On Weights Which Admit Reproducing Kernel of Szegő Type

Żynda, Tomasz Łukasz

doi:10.3103/s1068362320050064

Cited by 6 publications

(3 citation statements)

References 7 publications

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“…This paper is a continuation of [10]. For the reader's convenience we will recall preliminaries from that article.…”

Section: Introductionmentioning

confidence: 99%

Continuous Dependence of Szegő Kernel on a Weight of Integration

Żynda,

Pasternak-Winiarski,

Sadowski

et al. 2023

Complex Anal. Oper. Theory

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The weighted Szegő kernel was investigated in a few papers (see Nehari in J d’Analyse Mathématique 2:126–149, 1952; Alenitsin in Zapiski Nauchnykh Seminarov LOMI 24:16–28, 1972; Uehara and Saitoh in Mathematica Japonica 29:887–891, 1984; Uehara in Mathematica Japonica 42:459–469, 1995). In all of these, however, only continuous weights were considered. The aim of this paper is to show that the Szegő kernel depends in a continuous way on a weight of integration in the case when the weights are not necessarily continuous. A topology on the set of admissible weights will be constructed and Pasternak’s theorem (see Pasternak-Winiarski in Studia Mathematica 128:1, 1998) on the dependence of the orthogonal projector on a deformation of an inner product will be used in the proof of the main theorem.

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“…This paper is a continuation of [10]. For the reader's convenience we will recall preliminaries from that article.…”

Section: Introductionmentioning

confidence: 99%

Continuous Dependence of Szegő Kernel on a Weight of Integration

Żynda,

Pasternak-Winiarski,

Sadowski

et al. 2023

Complex Anal. Oper. Theory

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“…There are, however, infinite-dimensional Hilbert spaces of functions which are not equipped with a reproducing kernel. Some examples can be found in [5] or [7].…”

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confidence: 99%

“…We could also consider the case of weights not bounded from above, but in such a situation some polynomials may not be elements of our space. Note also that in the case of weighted Bergman and Szegö spaces (see [5,7]), if a weight "goes to zero" at some point too quickly, then there is no reproducing kernel of the corresponding weighted space. In our case any weight is admissible, i.e.…”

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confidence: 99%

Three types of reproducing kernel Hilbert spaces of polynomials

Żynda

2023

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In this paper we will investigate reproducing kernel Hilbert spaces of polynomials of degree at most n with three different inner products: given by an integral with a weight, given by the sum of products of values of a polynomial at n + 1 points and given by the sum of products of coefficients of the same power. In the first case we will show that the reproducing kernel depends continuously on deformation of an inner product in a precisely defined sense. In the second and third case we will give a formula for the reproducing kernel.

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Comparison of Two Topologies on Weighted Szegő Space

Żynda

Pasternak-Winiarski

2023

Trends in Mathematics

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On Weights Which Admit Reproducing Kernel of Szegő Type

Cited by 6 publications

References 7 publications

Continuous Dependence of Szegő Kernel on a Weight of Integration

Continuous Dependence of Szegő Kernel on a Weight of Integration

Three types of reproducing kernel Hilbert spaces of polynomials

Comparison of Two Topologies on Weighted Szegő Space

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