Estimates of the Singular Numbers of the Carleson Imbedding Operator

Parfenov, O. G.

doi:10.1070/sm1988v059n02abeh003148

Cited by 36 publications

(38 citation statements)

References 9 publications

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“…Moreover, it works more generally for Schur functions whose image lies in polygon inscribed in the unit disk. This upper bound appears, in a different context and under a very cryptic form, in [19]. First note the following simple lemma.…”

Section: Definition 11 (Lens Maps)mentioning

confidence: 94%

Some new properties of composition operators associated with lens maps

Lefèvre

Queffélec

et al. 2012

Isr. J. Math.

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Section: Definition 11 (Lens Maps)mentioning

confidence: 94%

Some new properties of composition operators associated with lens maps

Lefèvre

Queffélec

et al. 2012

Isr. J. Math.

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“…For K, a compact subset of C disjoint from T, and f, a general analytic function in C "K, remarkable results of Parfenov and Prokhorov [29,30] (formerly Gonchar's conjecture) assert that Theorem 1 remains true if supp(+) is replaced in (2.11) by K and lim sup replaced by lim inf . Results of the type given in Theorem 1 and also in the Theorems 3 and 4 have been proved in [15], but there the best rational approximants are defined on a real interval E disjoint from supp(+).…”

Section: Resultsmentioning

confidence: 99%

Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

Baratchart

Stahl²,

Wielonsky

2001

Journal of Approximation Theory

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Let f (z)= (t&z)&1 d+(t) be a Markov function, where + is a positive measure with compact support in R. We assume that supp(+)/(&1, 1), and investigate the best rational approximants to f in the Hardy space H 0 2 (V), where V :=[z # C | |z|>1] and H 0 2 (V) is the subset of functions f # H 2 (V) with f ( )=0. The central topic of the paper is to obtain asymptotic error estimates for these approximants. The results are presented in three groups. In the first one no specific assumptions are made with respect to the defining measure + of the function f. In the second group it is assumed that the measure + is not too thin anywhere on its support so that the polynomials p n , orthonormal with respect to the measure +, have a regular n th root asymptotic behavior. In the third group the defining measure + is assumed to belong to the Szego class. For each of the three groups, asymptotic error estimates are proved in the L 2 -norm on the unit circle and in a pointwise fashion. Also the asymptotic distribution of poles, zeros, and interpolation points of the best L 2 approximants are studied. Academic PressKey Words: best rational approximation in the L 2 -norm on the unit circle; asymptotic error estimates; Markov's theorem.

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“…If G is a domain (m = 1), it follows from the result of Zaharjuta and Skiba regarding the n-widths (see [20] and also [9]) that…”

Section: Lemma 8 Let G Be An M-domain and Supposementioning

confidence: 96%

“…Estimate (1.3) is known as Gonchar's conjecture [7]. Parfenov [9] gives a proof of (1.1) and (1.3) for the case where E is a continuum with connected complement. In [12], the second author proves (1.1) and (1.3) for an arbitrary compact set E. This paper is organized as follows.…”

Section: Corollary 2 Suppose E and F Are Disjoint Compact Subsets Ofmentioning

confidence: 99%

On estimates for the ratio of errors in best rational approximation of analytic functions

Kouchekian

Прохоров

2008

Trans. Amer. Math. Soc.

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Abstract. Let E be an arbitrary compact subset of the extended complex plane C with nonempty interior. For a function f continuous on E and analytic in the interior of E denote by ρ n (f ; E) the least uniform deviation of f on E from the class of all rational functions of order at most n. In this paper we show that if f is not a rational function and if K is an arbitrary compact subset of the interior of E, then n k=0 (ρ k (f ; K)/ρ k (f ; E)), the ratio of the errors in best rational approximation, converges to zero geometrically as n → ∞ and the rate of convergence is determined by the capacity of the condenser (∂E, K). In addition, we obtain results regarding meromorphic approximation and sharp estimates of the Hadamard type determinants.

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Estimates of the Singular Numbers of the Carleson Imbedding Operator

Cited by 36 publications

References 9 publications

Some new properties of composition operators associated with lens maps

Some new properties of composition operators associated with lens maps

Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

On estimates for the ratio of errors in best rational approximation of analytic functions

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