On Higher-Order Szegő Theorems with a Single Critical Point of Arbitrary Order

We prove the following higher-order Szeg? theorem: If a measure on the unit circle has absolutely continuous part \(w(\theta )\) and Verblunsky coefficients \(\alpha \) with square-summable variation, then for any positive integer m,

Open image in new window

is finite if and only if \(\alpha \in \ell ^{2m+2}\). This is the first known equivalence result of this kind in the regime of very slow decay, i.e., with \(\ell ^p\) conditions with arbitrarily large p. The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach.

     

On the Szegő Metric

We introduce a new biholomorphically invariant metric based on Fefferman’s invariant Szeg? kernel and investigate the relation of the new metric to the Bergman and Carathéodory metrics. A key tool is a new absolutely invariant function assembled from the Szeg? and Bergman kernels.     

Scattering Theory for CMV Matrices: Uniqueness,Helson–Szegő and Strong Szegő Theorems

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.     

Sensitivity analysis for Szegő polynomials

Szegő polynomials are orthogonal with respect to an inner product on the unit circle. Numerical methods for weighted least-squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of Szegő polynomials. This paper discusses the conditioning of several mappings involving Szegő polynomials and, thereby, sheds light on the sensitivity of some approximation problems involving trigonometric polynomials. This Research supported in part by NSF grant DMS-0107858.     

Szegő limit theorems

The first Szeg limit theorem has been extended by Bump–Diaconis and Tracy–Widom to limits of other minors of Toeplitz matrices. We use a more geometric method to extend their results still further. Namely, we allow more general measures and more general determinants. We also give a new extension to higher dimensions, which extends a theorem of Helson and Lowdenslager.     

On a Conjecture of Shapiro's

ＯｎａＣｏｎｊｅｃｔｕｒｅｏｆＳｈａｐｉｒｏ＇ｓ陈志国ＯｎａＣｏｎｊｅｃｔｕｒｅｏｆＳｈａｐｉｒｏ＇ｓ￥ＣｈｅｎＺｈｉｇｕｏ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ）Ａｂａｔｒａｃｔ：Ｔｈｉｓｐｒｏｂｌｅｍｗａｓ...     

On a Conjecture of Halperin

Ｏｎ　ａ　Ｃｏｎｊｅｃｔｕｒｅ　ｏｆ　ＨａｌｐｅｒｉｎＨｕａｎｇＨｕａｌｅ（黄华乐）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＡｃａｄｅｒｎｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ，１０００８０）Ａｂｓｔｒａｃｔ：Ｉｎ［１］，Ｈａｌｐｅｒｉｎｒａｉｓｅ...     

On the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.     

Szegő’s theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg?’s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.     

Bernstein—Szegő's Theorem for Sobolev Orthogonal Polynomials

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product where ρ ₀ and ρ ₁ are weights which satisfy Szegő's condition, supported on a smooth Jordan closed curve or arc. December 14, 1997. Date revised: September 21, 1998. Date accepted: November 16, 1998.     

Spectral asymptotics for Kac–Murdock–Szegő matrices

Szeg?’s First Limit Theorem provides the limiting statistical distribution of the eigenvalues of large Toeplitz matrices. Szeg?’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit Theorems to Kac–Murdock–Szeg? (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature.     

Proof Of A Conjecture Of Erdős On Triangles In Set-Systems

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and . Let be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi. We also show that equality holds if and only if consists of all r-element subsets containing a fixed element. Analogous results are obtained for nonuniform families.     

On a problem of Erdős

Let s≥2 be an integer. Denote by f ₁(s) the least integer so that every integer l>f ₁(s) is the sum of s distinct primes. Erd?s proved that f ₁(s)<p ₁+p ₂+?+p _s +Cslogs, where p _i is the ith prime and C is an absolute constant. In this paper, we prove that f ₁(s)=p ₁+p ₂+?+p _s +(1+o(1))slogs=p ₂+p ₃+?+p _s+1+o(slogs). This answers a question posed by P. Erd?s.     

On a Conjecture of S.Halperin

If X is a finite simply connected CW complex, then H_*(X,Q) is finitedimensional, let n_X = max{i|H_i(X,Q)≠0}. On the other hand,π_i(X) is the directsum of finitely many copies of Z and finite Abelian group. We call an interval[k,l] a torsion gap for X if π_k(X) and π_l (X) both coutain copies of Z, andπ_i (X)(k相似文献