Szegö theorem, carathéodory domains, and boundedness of calculating functionals

ABDULLAYEV F., Dovgoshei A.

Mathematical Notes, vol.77, no.1-2, pp.3-14, 2005 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 77 Issue: 1-2
  • Publication Date: 2005
  • Doi Number: 10.1007/s11006-005-0001-7
  • Journal Name: Mathematical Notes
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.3-14
  • Keywords: Borel measure, BPE point of polynomials, Calculating functional, Compact support, Harmonic measure, Lebesgue decomposition, Lebesgue measure, Polynomial approximation
  • Uşak University Affiliated: No

Abstract

Suppose that G is a bounded simply connected domain on the plane with boundary Γ, z0 ε G, ω is the harmonic measure with respect to z0 on Γ, μ is a finite Borel measure with support supp(μ) ⊆ Γ, μa + μs is the decomposition of μ with respect to ω, and t is a positive real number. We solve the following problem: for what geometry of the domain G is the condition ∫ ln (dμa/dω) dω = - ∞ equivalent to the completeness of the polynomials in Lt(μ) or to the unboundedness of the calculating functional p → p(z0), where p is a polynomial in Lt(μ)? We study the relationship between the densities of the algebras of rational functions in Lt(μ) and C(Γ). For t=2, we obtain a sufficient criterion for the unboundedness of the calculating functional in the case of finite Borel measures with support of an arbitrary geometry. © 2005 Springer Science+Business Media, Inc.