Convergence of arguments of Blaschke products in 𝐿_{𝑝}-metrics

Rybkin, A. V.

doi:10.1090/S0002-9939-1991-1010000-3

Convergence of arguments of Blaschke products in $L_ p$-metrics
HTML articles powered by AMS MathViewer

by A. V. Rybkin

Proc. Amer. Math. Soc. 111 (1991), 701-708

DOI: https://doi.org/10.1090/S0002-9939-1991-1010000-3

PDF | Request permission

Abstract:

It is shown that the naturally defined argument of a Blaschke product is a function which is the harmonic conjugate of an integrable function of constant sign. A direct construction of this function is obtained. This fact allows us to investigate the relation between conditions on the zeros of a Blaschke product and the convergence of the arguments of its partial finite subproducts in ${L_p}$-metrics, $0 < p \leq \infty$.

References

E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
Peter Colwell, Blaschke products, University of Michigan Press, Ann Arbor, MI, 1985. Bounded analytic functions. MR 779463
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
S. V. Hruščev and S. A. Vinogradov, Inner functions and multipliers of Cauchy type integrals, Ark. Mat. 19 (1981), no. 1, 23–42. MR 625535, DOI 10.1007/BF02384467
Paul Koosis, Introduction to $H_{p}$ spaces, London Mathematical Society Lecture Note Series, vol. 40, Cambridge University Press, Cambridge-New York, 1980. With an appendix on Wolff’s proof of the corona theorem. MR 565451
C. N. Linden, On Blaschke products diverging everywhere on the boundary of the unit disc, Proc. Amer. Math. Soc. 55 (1976), no. 1, 62–64. MR 393494, DOI 10.1090/S0002-9939-1976-0393494-8
N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1

The spectral shift function for a dissipative and selfadjoint operator, and trace formulas for resonances

53