A geometric test for completeness

Tests of a geometric character are found for completeness and incompleteness of systems of the form $\{[W(\pm z)]^{2n}\}$, $n=0,1,2,\dots$, in the space of functions analytic in a simply connected domain $D\subset\mathbf C$ symmetric with respect to the origin. Using these results, a number of concrete problems are solved.
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