Let φ be an analytic function in the Hardy space H~p on the open unit disc Δ = {z: |z| < 1} for 0 < p ≤ ∞. It is classical that φ has a factorization φ = BSF, where B is a Blaschke product, S is a singular function, and F is an outer function. Specifically, these factors are B(z)=z~mΠμ_k(Z_k-z)/(1-zz_k), μ_k=(|z_k|)/(z_k), where m is the order of the zero of φ at the origin and z_1 , z_2, ... are the zeros of φ in Δ {0}; S(z)=exp{-∫_0~(2π) (e~(it)+z)/(e~(it)-z)dv(t)}, where v is a nonnegative measure singular with respect to Lebesgue measure; and F(z) = λ exp{1/(2π)∫_0~(2π)(e~(it)+z)/(e~(it)-z)log|φ(e~(it))| dt}, where λ is a unimodular constant. See [2] for a full description of these functions and their properties.
展开▼
