## A Study in Polynomial Approximation in the Complex Domain |

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_ I _ ANALYTIC CURVES Bergman kernel Carleman polynomials Carleman region Cauchy's integral formula closed subdomain continuous function convex hull defined Definition h.1 domain function eigen value ﬁg ﬁn fixed fk(z fn(z Fourier coefficient Gn(z hn(z Holder condition Holder's inequality I I I I I I I Q I O I I Q I I Q Q Ig(z integral ldzl leading coefficient Lemma limit points matrix maximum principle notation Parseval's theorem pk(z polynomial of degree polynomials of precise precise degree prove Q I I Q I Q qm(z qn(z rectifiable Jordan curve represent sequence of polynomials simply connected tn(z transfinite diameter