DOIAbstract
AbstractThe Smirnov Class N+ of analytic functions on the disk has traditionally been studied with the complete metric topology given by ρ(f,g)= sup0<r<112π∫Tlog(1+∥f(reiθ)−g(reiθ)∥)dθ.However, the space can be represented as the union of certain weighted Hardy spaces H2(w) (the closure of the polynomials in L2(w dθ)), and hence given the inductive limit topology H.We give a new proof of Yanagihara's characterisation of the dual of (N+, ϱ). We prove that (N+, H) is an (incomplete) metrizable topological algebra in which Fourier series converge. We then study the individual spaces H2(w), prove asymptotic versions of Szegö's theorem and the Helson-Szegö theorem on these spaces, and characterise the universal multipliers of their duals
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