## On exponential representations of analytic functions in the upper half-plane with positive imaginary part |

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absolute moments absolutely continuous admits a canonical analytic ANALYTIC FUNCTIONS angle assertion asymptotic expansions behavior belong to TTl bounded function canonical representation rel characteristic function class of functions closed subinterval comparison theorem converges correspondence 3.5 corresponding measure defined denote density du(X dv(X equivalent exists and equals exponential representation factorization of cp finite function cp give given hence imaginary inequalities J-co last equality Lebesgue measure left half-axis Lemma Lemma 11 lim cp lim inf lim sup limit Log cp lTl+(k main theorems obtain Obviously parameter point measure proof of Theorem properties real axis real number regular and positive regular path relations representation of cp respect to Lebesgue right half-axis satisfies Section sequence singular integral singular measure statement Stieltjes integrals sufficient condition Theorem A rel total mass transformed TTl(l uniform convergence uniformly bounded unit circle upper half-plane Verblunsky write X2+l