Spectral estimates of Cauchy's transform in L2(Ω)

J. Arazy; D. Khavinson

doi:10.1007/BF01203120

Spectral estimates of Cauchy's transform in L²(Ω)

J. Arazy, D. Khavinson

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the singular numbers of the Cauchy transform {Mathematical expression} on L²(D) are asymptotically {Mathematical expression}, while s_n(C_|L_a²(D))≈1/n (where L_a²(D) is the subspace of analytic functions in L²(D)). Also, the singular numbers of the logarithmic potential {Mathematical expression} on L²(D) are asympotically s_n(L)≈1/n, while s_n(L_|La²(D))≈1/n². Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform from L_L²(μ) into L²(ν) where μ and ν are rotation-invariant measures on {Mathematical expression}.

Original language	English
Pages (from-to)	901-919
Number of pages	19
Journal	Integral Equations and Operator Theory
Volume	15
Issue number	6
DOIs	https://doi.org/10.1007/BF01203120
State	Published - Nov 1992

Keywords

AMS Classification Numbers: 47B10, 46E22

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

Access to Document

10.1007/BF01203120

Cite this

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title = "Spectral estimates of Cauchy's transform in L2(Ω)",

abstract = "We prove that the singular numbers of the Cauchy transform {Mathematical expression} on L2(D) are asymptotically {Mathematical expression}, while sn(C|La2(D))≈1/n (where La2(D) is the subspace of analytic functions in L2(D)). Also, the singular numbers of the logarithmic potential {Mathematical expression} on L2(D) are asympotically sn(L)≈1/n, while sn(L|La2(D))≈1/n2. Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform from LL2(μ) into L2(ν) where μ and ν are rotation-invariant measures on {Mathematical expression}.",

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T1 - Spectral estimates of Cauchy's transform in L2(Ω)

AU - Arazy, J.

AU - Khavinson, D.

PY - 1992/11

Y1 - 1992/11

N2 - We prove that the singular numbers of the Cauchy transform {Mathematical expression} on L2(D) are asymptotically {Mathematical expression}, while sn(C|La2(D))≈1/n (where La2(D) is the subspace of analytic functions in L2(D)). Also, the singular numbers of the logarithmic potential {Mathematical expression} on L2(D) are asympotically sn(L)≈1/n, while sn(L|La2(D))≈1/n2. Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform from LL2(μ) into L2(ν) where μ and ν are rotation-invariant measures on {Mathematical expression}.

AB - We prove that the singular numbers of the Cauchy transform {Mathematical expression} on L2(D) are asymptotically {Mathematical expression}, while sn(C|La2(D))≈1/n (where La2(D) is the subspace of analytic functions in L2(D)). Also, the singular numbers of the logarithmic potential {Mathematical expression} on L2(D) are asympotically sn(L)≈1/n, while sn(L|La2(D))≈1/n2. Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform from LL2(μ) into L2(ν) where μ and ν are rotation-invariant measures on {Mathematical expression}.

KW - AMS Classification Numbers: 47B10, 46E22

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DO - 10.1007/BF01203120

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