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

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.

Boundary Interpolation by Finite Blaschke Products

2006 ◽  
Vol 27 (1) ◽  
pp. 75-98 ◽  
Author(s):  
Pamela Gorkin ◽  
Robert C. Rhoades
Keyword(s):  
Blaschke Products ◽  
Boundary Interpolation ◽  
Finite Blaschke Products

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.

What sort of representation is conscious?

2002 ◽  
Vol 25 (3) ◽  
pp. 336-337 ◽  
Author(s):  
Zoltan Dienes ◽  
Josef Perner
Keyword(s):  
Mental Representations ◽  
Minimal Degree ◽  
Central Claim

We consider Perruchet & Vinter's (P&V's) central claim that all mental representations are conscious. P&V require some way of fixing their meaning of representation to avoid the claim becoming either obviously false or unfalsifiable. We use the framework of Dienes and Perner (1999) to provide a well-specified possible version of the claim, in which all representations of a minimal degree of explicitness are postulated to be conscious.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

Comparison of boundary interpolation methods on the geometrical modeling of viscosity for CaO-Al2O3-SiO2 melts

2021 ◽  
Vol 562 ◽  
pp. 120782
Author(s):  
Xiaomin Zha ◽  
Dexiang Hou ◽  
Zhigang Yu ◽  
Jieyu Zhang ◽  
Kuochih Chou
Keyword(s):  
Boundary Interpolation ◽  
Geometrical Modeling ◽  
Interpolation Methods

scholarly journals On varieties of almost minimal degree I: Secant loci of rational normal scrolls

2010 ◽  
Vol 214 (11) ◽  
pp. 2033-2043 ◽  
Author(s):  
M. Brodmann ◽  
E. Park
Keyword(s):  
Minimal Degree ◽  
Secant Loci
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