scholarly journals Boundary interpolation and approximation by infinite Blaschke products

2010 ◽  
Vol 107 (2) ◽  
pp. 305
Author(s):  
Isabelle Chalendar ◽  
Pamela Gorkin ◽  
Jonathan R. Partington
Keyword(s):  
General Situation ◽  
Blaschke Products ◽  
Boundary Interpolation ◽  
General Domain ◽  
Interpolation Problems ◽  
Constructive Proofs ◽  
Set Of Points ◽  
Interpolating Blaschke Products ◽  
Tangential Limits

This paper considers the problem of boundary interpolation (in the sense of non-tangential limits) by Blaschke products and interpolating Blaschke products. Simple and constructive proofs, which also work in the more general situation of $H^\infty(\Omega)$ where $\Omega$ is a more general domain, are given of a number of results showing the existence of Blaschke products solving certain interpolation problems at a countable set of points on the circle. A variant of Frostman's theorem is also presented.

Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

1962 ◽  
Vol 14 ◽  
pp. 334-348 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Blaschke Product ◽  
Common Knowledge ◽  
Holomorphic Functions ◽  
Blaschke Products ◽  
Radial Limit ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

Blaschke products in Lipschitz spaces

2009 ◽  
Vol 52 (3) ◽  
pp. 689-705 ◽  
Author(s):  
Miroljub Jevtić
Keyword(s):  
Blaschke Products ◽  
Lipschitz Spaces ◽  
Interpolating Blaschke Products

AbstractWe study the membership of Blaschke products in Lipschitz spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz angle. We prove several theorems that complement or extend the earlier works of Ahern and the author.

Boundary Interpolation with Blaschke Products of Minimal Degree

2006 ◽  
Vol 6 (2) ◽  
pp. 493-511 ◽  
Author(s):  
Gunter Semmler ◽  
Elias Wegert
Keyword(s):  
Minimal Degree ◽  
Blaschke Products ◽  
Boundary Interpolation

IDEALS GENERATED BY INTERPOLATING BLASCHKE PRODUCTS

Analysis ◽  
1994 ◽  
Vol 14 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Raymond Mortini
Keyword(s):  
Blaschke Products ◽  
Interpolating Blaschke Products

Cluster sets of interpolating Blaschke products

2005 ◽  
Vol 96 (1) ◽  
pp. 369-395 ◽  
Author(s):  
Pamela Gorkin ◽  
Raymond Mortini
Keyword(s):  
Blaschke Products ◽  
Cluster Sets ◽  
Interpolating Blaschke Products

scholarly journals On partial isometries with circular numerical range

2021 ◽  
Vol 8 (1) ◽  
pp. 176-186
Author(s):  
Elias Wegert ◽  
Ilya Spitkovsky
Keyword(s):  
Blaschke Product ◽  
Partial Isometry ◽  
Shift Operator ◽  
Numerical Range ◽  
Elliptic Functions ◽  
Blaschke Products ◽  
Partial Isometries ◽  
Unitary Similarity ◽  
Boundary Interpolation ◽  
Poncelet Polygons

Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

scholarly journals Positive Polynomials and Boundary Interpolation with Finite Blaschke Products

2021 ◽  
Author(s):  
Sergei Kalmykov ◽  
Béla Nagy
Keyword(s):  
Unit Circle ◽  
Specific Form ◽  
The Other ◽  
Geometric Representation ◽  
Polynomial Equations ◽  
Blaschke Products ◽  
Positive Polynomials ◽  
Boundary Interpolation ◽  
Finite Blaschke Products ◽  
System Of Polynomial Equations

AbstractThe famous Jones–Ruscheweyh theorem states that n distinct points on the unit circle can be mapped to n arbitrary points on the unit circle by a Blaschke product of degree at most $$n-1$$ n - 1 . In this paper, we provide a new proof using real algebraic techniques. First, the interpolation conditions are rewritten into complex equations. These complex equations are transformed into a system of polynomial equations with real coefficients. This step leads to a “geometric representation” of Blaschke products. Then another set of transformations is applied to reveal some structure of the equations. Finally, the following two fundamental tools are used: a Positivstellensatz by Prestel and Delzell describing positive polynomials on compact semialgebraic sets using Archimedean module of length N. The other tool is a representation of positive polynomials in a specific form due to Berr and Wörmann. This, combined with a careful calculation of leading terms of occurring polynomials finishes the proof.

scholarly journals Maximal subalgebra of Douglas algebra

1988 ◽  
Vol 11 (4) ◽  
pp. 735-741
Author(s):  
Carroll J. Gullory
Keyword(s):  
Blaschke Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Blaschke Products ◽  
Maximal Subalgebra ◽  
Interpolating Blaschke Product ◽  
Douglas Algebra ◽  
Necessary And Sufficient ◽  
Interpolating Blaschke Products ◽  
Douglas Algebras

Whenqis an interpolating Blaschke product, we find necessary and sufficient conditions for a subalgebraBofH∞[q¯]to be a maximal subalgebra in terms of the nonanalytic points of the noninvertible interpolating Blaschke products inB. If the setM(B)⋂Z(q)is not open inZ(q), we also find a condition that guarantees the existence of a factorq0ofqinH∞such thatBis maximal inH∞[q¯]. We also give conditions that show when two arbitrary Douglas algebrasAandB, withA⫅Bhave property thatAis maximal inB.

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