An \(L^{p}\) analog to AAK theory for \(p\geq 2\)
type: Item
instance of: scholarly article
title: An \(L^{p}\) analog to AAK theory for \(p\geq 2\)
zbMATH Open document ID: 1050.47021
published in: Journal of Functional Analysis
publication date: 2002-09-08T00:00:00Z
review text: This paper deals with Hankel operators from the \(H^ s \) space of the unit circle into \({\overline H}^2_0 \), where \(1/p+1/s=1/2 \). Interpolating between the two natural cases \(p=2 \) and \(p=\infty \), the authors develop an \(L^ p \) analogue of the Adamyan-Arov-Kreĭn theory. In particular, they show that the distance to the set of operators of rank at most \(m \) is equal to the distance to the set of Hankel operators of rank at most \(m \). They also explain the relationship between the \(m\)-th singular value and the best approximation by meromorphic functions with at most \(m \) poles in the disk. Various applications to rational approximation are also given. There are related papers by \textit{C. Le Merdy} [Bull. Lond. Math. Soc. 25, 275--281 (1993; Zbl 0793.47024)] and by \textit{V. A. Prokhorov} [J. Approximation Theory 116, 380--396 (2002; Zbl 1017.47024)]. For the extensive theory of Hankel operators and related topics, we refer the reader to the monographs by \textit{S.~C.~Power} [``Hankel operators on Hilbert space'' (Research Notes in Mathematics 64, Pitman) (1982; Zbl 0489.47011)] and \textit{J.~R.~Partington} [``An introduction to Hankel operators'' (London Mathematical Society Student Texts 13, Cambridge University Press) (1988; Zbl 0668.47022)], and especially to the recent one by \textit{V.V.~Peller} [``Hankel operators and their applications'' (Springer Monographs in Mathematics, Springer Verlag, New York) (2003; Zbl 1030.47002)]. The latter contains a detailed account of the theory, including a summary of some results of the paper under review (see Chapter~4, Concluding Remarks on pp. 145--146). The paper under review contains a wealth of material. The list of sections describes very well the content: 1. Introduction, 2. Analytic Approximation, 3. A Hankel Operator Approach, 4. Extremal Functions and Maximizing Vectors, 5. Meromorphic Approximation, 6. Interpolation Between \(p=2 \) and \(p=\infty \), 7. Some Ljusternik-Schnirelman (Lyuster…
reviewed by: Dragan Vukotić
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 1797584
zbMATH Keywords: Hankel operators
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Adamyan-Arov-Krein theorem
zbMATH Keywords: singular numbers
zbMATH Keywords: best approximation
zbMATH Keywords: meromorphic approximation
author: L. Baratchart
author: Fabien Seyfert
MaRDI profile type: Publication
full work available at URL: jfan.2001.3860
OpenAlex ID: W2011779177
cites work: ANALYTIC PROPERTIES OF SCHMIDT PAIRS FOR A HANKEL OPERATOR AND THE GENERALIZED SCHUR-TAKAGI PROBLEM
cites work: Q4317796
cites work: Best rational approximation to Markov functions
cites work: Q5547883
cites work: On interpolation by rational functions
cites work: Q3139442
cites work: Best meromorphic approximation of Markov functions on the unit circle.
cites work: A criterion for uniqueness of a critical point in \(H_ 2\) rational approximation
cites work: Asymptotic uniqueness of best rational approximants of given degree to Markov functions in \({\mathcal L}^2\) of the circle
cites work: Rational approximation in the real Hardy space \(H_ 2\) and Stieltjes integrals: A uniqueness theorem
cites work: Q4039801
cites work: Q3803596
cites work: Q3999276
cites work: Abstract, weighted, and multidimensional Adamjan-Arov-Krein theorems, and the singular numbers of Sarason commutants
cites work: Q3999241
cites work: Q5618011
cites work: Q5328270
cites work: The n-width of sets of analytic functions
cites work: Q5508985
cites work: Q3922907
cites work: Q3788266
cites work: Q4134602
cites work: Q4039983
cites work: The CF table
cites work: Q5727906
cites work: On some extremal problems of the theory of analytic functions
cites work: Foundations of the theory of extremal problems for bounded analytic functions and various generalizations of them
cites work: Q3874458
cites work: Q5649693
cites work: THE DISTRIBUTION OF POLES OF RATIONAL FUNCTIONS OF BEST APPROXIMATION AND RELATED QUESTIONS
cites work: Extremum problems in the theory of analytic functions
cites work: The Adamjan-Arov-Krein Theorem on Hp -Spaces (2 ⩽ p ⩽ ∞) and on the Disc Algebra
cites work: On bounded bilinear forms
cites work: Q3713006
cites work: Continuity of the Operator of Best Uniform Approximation by Bounded Analytic Functions
cites work: ESTIMATES OF THE SINGULAR NUMBERS OF THE CARLESON IMBEDDING OPERATOR
cites work: Q3994887
cites work: Q4711043
cites work: Boundedness properties of the operators of best approximation by analytic and meromorphic functions
cites work: Hankel operators, best approximations, and stationary Gaussian processes
cites work: Superoptimal analytic approximations of matrix functions
cites work: Q3951096
cites work: Q3835682
cites work: Q4875343
cites work: On certain extremum problems for analytic functions
cites work: Q3711178
cites work: Q4360405
cites work: Algebras of functions on the unit circle
cites work: On covariance function tests used in system identification
cites work: The convergence of Padé approximants to functions with branch points
cites work: Q4317779
cites work: Inner and Outer Functions on Riemann Surfaces
cites work: Q5762274
cites work: Q4040100

An \(L^{p}\) analog to AAK theory for \(p\geq 2\) (cites work)
cites work: Q3706043

An \(L^{p}\) analog to AAK theory for \(p\geq 2\) (cites work)
cites work: Q3995594

An \(L^{p}\) analog to AAK theory for \(p\geq 2\) (DOI)
DOI: 10.1006/JFAN.2001.3860