Radon-Nikodym property.
11]). n Let D be a countable dense subset of Y . Suppose that y = i=1 αi yi for some n ∈ N, αi ∈ Q + iQ and yi ∈ D. e. 4) i=1 for almost all s ∈ R+ . This holds for all possible n ∈ N, αi ∈ Q + iQ and yi ∈ D, but there are only countably many such possibilities. 4) holds for all s ∈ R+ \ Ω and all n ∈ N, αi ∈ Q + iQ and yi ∈ D. 4) holds for all αi ∈ C. This shows that for s ∈ R+ \ Ω, the map y → gy (s) from D to C extends to a unique f (s) ∈ Y ∗ = X with f (s) ≤ 1. For y ∈ D, y, f (·) is measurable and bounded, and t y, F (t) = y, f (s) ds.
5 b) and Appendix D) does not have the Radon-Nikodym property. 9. The space c0 does not have the Radon-Nikodym property. Proof. Let F (t) := (Fn (t))n∈N , where Fn (t) := c0 is Lipschitz continuous since t |Fn (t) − Fn (s)| = s 1 n sin(nt) (n ∈ N). Then F : R+ → cos(nr) dr ≤ |t − s| (t, s ≥ 0, n ∈ N). 3. CONVOLUTIONS 21 However, F is nowhere diﬀerentiable, since Fn (t) = cos(nt) and (cos(nt))n∈N ∈ c0 . Thus, c0 does not have the Radon-Nikodym property. 6 that the space l1 = c∗0 has the Radon-Nikodym property.