Boundary Interpolation by Finite Blaschke Products

Given 2n distinct points z₁, z′₁, z₂, z′₂, ..., z_n, z′_n (in this order) on the unit circle, and n points w₁, ..., w_n on the unit circle, we show how to construct a Blaschke product B of degree n such that B(z_j) = w_j for all j and, in addition, B(z′_j) = B(z′_k) for all j and k. Modifying this example yields a Blaschke product of degree n - 1 that interpolates the z_j's to the w_j's. We present two methods for constructing our Blaschke products: one reminiscent of Lagrange's interpolation method and the second reminiscent of Newton's method. We show that locating the zeros of our Blaschke product is related to another fascinating problem in complex analysis: the Sendov Conjecture. We use this fact to obtain estimates on the location of the zeros of the Blaschke product.     

Blaschke product interpolation and its application to the design of digital filters

A new proof is given thatn distinct points on the unit circle can be mapped inton arbitrary points on the unit circle of the complex plane by a finite Blaschke product. A result of this proof is that the mapping can be done with at mostn?1 factors in the product. The problem is studied in the context of its application to frequency transformations used to design digital filters.     

Cluster sets of interpolating Blaschke products

We construct interpolating Balschke products whose radial cluster sets at a given point of the unit circle can be prescribed to be one of the following: the closed unit disk; an arbitrary closed arc on the unit circle; an arbitrary interval of the form [x, y], wherexy ≠ 0 and −1≤1x≤y≤1. We also show that there does not exist an interpolating Blaschke product having [0,y] or [x, 0] as a radial cluster set. On the other hand, there do exist finite products of interpolating Blaschke products that have [0, 1] as a radial cluster set. Research supported by the RIP-program Oberwolfach, 2002/2003.     

Blaschke products and singular functions with prescribed boundary values

This paper shows that there exists a Blaschke product having a prescribed radial limit at each point of a prescribed finite subset of the unit circle. In addition, an analogue for singular inner functions is proved; and an extension dealing with tangential limits is established.     

Maximal Blaschke products

We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari–Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.     

On Finite Blaschke Products whose Restrictions to the unit circle are Exact Endomorphisms

An easily checked sufficient condition is given for the restrictionof a finite Blaschke product to the unit circle to be an exactendomorphism. A formula for the entropy of such restrictionswith respect to the unique finite invariant measure equivalentto Lebesgue measure is given and it is shown that if such arestriction has maximal entropy then it is conformally equivalentto the product of a rotation and a power.     

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

A result of D.J. Newman asserts that a uniformly separated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characterizations of such sequences. If the zeros of a Blaschke product B lie in a Stolz angle, then for all and it has recently been shown that this result cannot be improved. Also, Newman's result can be used to prove that if B is an interpolating Blaschke product whose zeros lie in a Stolz angle, then 相似文献   

Abschätzung der Normen von Operatoren analytischer Funktionen

This article investigates the norms of certain interpolation operators of analytic functions on the unit disc. In particular, it is shown that the norms of interpolation operators being the identical operator for all n -degree polynomials have a lower bound of order ln n . This result is compared with a recent result regarding trigonometric interpolation of continuous functions on the unit circle. It is shown that opposed to the operators of analytic functions on the unit disc, the method of oversampling can be applied in order to uniformly bound the interpolation operators. Moreover, some practical implications with regard to communication engineering are discussed. It is concluded that in practice the results lead to non-linear interpolation operators.     

Polynomial Interpolation on the Unit Sphere and on the Unit Ball

The problem of interpolation on the unit sphere S ^d by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B ^d by polynomials of degree n. As a consequence several explicit sets of points on S ^d are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R ² for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.     

The scalar Nevanlinna-Pick interpolation problem with boundary conditions

We show that if the Nevanlinna-Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna-Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H_∞ norm solutions. Finally, some numerical examples are given.     

Harmonic interpolation of Hermite type based on Radon projections with constant distances

We study harmonic interpolation of Hermite type of harmonic functions based on Radon projections with constant distances of chords. We show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords coalesce to some points on the unit circle, we prove that the interpolation polynomials tend to a Hermite interpolation polynomial at the coalescing points.     

Conditional expectation and the Bergman projection

For any σ-algebra of measurable subsets of the unit disk generated by a finite Blaschke product, we prove that the associated conditional expectation operator commutes with the Bergman projection operator if and only if the σ-algebra is generated by a monomial. In the process, a formula for the conditional expectation operator (under certain assumptions) is obtained. When compared with earlier results of A.B. Aleksandrov concerning conditional expectation associated with σ-algebras of measurable subsets of the circle, our results exhibit a stark contrast between the way conditional expectation operators act in the Bergman and Hardy space settings.     

Bedrosian Identity in Blaschke Product Case

This paper offers a characterization of amplitude functions in L²(\mathbb R){L^2(\mathbb R)} satisfying the Bedrosian identity in the case that the phase functions are determined by the boundary value on the unit circle of finite Blaschke products.     

On Toeplitz operators similar to unilateral shifts

Let T be a Toeplitz operator on the Hardy space H² on the unit circle, and let the symbol of T be of the form ϕ/ψ, where ϕ is an inner function, ψ is a finite Blaschke product, and deg ψ ≤ deg ϕ. D. N. Clark proved that such an operator T is similar to an isometry. In this paper, we find necessary and sufficient conditions under which such an operator T is similar to a unilateral shift. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 85–104.     

Multiple interpolation by Blaschke products

Basic result: let {z_n} be a sequence of points of the unit disc and {k_n} be a sequence of natural numbers, satisfying the conditions: Then for any bounded sequence of complex numbers there exists a sequence such that the function interpolates : where B is the Blaschke product with zeros at the points _n ^(k)}, M is a constant,. if N=1 this theorem is proved by Earl (RZhMat, 1972, 1B 163). The idea of the proof, as in Earl, is that if the zeros {_n ^(k)} run through neighborhoods of the points z_n, then the Blaschke products with these zeros interpolate sequences , filling some neighborhood of zero in the space Z. The theorem formulated is used to get interpolation theorems in classes narrower than H.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 195–202, 1977.I thank S. A. Vinogradov for helpful discussions and interest in the work.     

有限BIR分解在相似下唯一的一类算子

本文证明了H~p(T)(1≤p≤∞)空间上以n阶Blaschke乘积为符号的Toeplitz算子相似于⊕_nT_z,从而证明了这类算子的有限BIR分解在相似下唯一．     

An improved order of approximation for thin-plate spline interpolation in the unit disc

Summary. We show that the -norm of the error in thin-plate spline interpolation in the unit disc decays like , where , under the assumptions that the function to be approximated is and that the interpolation points contain the finite grid . Received February 13, 1998 / Published online September 24, 1999     

Harmonic mappings onto stars

A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.     

On the Radial Behavior of Universal Taylor Series

In this work we deal with universal Taylor series in the open unit disk, in the sense of Nestoridis; see [12]. Such series are not (C,k) summable at every boundary point for every k; see [7], [11]. In the opposite direction, using approximation theorems of Arakeljan and Nersesjan we prove that universal Taylor series can be Abel summable at some points of the unit circle; these points can form any closed nowhere dense subset of the unit circle.