Toeplitz Matricesin Other (Other) by Dgoodz19
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Proof. As usual, by factoring out the Blaschke product we may assume that G(z) 0 for all z D and that, therefore, there exists an analytic function h in D with G(z) exp h(z). Since G(z) exp(Re h(z)), we have log G(z) Re h(z). Let z reit. Since G (ei) F (ei) f (ei) a.e. by Corollary and log F (reit) 1 2 P ( t) log f (ei) d, 2 0 r the theorem will be established if it is shown that 2 log G(reit) 1 P ( t) log G(ei) d
for reit D. 2 0 r As we have in a number of previous proofs, we define the function hs for each s (0, 1) by hs(z) h(sz). Each hs is in H, so we can write
hs(reit) 1 2
Pr( t)h(sei) d
by the Poisson integral formula (Theorem ) (since h s(ei) ...