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Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]

100%

Saff E. B., Stahl H.

Banach Center Publications

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1995

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tom 31

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nr 1

329-348

EN

Let $r_n* ∈ ℛ_{nn}$ be the best rational approximant to $f(x) = x^α$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of $r_n*$ lie on the negative axis $ℝ_{<0}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function $e_n = f - r_n*$ on [0,1], and survey related convergence results.

2

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Best approximation in spaces of bounded linear operators

77%

Lewicki G.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Chapter 0...............................................................................................................................................................................5 0.1. Introduction..................................................................................................................................................................5 0.2. Preliminary results.......................................................................................................................................................9 Chapter I..............................................................................................................................................................................16 I.1. Best approximation in finite-dimensional subspaces of ℒ(B,D)....................................................................................16 I.2. Kolmogorov's type criteria for spaces of compact operators; general case.................................................................26 I.3. Criteria for the space $K(C_K(T))$.............................................................................................................................30 I.4. The case of sequence spaces....................................................................................................................................38 Chapter II.............................................................................................................................................................................43 II.1. Extensions of linear operators from hyperplanes of $l^{(n)}_∞$.................................................................................43 II.2. Minimal projections onto hyperplanes of $l^{(n)}_1$...................................................................................................52 II.3. Strongly unique minimal projections onto hyperplanes of $l^{(n)}_∞$ and $l^{(n)}_1$...............................................59 II.4. Minimal projections onto subspaces of $l^{(n)}_∞$ of codimension two......................................................................71 II.5. Uniqueness of minimal projections onto subspace of $l^{(n)}_∞$ of codimension two................................................75 II.6. Strong unicity criterion in some space of operators....................................................................................................79 Chapter III.............................................................................................................................................................................83 III.1. Extensions of linear operators from finite-dimensional subspaces I...........................................................................83 III.2. Extensions of linear operators from finite-dimensional subspaces II..........................................................................90 III.3. Algorithms for seeking the constant $W_m$..............................................................................................................97 References..........................................................................................................................................................................99 Index..................................................................................................................................................................................102 Index of symbols................................................................................................................................................................102

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Best approximations, fixed points and parametric projections

75%

Cardinali T.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2002

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tom 22

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nr 2

243-260

EN

If f is a continuous seminorm, we prove two f-best approximation theorems for functions Φ not necessarily continuous as a consequence of our version of Glebov's fixed point theorem. Moreover, we obtain another fixed point theorem that improves a recent result of [4]. In the last section, we study continuity-type properties of set valued parametric projections and our results improve recent theorems due to Mabizela [11].

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Optimal cubature formulas in a reflexive Banach space

63%

Vaskevich V.

Open Mathematics

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2003

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tom 1

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nr 1

79-85

EN

Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a finite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given reflexive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to infinity.

Ograniczanie wyników

1 Banach Center Publications

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Open Mathematics

1 Cardinali T.

1 Lewicki G.

1 Saff E. B.

1 Stahl H.

1 Vaskevich V.

1 2003

1 2002

1 1995

1 1994

Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]

Best approximation in spaces of bounded linear operators

Best approximations, fixed points and parametric projections

Optimal cubature formulas in a reflexive Banach space