A Note on Bounded Interpolation in the Unit Disc

This report is in two self-contained parts. The first is a collection of research problems in complex analysis. Most of the problems were posed by participants at the London Mathematical Society Symposium on Potential Theory and Conformal Mapping held at the University of Durham from 2 July to 12 July, 1976 and sponsored by the Science Research Council. The problems marked with an asterisk were posed at problem sessions held at Queen Elizabeth College, London during the academic year 1975-76. The problems are numbered consecutively with the problems in reports of two earlier symposia, references [A] and [B] below, though two additional sections, on spaces of analytic functions and on interpolation and approximation, have been added.The second part of the report is a summary of a lecture given at the symposium by Professor W. K. Hayman, reporting progress on problems in the two previous reports. The authors wish to thank Professor Hayman for his assistance in compiling this report, and the members of the symposium for suggesting so many interesting and varied problems. On behalf of all the participants we express our appreciation to the Science Research Council for its generous support. PART I: NEW PROBLEMS 1. Meromorphic Functions 1.30: Can one establish an upper bound on the number of finite asymptotic values of a meromorphic function /(z) in C, taking into account both the order of/ and the angular measure of its tracts? (W. Al Katifi) 2. Entire Functions *2.47: Let E p be the linear space of entire functions / such that for some A > 0, B > 0; K p the family of functions k(z) positive and continuous on C with exp(yl|z| p ) = o(k(z)) as \z\ -• oo, for all A > 0; S a subset of C ; and || || fc s , || || fc the semi-norms defined f o r / e £ p , keK p by k,s = sup 5^ I k(z) l\M\ II/IU = sup -c [ k(z)We say that S is a sufficient set for E p if the topologies defined by the semi-norms {|| \\ k , keK p ], {|| \\ k>s ,keK p } coincide [13].

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“…This is certainly the case if the Blaschke sequence is uniformly separated, i.e., z m -z n (see e.g. [12]).…”

Section: < Imentioning

confidence: 98%

Research Problems in Complex Analysis

Anderson

Barth

Brannan

1977

Bulletin of the London Mathematical Society

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“…In this case, for any proper open subset Ω of C containing the spectrum of A, there is a function f that attains sup f ∈H(Ω) f (A) / f Ω , where H(Ω) denotes the set of analytic functions in Ω. Furthermore, if Ω is simply connected, the form of f is known [3,7,9]:…”

Section: Introductionmentioning

confidence: 99%

Some Extensions of the Crouzeix--Palencia Result

Caldwell¹,

Greenbaum²,

Li³

2018

SIAM J. Matrix Anal. & Appl.

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In [The Numerical Range is a (1 + √ 2)-Spectral Set, SIAM J. Matrix Anal. Appl. 38 (2017), pp. 649-655], Crouzeix and Palencia show that the closure of the numerical range of a square matrix or linear operator A is a (1 + √ 2)-spectral set for A; that is, for any function f analytic in the interior of the numerical range W (A) and continuous on its boundary, the inequalitywhere the norm on the left is the operator 2-norm and f W (A) on the right denotes the supremum of |f (z)| over z ∈ W (A). In this paper, we show how the arguments in their paper can be extended to show that other regions in the complex plane that do not necessarily contain W (A) are K-spectral sets for a value of K that may be close to 1 + √ 2. We also find some special cases in which the constant (1 + √ 2) for W (A) can be replaced by 2, which is the value conjectured by Crouzeix.

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“…Finally, since f (A) attains its norm at x, we have f (A) * f (A)x = f (A) 2 x, which gives (14). There is an analogous result for w-extremal functions f for (A, Ω).…”

Section: Orthogonality Properties For Extremal Functionsmentioning

confidence: 69%

Crouzeix's Conjecture and related problems

Bickel,

Gorkin,

Greenbaum

et al. 2020

Preprint

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In this paper, we establish several results related to Crouzeixs conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.

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A Note on Bounded Interpolation in the Unit Disc

Cited by 14 publications

References 5 publications

Research Problems in Complex Analysis

Research Problems in Complex Analysis

Some Extensions of the Crouzeix--Palencia Result

Crouzeix's Conjecture and related problems

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