By Robert C. Gunning, Hugo Rossi
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Extra info for Analytic Functions of Several Complex Variables
The following are equivalent. (i) There is u ∈ BV (ޒ+ ; X) such that f (λ) = du(λ) for all λ > 0; (ii) f ∈ C ∞ ((0, ∞); X) and ∞ sup λn |f (n) (λ)|/n! =: m1 (f ) < ∞. λ>0 n=0 If this is the case then Var u|∞ 0 = m1 (f ). 2 the following lemma is useful. 1 Suppose fn ∈ C 2 (J; X) satisﬁes fn → f in C(J; X) and |f¨n |0 ≤ M < ∞, for all n ∈ ގ, where J = [0, a]. Then fn → f in C 1 (J; X), f˙ ∈ Lip(J; X), and |f˙|Lip ≤ M . 20) Preliminaries 7 with c(a) = (a/4+4/a). 20) applied to g = fn −fm implies fn → f in C 1 (J; X), and from |f˙n (t) − f˙n (s)| ≤ |f¨n |0 |t − s| ≤ M (t − s), t, s ∈ J, with n → ∞ the result follows.
This shows the inclusion suppDf ⊂ σ(f ). 8. For this purpose let ∞ ψ ∈ C0∞ ( )ޒbe such that supp ψ ⊂ (−1, 1) , ψ ≥ 0, and −∞ ψ(ρ)dρ = 1; deﬁne ψn (ρ) = nψ(nρ). Then ψn → 1 as n → ∞, uniformly for bounded t, and ψn ∈ S. Since f is polynomially growing, fn = ψn f belongs to L1 ( ;ޒX), fn → f uniformly for bounded t, and |fn | ≤ |f | = g. 8 since f is growing polynomially. 1, (ii) yields σ(fn ) = suppfn ⊂ Bε (suppDf ) for n ≥ n(ε). In fact, if ϕ ∈ S is such that supp ϕ ⊂ Bε/2 (ρ0 ), then ∞ −∞ ∞ fn (ρ)ϕ(ρ)dρ = −∞ ∞ = −∞ ∞ fn (t)ϕ(t)dt = −∞ f (t)ψn (t)ϕn (t)dt f (t)ψn ∗ ϕ(t)dt = [Df , ψn ∗ ϕ]; with supp (ψn ∗ ϕ) ⊂ supp ϕ + supp ψn ⊂ Bε/2 (ρ0 ) + (−1/n, 1/n) we conclude [Df , ψn ∗ ϕ] = 0 if dist (ρ0 , supp Df ) ≥ ε and ε/2 > 1/n.
Let R+ (t) = R(t)e0 (t), R− (t) = R(−t) − R+ (−t), t ∈ ޒ, where e0 (t) denotes the Heaviside function. Then R+ and R− vanish for t < 0, hence their Laplace transforms are well-deﬁned, analytic for Re λ > 0, bounded and continuous on ¯ + , and tend to zero as |λ| → ∞. e. G(λ) is entire by Morera’s theorem, (cp. eg. Conway ), bounded and G(λ) → 0 as |λ| → ∞. e. R = R+ by uniqueness of the Fourier transform. 5 The Spectrum of Functions of Subexponential Growth Let f ∈ L1loc ( ;ޒX) be of subexponential growth, where X denotes a complex Banach space; by this we mean ∞ −∞ e−ε|t| |f (t)|dt < ∞, for each ε > 0.
Analytic Functions of Several Complex Variables by Robert C. Gunning, Hugo Rossi