Multipliers and Divisors of Cauchy—Stieltjes Integrals

Abstract

Let U be an open unit circle on the complex plane, let ә U be its boundary, and let z be the complex variable. In the present article we investigate the multiplicative properties of functions g representable in U by an integral of the Cauchy—Stieltjes type:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacI % cacqaHcpGvcaGGPaGaeyypa0Zaa8quaeaadaWcaaqaaiaadsgacaWG % nbGaaiikaiaadshacaGGPaaabaGaamiDaiabgkHiTiabek8awbaaaS % qaaiabgkGi2kablQIivbqab0Gaey4kIipakiaacIcacqaHcpGvcqGH % iiIZcqWIQisvcaGGPaGaaiilaaaa!4DDD!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$g(\varsigma ) = \int\limits_{\partial \cup } {\frac{{dM(t)}}{{t - \varsigma }}} (\varsigma \in \cup ),$$

(1)

in which M is a complex Borel measure on U. We denote the set of all such functions g by K. The set K is linear under the ordinary operations of addition of functions and multiplication of a function by a complex number.