The Rudin kernels on an arbitrary domain

Saburou Saitoh

Vol. 83 (1979), No. 1, 273–284

DOI: 10.2140/pjm.1979.83.273

Abstract

Let {G_n} (G₀ ∋ x,t) denote a regular exhaustion of an arbitrary domain G in the complex plane. For fixed x, t(∈ G), let R_t⁽ⁿ⁾(z,x), L_t⁽ⁿ⁾(z,x) and L_t⁽ⁿ⁾(z,x) denote the Rudin kernels of G_n, respectively. The convergence of the sequences {R_t⁽ⁿ⁾(z,x)}, {L_t⁽ⁿ⁾(z,x)} and {L_t⁽ⁿ⁾(z,x)} is discussed and some properties with respect to their limit functions are investigated. In the final Section, it is pointed oul that in the case of an arbitrary hyperbolic Riemann surface, the circumstances are quite different, in general.

Mathematical Subject Classification 2000

Primary: 30C35

Secondary: 30F99

Milestones

Received: 1 July 1974

Published: 1 July 1979