# Download e-book for kindle: Szego's Theorem and Its Descendants: Spectral Theory for L2 by Barry Simon

By Barry Simon

This booklet offers a complete evaluation of the sum rule method of spectral research of orthogonal polynomials, which derives from Gábor Szego's vintage 1915 theorem and its 1920 extension. Barry Simon emphasizes beneficial and adequate stipulations, and offers mathematical heritage that earlier has been on hand purely in journals. issues comprise heritage from the idea of meromorphic services on hyperelliptic surfaces and the examine of protecting maps of the Riemann sphere with a finite variety of slits got rid of. this permits for the 1st book-length remedy of orthogonal polynomials for measures supported on a finite variety of durations at the actual line.

as well as the Szego and Killip-Simon theorems for orthogonal polynomials at the unit circle (OPUC) and orthogonal polynomials at the actual line (OPRL), Simon covers Toda lattices, the instant challenge, and Jacobi operators at the Bethe lattice. contemporary paintings on functions of universality of the CD kernel to procure exact asymptotics at the high-quality constitution of the zeros is additionally integrated. The booklet areas precise emphasis on OPRL, which makes it the fundamental better half quantity to the author's past books on OPUC.

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**Extra info for Szego's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials **

**Example text**

6) j =1 Proof. 6) is immediate. 7) j = 0, 1, . . 8) We will use n (z; dµ) if we want the dµ dependence to be explicit. Thus, if fj = z j −1 , j = 1, . . , N , and gj = j −1 , j = 1, . . 2). 2. 3. 12) N→∞ Proof. Since n is orthogonal to any polynomial of degree n − 1, it minimizes { n + g | deg(g) ≤ n − 1}. 11). Since n is decreasing and positive, it has a limit and, of course, ( 0 2 . . n 2 )1/n then converges to lim n 2 . 4. 13) Remarks and Historical Notes. Szeg˝o’s great 1920–1921 paper [430] was the ﬁrst systematic exploration of OPUC, although he had earlier discussed OPs on curves [429].

2) that it has a simple direct proof. 14). dµN strips N α’s off the “bottom” while dµ(N) leaves the bottom N α’s and sets the others to zero. 2). 11). 11), but I know no direct proof. All that one gets from general principles is a semicontinuity. 3). Let dµ , dµ be nontrivial probability measures on ∂D so that dµ → dµ weakly (in the dual topology deﬁned by C(∂D)). 30) is trivial. 32) j =0 Here positivity saves us! 30). 1. To summarize, the steps involved (which will reappear in Chapters 3, 4, and 9) are: (1) Prove a step-by-step sum rule with positive terms from some kind of Jensen equality.

Sz is a bijection between nontrivial even probability measures on ∂D and nontrivial probability measures on [−2, 2]. 1. Let dρ = Sz(dµ) for nontrivial probability measures on [−2, 2] and ∂D. Let Pn , pn be the monic and orthonormal OPRL for dρ and n , ϕn the monic and orthonormal OPUC for dµ. 9) Sketch. 6). Every such Laurent polynomial has the form Qn (z + 1z ) for Qn (·) of degree n. Since 2n (0) = −α¯ 2n−1 , ∗2n (z) = −α2n−1 z 2n + · · · , so Qn is monic. 10) =0 since 2n ⊥ {z, . . , z } and ∗2n ⊥ {z, .