Two problems on removable sets for analytic functions

Abstract

Introduction. Two elementary theorems on continuation of analytic functions across a compactum CC=R 2 are stated, and to each an example is given to show that the sufficient condition is best possible. In I. we use methods from the theory of singular integrals to estimate a certain sum of analytic functions. In II. we use Fourier analysis in R 2 to estimate certain sums of functions that seem immune to direct methods (especially their higher derivatives). We make explicit use of the symbols of the operators O/Oy and 0/O~; we are impelled to a complicated construction of the exceptional set by the necessity of evaluating certain integrals arising as Fourier transforms.