Decomposing finite Blaschke products

Daepp, Ulrich; Gorkin, Pamela; Shaffer, Andrew; Sokolowsky, Benjamin; Voss, Karl

doi:10.1016/j.jmaa.2015.01.039

Cited by 10 publications

(4 citation statements)

References 17 publications

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“…The equivalence of (i) and (ii) in both theorems is proven in [2]. The equivalence of (ii) and (iii) in both theorems follows from Theorem 1.…”

Section: The Hexagon Casementioning

confidence: 71%

“…The next result will be very helpful in that regard. It is essentially an OPUC version of a result from [2].…”

Section: Background and Notationmentioning

confidence: 99%

“…Our focus is on finding those A ∈ S n−1 whose numerical range is bounded not just by an n-Poncelet curve but by an n-Poncelet ellipse. The relationship between numerical ranges of A ∈ S n−1 and Poncelet ellipses is a subject of significant ongoing research (see [2,3,4,7,8,12,13,15,17]). The following theorem (from [17]) is the starting point of our investigation and shows that the ellipses that we are looking for do in fact exist.…”

Section: Background and Notationmentioning

confidence: 99%

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On foci of ellipses inscribed in cyclic polygons

Hunziker¹,

Martínez–Finkelshtein²,

Poe³

et al. 2021

Preprint

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Given a natural number n ≥ 3 and two points a and b in the unit disk D in the complex plane, it is known that there exists a unique elliptical disk having a and b as foci that can also be realized as the intersection of a collection of convex cyclic n-gons whose vertices fill the whole unit circle T. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of n. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.

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“…The equivalence of (i) and (ii) in both theorems is proven in [2]. The equivalence of (ii) and (iii) in both theorems follows from Theorem 1.…”

Section: The Hexagon Casementioning

confidence: 71%

“…The next result will be very helpful in that regard. It is essentially an OPUC version of a result from [2].…”

Section: Background and Notationmentioning

confidence: 99%

Section: Background and Notationmentioning

confidence: 99%

See 1 more Smart Citation

On foci of ellipses inscribed in cyclic polygons

Hunziker¹,

Martínez–Finkelshtein²,

Poe³

et al. 2021

Preprint

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show abstract

“…1 question, is impractical. Various, more practical, criteria were given in [8] that also yield further insight as to what really makes a Blaschke product decomposable and how to decompose it.…”

Section: Introductionmentioning

confidence: 99%

Clark measures and a theorem of Ritt

Chalendar¹,

Gorkin²,

Partington³

et al. 2018

Math. Scand.

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We determine when a finite Blaschke product $B$ can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for $B$. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute.

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