Coefficient stripping in the matricial Nehari problem

Kasahara, Yukio; Bingham, Ν. H.

doi:10.1016/j.jat.2017.04.002

Cited by 3 publications

(7 citation statements)

References 15 publications

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“…The implication (v) ⇒ (iv) can be fully proved by using the authors' results in . Let

γ = (γ_{1}, γ_{2}, \dots)

be a Nehari sequence.…”

Section: Proof and Relevant Resultsmentioning

confidence: 90%

“…So, it is associated with a Schur function

f_{n}

, via a measure

μ_{n}

satisfying . Then, by [, Theorem 1.1], the functions

f_{0}, f_{1}, f_{2}, \dots

obey Schur's algorithm with a constraint

κ_{n}^{L} h_{L} false(0 false) > 0, h_{R} false(0 false) κ_{n}^{R} > 0 .

As before,

κ_{n}^{L}

and

κ_{n}^{R}

are the leading coefficients of the orthonormal polynomials

φ_{n}^{L}

and

φ_{n}^{R}

for

μ = μ_{0}

, and the constraint uniquely determines

ρ_{n}^{L}

and

ρ_{n}^{R}

. By simple computation, the formula can be inverted as

f_{n} = {\{{(ρ_{n}^{R})}^{*}\}}^{- 1} true(z f_{n + 1} - ζ_{n}^{*} true) {(1 - ζ_{n} z f_{n + 1})}^{- 1} ρ_{n}^{L},

where

ζ_{n} = - ρ_{n}^{L} α_{n}^{*} {{(ρ_{n}^{R})}^{*}}^{- 1} = - {(ρ_{n}^{L})

…”

Section: Proof and Relevant Resultsmentioning

confidence: 99%

“…Let

γ = (γ_{1}, γ_{2}, \dots)

be a Nehari sequence. By [, Lemma (i)], a truncated sequence

{boldγ}_{n} \equiv false(γ_{n + 1}, γ_{n + 2}, \dots false)

remains a Nehari sequence for all

n = 0, 1, 2, \dots

. So, it is associated with a Schur function

f_{n}

, via a measure

μ_{n}

satisfying .…”

Section: Proof and Relevant Resultsmentioning

confidence: 99%

“…and so, by uniqueness of outer functions, The matrix valued function , called a -generating matrix, plays a crucial role in the Nehari problem, providing a linear fractional parametrization of the solution set  ( ) ; see [3,4,14]. Let 2 be a Hilbert space of -valued functions on with inner product ⟨ , ⟩ 2 = ∫ * .…”

Section: Wiener's Lemma a Function In ℳ Is Invertible Therein If It mentioning

confidence: 99%

“…Recently, based on the investigation for the Nehari problem by Adamjan-Arov-Kreȋn [2] and Adamjan [1], the authors [13,14] pointed out that matrix measures (with some constraint) are also parametrized by alternative sequences = ( 1 , 2 , …) of strictly contractive matrices (with some constraint), called Nehari sequences, and established some basic results on the correspondence among , and . This note presents Baxter's theorem in terms of a Nehari sequence, namely, has the properties just mentioned if and only if is summable, answering an open question posed by the second author [6].…”

Section: Introductionmentioning

confidence: 99%

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Matricial Baxter's theorem with a Nehari sequence

Kasahara

Bingham

2018

Mathematische Nachrichten

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In the theory of orthogonal polynomials, (non‐trivial) probability measures on the unit circle are parametrized by the Verblunsky coefficients. Baxter's theorem asserts that such a measure is absolutely continuous and has positive density with summable Fourier coefficients if and only if its Verblusnky coefficients are summable. This note presents a version of Baxter's theorem in the matrix case from a viewpoint of the Nehari problem.

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“…The implication (v) ⇒ (iv) can be fully proved by using the authors' results in . Let