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Abramowitz and Stegun Amer analytic and nonvanishing analytic continuation analytic function argC(z assume asymp asymptotic behavior bounded and analytic C(zQ Choose j'>0 sufficiently class of functions closed half plane conformal map constant containing a half continued analytically coset defined denote dw/w eizTT elzTT equation examples FINITE CONVOLUTION OPERATORS form C(z func function C(z half plane y>y hypothesis infinity of order initial branch integer Invariant subspaces Kalisch Lemma let C(z Let G linear subspace locally conformal log(-iz logarithmic m-valent maps some half maps the half Math monodromy theorem Nauk order of growth Paley-Wiener Theorem plane x^y problem of unicellularity proof of Theorem proved in 12 punctured disk range of C(z ReG(z region regular type regularly varying result is proved Sahnovic slowly varying Spectral analysis strong sense sufficiently large symbol C(z Theorem 2.3 tion totic univalent universal covering space varying at infinity Volterra operators