By Harald Bohr

ISBN-10: 082840027X

ISBN-13: 9780828400275

Stimulated by way of questions about which features should be represented through Dirichlet sequence, Harald Bohr based the speculation of virtually periodic features within the Twenties. this pretty exposition starts with a dialogue of periodic services ahead of addressing the virtually periodic case. An appendix discusses nearly periodic features of a posh variable. this can be a attractive exposition of the idea of virtually Periodic capabilities written through the author of that thought; translated via H. Cohn.

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**Sample text**

However, it is a Runge domain. H /; K is a compact subset of H , and > 0: By Theorem 15, f has a holomorphic continuation to the polydisc D2 ; which we continue to denote by f: In the polydisc, we may expand f in power series. The partial sums are polynomials which converge uniformly to f on compact subsets of the polydisc and, in particular, on K: Thus, H is a Runge domain. In the other direction, we have mentioned in the introduction that in C every domain is a domain of holomorphy, but not all domains are Runge domains.

9) . 6) and consequently by Theorem 44 u is convex. x0 ; h0 / < 0 for some x0 2 and some h0 6D 0. Since u 2 C 2 . /, the function Qu . ; h0 / is continuous in . y; h0 / < 0 for every y in the ı-neighborhood of x0 . Let h D ch0 , with c so small that jhj < ı, and set x D x0 Ch. 0; 1/. x0 / h: By Theorem 44, u is not convex in t u . We now state an analogous characterization of plurisubharmonic functions. Theorem 46. Let be an open set in Cn . A real-valued function u 2 C 2 . / is plurisubharmonic in if and only if Lu 0, that is, Â @2 u @zj @zk Ã 0: Proof.

5) C pm D 1. Proof. The proof is by induction. For m D 1 the assertion is trivial and for m D 2 it is the definition of convexity. Suppose the assertion is true for m and let xj and pj be as in the theorem with j D 1; ; m C 1. We may assume that 0 < pmC1 < 1. 1 pmC1 /y; where X pj p1 x1 C pm xm D xj ; 1 pmC1 1 pmC1 j D1 m yD and m X pj D 1: 1 pmC1 j D1 Therefore, since is convex, y 2 . x1 / C pmC1 where the next-to-last equality is by the induction hypothesis. xmC1 /; t u 8 Plurisubharmonic Functions 47 Having characterized continuous convex functions, we now characterize differentiable convex functions.

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