An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables

Pascoe, J. E.

doi:10.1017/s0013091517000396

Cited by 15 publications

(21 citation statements)

References 12 publications

(24 reference statements)

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“…Some RIFs

ϕ

possess non‐tangential regularity at points on

T^{2}

that is stronger than the notion of a B‐point. To account for this, we recall the definition of the intermediate Löwner class, denoted

L^{J_{-}}

from . Technically, defines

L^{J_{-}}

using behavior at

(\infty, \infty)

.…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

“…To account for this, we recall the definition of the intermediate Löwner class, denoted

L^{J_{-}}

from . Technically, defines

L^{J_{-}}

using behavior at

(\infty, \infty)

. Here, we send

false(w_{1}, w_{2} false) \mapsto false(- \frac{1}{w_{1}}, - \frac{1}{w_{2}} false)

in the conditions and so, define

L^{J_{-}}

using behavior at

(0, 0)

as follows.…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

“…Definition Let

J

be a non‐negative integer, let

j = false(j_{1}, j_{2} false) \in {double-struckN}^{2}

, and set

false| j false| = j_{1} + j_{2}

. A two‐variable Pick function

g

is in the intermediate Löwner class at

(0, 0)

, denoted

L^{J_{-}}

, if

{trueprefixlim}_{s \to 0} false| g (i s, i s) false| = 0

and if there exists a multi‐index system of real numbers

{false{ρ_{j} false}}_{false| j false| ⩽ 2 J - 2}

such that

0true g (w) = \sum_{false| j false| ⩽ 2 J - 2} ρ_{j} w^{j} + O ((), {∥ w ∥}^{2 J - 1}) non-tangentially .

We follow and say non‐tangentially means that for each

c \in R

, the formula holds for all

w

sufficiently small that also satisfy

‖\frac{1}{w}‖ ⩽ c min \{- Im ((), \frac{1}{w_{1}}), - Im ((), \frac{1}{w_{2}})\} .

…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

“…To simplify notation, we often assume that the point

τ

under consideration is

(1, 1)

g false(w false) : = g_{(1, 1)} false(w false) = f (- \frac{1}{w_{1}}, - \frac{1}{w_{2}}),

where

f

is defined in and has a Type 1 Nevanlinna representation as in with associated

scriptH

A

Y

, and

α

. Working through the proof of [, Proposition 3.2], one can prove the following. Lemma Let

ϕ

be an RIF on

D^{2}

and assume

τ = (1, 1)

is a

B^{J}

point for

ϕ

.…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

See 3 more Smart Citations

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

Bickel

Pascoe

Sola

2017

Proc. London Math. Soc.

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Abstract. We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative H p membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in H p for p ≥ 3 2 and that higher nontangential regularity of a rational inner function paradoxically reduces the H p integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for H p . Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.

show abstract

“…Some RIFs

ϕ

possess non‐tangential regularity at points on

T^{2}

that is stronger than the notion of a B‐point. To account for this, we recall the definition of the intermediate Löwner class, denoted

L^{J_{-}}

from . Technically, defines

L^{J_{-}}

using behavior at

(\infty, \infty)

.…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

“…To account for this, we recall the definition of the intermediate Löwner class, denoted

L^{J_{-}}

from . Technically, defines

L^{J_{-}}

using behavior at

(\infty, \infty)

. Here, we send

false(w_{1}, w_{2} false) \mapsto false(- \frac{1}{w_{1}}, - \frac{1}{w_{2}} false)

in the conditions and so, define

L^{J_{-}}

using behavior at

(0, 0)

as follows.…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

“…Definition Let

J

be a non‐negative integer, let

j = false(j_{1}, j_{2} false) \in {double-struckN}^{2}

, and set

false| j false| = j_{1} + j_{2}

. A two‐variable Pick function

g

is in the intermediate Löwner class at

(0, 0)

, denoted

L^{J_{-}}

, if

{trueprefixlim}_{s \to 0} false| g (i s, i s) false| = 0

and if there exists a multi‐index system of real numbers

{false{ρ_{j} false}}_{false| j false| ⩽ 2 J - 2}

such that

0true g (w) = \sum_{false| j false| ⩽ 2 J - 2} ρ_{j} w^{j} + O ((), {∥ w ∥}^{2 J - 1}) non-tangentially .

We follow and say non‐tangentially means that for each

c \in R

, the formula holds for all

w

sufficiently small that also satisfy

‖\frac{1}{w}‖ ⩽ c min \{- Im ((), \frac{1}{w_{1}}), - Im ((), \frac{1}{w_{2}})\} .

…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

“…To simplify notation, we often assume that the point

τ

under consideration is

(1, 1)

g false(w false) : = g_{(1, 1)} false(w false) = f (- \frac{1}{w_{1}}, - \frac{1}{w_{2}}),

where

f

is defined in and has a Type 1 Nevanlinna representation as in with associated

scriptH

A

Y

, and

α

. Working through the proof of [, Proposition 3.2], one can prove the following. Lemma Let

ϕ

be an RIF on

D^{2}

and assume

τ = (1, 1)

is a

B^{J}

point for

ϕ

.…”

Section: Preliminaries: the Structure Of Rifsmentioning

confidence: 99%

See 2 more Smart Citations

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

Bickel

Pascoe

Sola

2017

Proc. London Math. Soc.

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show abstract

“…In [11], G. Knese describes boundary behavior of rational inner functions from the bidisk into the disk. In [12], J. E. Pascoe develops a method for constructing rational inner functions of a given level of regularity at the boundary. We anticipate that this work will lead to further extension of the generalized Hilbert space model approach to a larger set of boundary singularities.…”

mentioning

confidence: 99%

Analytic functions on the bidisk at boundary singularities via Hilbert space methods

Tully‐Doyle¹

2017

Operators and Matrices

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Abstract. We investigate the behavior of a generalized Hilbert space model of a function in the Schur class of the bidisk at singular boundary points that satisfy a growth condition. We examine the relationship between the boundary behavior of Schur functions and the geometry of corresponding generalized Hilbert space models. We describe a geometric condition on an associated operator that classifies the behavior of the directional derivative of the underlying Schur function at a carapoint.Mathematics subject classification (2010): 32A30, 32S05, 30E20, 47A56, 47A57.

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A note on polydegree (n, 1) rational inner functions, slice matrices, and singularities

Sola

2022

Arch. Math.

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We analyze certain compositions of rational inner functions in the unit polydisk $$\mathbb {D}^{d}$$ D d with polydegree (n, 1), $$n\in \mathbb {N}^{d-1}$$ n ∈ N d - 1 , and isolated singularities in $$\mathbb {T}^d$$ T d . Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a d-dimensional version of a question posed in Bickel et al. (Am J Math 144: 1115–1157, 2022) in the affirmative.

show abstract

An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables

Cited by 15 publications

References 12 publications

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

Analytic functions on the bidisk at boundary singularities via Hilbert space methods

A note on polydegree (n, 1) rational inner functions, slice matrices, and singularities

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