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- W2095445921 abstract "where the exponents are non-negative real numbers instead of integers). First of all, for p > I I, the integral 4(p; t) is finite, unless q (z) 0 identically. It therefore provides (for each pair of the parameters p> I ) a real valued homomorphism of the multiplicative set of nonzero elements of the integral domain Ao ( of all functions holomorphic in the sense of [II] on the unit disc A, generalized as in [II]). The second result is that this homomorphism depends continuously on q5, in the sense that if q5n(z) converges uniformly to q+(z) on (suitable) compact sets in A, (which are never required to touch the boundary), then kn(p; ;) -40(P; ?) We make use of some properties of equicontinuous families of holomorphic functions. These properties were first utilized in connection with almost periodic functions by H. Bohr [VI], and later used by B. Jessen [III] in studying a function which we show below to be k(p; 0). Jessen and Tornehave [IV] establish convexity properties of c/(p; 0), and connect the points of increase of the derivative with the zeros of q5(z). We shall explore these matters in general, elsewhere. 2. Measures for semi-groups of functions. We review briefly some matters discussed in [I, ??5-6]. There S is a topological space and H is a class of real valued non-negative continuous functions, closed under multiplication, and containing all positive constants. Moreover, there is a compact set B in S such that every element of H attains its maximum value (for all of S) on B." @default.
- W2095445921 created "2016-06-24" @default.
- W2095445921 creator A5044711567 @default.
- W2095445921 date "1957-09-01" @default.
- W2095445921 modified "2023-09-24" @default.
- W2095445921 title "The Boundary Integral of Log ∣φ∣ for Generalized Analytic Functions" @default.
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- W2095445921 doi "https://doi.org/10.2307/1992865" @default.
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