Meromorphic Extendibility and Rigidity of Interpolation

Let T be the unit circle, f be an α-Hölder continuous function on T, α > 1/2, and A be the algebra of continuous function in the closed unit disk D that are holomorphic in D. Then f extends to a meromorphic function in D with at most m poles if and only if the winding number of f + h on T is bigger or equal to −m for any h ∈A such that f + h 6= 0 on T. 1. MAIN RESULTS Let g be a non-vanishing continuous function on a simple Jordan curve T . Denote by wT (g ) the winding number of g (T ) around the origin. That is, 2πwT (g ) is equal to the change of the argument of g on T when the curve T is traversed in the positive direction with respect to D , the interior domain of T . Denote byA (D) the algebra of functions continuous on D and holomorphic in D . Motivated by the work of Alexander and Wermer [2] and Stout [12], Globevnik [4] obtained the following characterization of functions in the disk algebra A := A (D), where D is the unit disk. Theorem 1 (Globevnik [4]). A continuous function f on the unit circle T extends holomorphically throughD if and only if wT( f +q)≥ 0 for each polynomial q such that f +q 6= 0 on T. A shorter proof, based on the notion of badly-approximable functions, was obtained by Khavinson [8]. The polynomials are a dense subalgebra ofA . Thus, for any h ∈A such that f +h 6= 0 on T, there exists a polynomial q satisfying |h − q |< | f + h| on T. Then (1) wT( f + q) =wT( f + h + q − h) =wT( f + h)+wT ‚