By Gianni Gilardi

ISBN-10: 8838607273

ISBN-13: 9788838607271

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**Example text**

Decaying random Verblunsky coefficients Decaying random potentials were studied starting with Simon [86] who found the first example of Jacobi matrices with |an −1|+|bn | → 0 and µ purely singular. The pioneering results on decaying Verblunsky coefficients are due to Nikishin [69] and Teplyaev [105]. 6-7]. Now Verblunsky coefficients {αj (ω)} are assumed to be independent random but not necessarily identically distributed variables, which decay to zero in some sense; at a minimum, the mean value E(|αj (ω)|2 ) → 0.

Then {νnj (µ)} converges to the uniform measure on the circle of radius A. Simon suggested a new approach to this result based on CMV matrices instead of potential theory. 3): 1 Tr(C (n) )k = z k dνn (µ). 3 Let k ∈ Z+ |αn (µ1 ) − αn (µ2 )| = o(n), n → ∞. Then for each z k (dνn (µ1 ) − dνn (µ2 )) = 0. lim n→∞ D In particular, if νn (µ1 ) tends to ν and γ is any limit point of νn (µ2 ), then z k dγ = D z k dν. D A crucial feature of the Mhaskar – Saff theorem is its universality. Under its assumption the angular distribution is the same.

In type I we are looking for polynomials Qn,j of degree nj − 1 for each j = 1, . . , r such that r xk Qn,j (x)dµj (x) = 0, k = 0, 1, . . , |n| − 2. j=1 These orthogonality relations give |n| − 1 homogeneous linear equations for the |n| coefficients of the r polynomials Qn,j , so there is a non-trivial solution. If the rank of the system is |n| − 1, then the solution is unique up to a multiplicative factor, in which case the index n is called normal. This happens precisely if each Qn,j is of exact degree nj − 1.

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