Determining sets of frequencies in spectra of measures with finite energy

A set, is said to be determining if for any measure on the circumference such that one has the inequality, in the paper one shows that the upper density of any determining set is positive. On the other hand, there exist subsets of the set+ having arbitrarily small density.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 222–227, 1982.     

Simple proof of a theorem on removable singularities of analytic functions satisfying a Lipschitz condition

Let E be a compact subset of the complex plane ?, having positive planar Lebesgue measure. Then there exists a nonconstant function f, analytic in the domain ? É, satisfying the Lipschitz condition In this note there is given a simple proof of the theorem of N. X. Uy, formulated above. It is also proved that each bounded measurable function α, defined on the set E, can be revised on a set of small Lebesgue measure so that for the function ? obtained the Cauchy integral satisfies condition (1).     

The Euler–Lagrange Theory for Schur's algorithm: Wall pairs

This paper develops a techniques of Wall pairs for the study of periodic exposed quadratic irrationalities in the unit ball of the Hardy algebra.     

The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

In this paper the ideas of Algebraic Number Theory are applied to the Theory of Orthogonal polynomials for algebraic measures. The transferring tool are Wall continued fractions. It is shown that any set of closed arcs on the circle supports a quadratic measure and that any algebraic measure is either a Szegö measure or a measure supported by a proper subset of the unit circle consisting of a finite number of closed arcs. Singular parts of algebraic measures are finite sums of point masses.     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

A Singular Riesz Product in the Nevai Class and Inner Functions with the Schur Parameters in ∩p>2 l

There exist singular Riesz products dσ=∏^∞_κ=1 (1+Re(α_κζ^n_κ)) on the unit circle with the parameters (a_n)_n0 of orthogonal polynomials in L²(dσ) satisfying ∑^∞_n=0 |a_n|^p<+∞ for every p, p>2. The Schur parameters of the inner factor of the Cauchy integral ∫ (ζ−z)⁻¹ dσ(ζ), σ being such a Riesz product, belong to ∩_p>2 l^p.