Iterated finite blaschke products and annular solutions of a functional equation

It is well known thatT(r,f)is differentiable at least forr>r0. We show that, in fact,T(r,f)is differentiable for all but at most one value ofr, and ifT(r,f)fails to have a derivative for some value ofr, thenfis a constant times a quotient of finite Blaschke products.

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Commuting finite Blaschke products

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385799130165 ◽

1999 ◽

Vol 19 (3) ◽

pp. 549-552 ◽

Cited By ~ 3

Author(s):

CARLOS ARTEAGA

Keyword(s):

Fixed Points ◽

Blaschke Products ◽

Finite Blaschke Products

We consider the set of finite Blaschke products $F$ for which the fixed points on the circle $S^1$ are expanding and we prove that if $F'(x) \ne F'(y)$ for all different fixed points $x,y$ of $F$ on $S^1$, then $F$ commutes only with its own powers.

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

Bulletin of the London Mathematical Society ◽

10.1112/blms/15.4.343 ◽

1983 ◽

Vol 15 (4) ◽

pp. 343-348 ◽

Cited By ~ 12

Author(s):

N. F. G. Martin

Keyword(s):

Unit Circle ◽

Blaschke Products ◽

Finite Blaschke Products

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Positive Polynomials and Boundary Interpolation with Finite Blaschke Products

Computational Methods and Function Theory ◽

10.1007/s40315-021-00430-4 ◽

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.

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