A note on holomorphic functions and the Fourier-Laplace transform

Abstract

We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

“…Remark 6.6. A different approach to Proposition 6.2 (and also to the very definition of holomorphic Besov spaces) was proposed in [76], but some of the arguments given in the Appendix of that paper (e.g., p.266, lines [16][17][18]…”

Formulation of Time-Fractional Electrodynamics Based on Riemann-Silberstein Vector

“…However, since we operate outside L 2 (R), we may not directly apply classical methods and we should treat the function G r (ω) as a tempered distribution. Fortunately, there are tools that may be applied here as well: we are going to refer to the theorem proved in [68] and rephrased as Theorem 7 and discussed in [55].…”

“…Theorem 2 (see Theorem 3.8 in [68]). Suppose that F ∈ H(C + ) satisfies: (i) for each ρ 0 > 0 function F restricted to the set {s ∈ C : s > ρ 0 } is of order/type ≤ (2, 0) (ii) b = lim sup y→+∞ y −1 ln |(F(y)| is finite (iii) there exists R > 0 such that for all ρ ∈ (0, R] the function F ρ (ω) = F(ρ + iω) satisfies F ρ ∈ S .…”

Approximating Financial Time Series with Wavelets

“…It has many applications in the sciences and technology" (Korn & Korn 1967). More interesting information about the Fourier transform and the Laplace transform can be found, inter alia, in: (Phillips, Parr & Riskin 1995, Hilger 1999, Carlsson & Wittsten 2017.…”