By H. Lipson, W. Cochran
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3. Let 1 ≤ p ≤ ∞ and let F ∈ Lp (∂D). Define f on the disk by 1 f (re ) = 2π π iϑ −π F (t)Pr (ϑ − t) dt. Then f is harmonic on D, and we have the following behavior towards the boundary. e. as r ր 1. Here again, fr (ϑ) = f (reiϑ ). (b) fr ∈ Lp(−π, π) for every 0 ≤ r < 1, and sup0≤r<1 fr Lp (−π,π) < ∞. In fact, fr Lp(−π,π) is a increasing function in r, that is, r1 < r2 implies fr1 Lp (−π,π) ≤ fr2 Lp(−π,π) . (c) If 1 ≤ p < ∞, then fr → F in Lp(−π, π) as r ր 1. If p = ∞, then ∗ fr ⇀ F in L∞ (−π, π) as r ր 1.
Moreover, the quadratic term in the second factor shows that there is a frequency chirp: The instantaneous frequency ωinst of a pulse f is d arg f (x). 14) we have defined by ωinst (x) = dx ωinst (x) = ω0 + 8b2D2 x, 1 + 16D22 b2 that is, the instantaneous frequency varies linearly in x. 13). 3. DISPERSION 19 fourth order dispersion (for φ(4) ), and so on. The influence of Dν = φ(ν) (ω0) is just not as easily described for ν ≥ 3. , the dispersion coefficients up to D6 are taken into account. Chapter 2 Hardy Spaces, LTI Systems and the Paley-Wiener Theorem There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied.
2. Let 0 < r < 1. The function 1 − r2 Pr (ϑ) = 1 − 2r cos ϑ + r2 is called Poisson kernel for the disk. , it holds true that (a) Pr (ϑ) ≥ 0. (b) 1 2π π −π Pr (ϑ) dϑ = 1 for 0 ≤ r < 1. (c) If 0 < δ < π, then limrր1 supϑ∈[−π,π]\[−δ,δ] |Pr (ϑ)| = 0. 1. HARDY SPACES The next theorem states that by convolving a function from Lp(∂D) with the Poisson kernel, one obtains a function that is harmonic on the disk D. In the following, it is convenient to identify Lp(∂D) with Lp(−π, π). Especially, if ϑ ∈ [−π, π] and f ∈ Lp(∂D), then we also write f (ϑ) instead of f (eiϑ ).
Determination of Crystal Structures by H. Lipson, W. Cochran