On Generators of the Banach Algebra L1 0oo And Related Questions

Harold S Shapiro

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18 Nox it follows by a well Imovm theorora of Caratheodory that the inapping function P(z) is extendible so as to be a homeo- morphisra of the closed upper half -plane (including the boundary point oo) onto -^""^ + f~« Since f is rectifiable, F{x) is of bounded variation on (-00, 00), and the result follows from Theorem l\. „ VJe note that the norm of the function f obtained is less than tr times the perimeter of -O- « Concerning Remark 2 following Theorem Ij. , it is evident t...hat Theorem 5 implies a stronger theorem than Theorem l\. Which may be formulated in terms of the Riemann surfac e onto which F maps the upper half -plane, and this surface need not be one-sheeted as assiomed in Theorem [|. « Proof of Theorem 3 a V/e have established that, for the sub- algebra A of A consisting of those F vjhich are of boxinded variation on - 00

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