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A generalized Hölder inequality and a generalized Szego theorem. (English) Zbl 0696.60035

A generalization of the following theorem of Grenander and Szegö is given: Let G be a cyclic graph with E edges. On each edge e a continuous function \(f^ e(x)\) is defined with a Fourier-expansion \[ (*)\quad f^ e_ k=(2\pi)^{-1}\int^{\pi}_{-\pi}f^ e(x)\exp (ikt)dx,\quad k=0,\pm 1,... \] Set \[ (**)\quad S(G)=\sum_{-n\leq k_ 1,k_ 2,...k_{\ell}\leq n}\prod^{E}_{\ell =1}f^ e_{k_{\ell}- k_{\ell -1}}. \] Then we have \[ (***)\quad \lim_{n\to \infty}S_ u(G)/n=(2\pi)^{-1}\int^{\pi}_{-\pi}\prod^{E}_{\ell =1}f^ e(x)dx. \] The generalization of (***) is about replacing the cyclic graph G by more general ones.
Reviewer’s remark. Because of the many typographical errors I was unable to follow the details. For instance, in the definition of the Fourier coefficients (*), negative k’s also occur, where use is made of already in (**); but (*) is stated as “for \(j_{\nu}=1,2,...''\) Furthermore, (***) is a limit for \(n\to \infty\); but it is stated on p. 688 that (1.2) (which is (***)) becomes Parseval’s formula for \(n=2\).
Reviewer: P.Szüsz

MSC:

60F99 Limit theorems in probability theory
26D15 Inequalities for sums, series and integrals
94C15 Applications of graph theory to circuits and networks
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