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Problem set 10, Harmonic Analysis

Due April 12

Problem 1

For any $latex \xi\in\mathbb R$,
$latex \displaystyle \lim_{N\to\infty} \int_{-N}^N e^{-\pi (x+i\xi)^2}dx = \lim_{N\to\infty} \int_{-N}^N e^{-\pi x^2}dx = 1.$
(Hint: Consider the contour integral $latex \int_\gamma e^{-\pi z^2} dz = 0$ over the rectangle $latex \gamma$ with vertices $latex N, N+i\xi, -N+i\xi$ and $latex -N$.)

Problem 2

Let $latex u>0$. Then
  1. $latex \displaystyle \frac{1}{\pi} \int_{-\infty}^\infty \frac{e^{-2\pi iuv}}{1 + v^2} dv = \frac{1}{\sqrt\pi} \int_0^\infty \frac{1}{\sqrt s} e^{-s} e^{-\pi^2 u^2/s} ds$;
  2. $latex \displaystyle \frac{1}{\pi} \int_{-\infty}^\infty \frac{e^{-2\pi iuv}}{1 + v^2} dv = e^{-2\pi u}$.

Problem 3

Prove the Riemann-Lebesgue Lemma: If $latex f\in L^1(\mathbb R^d)$, then $latex \hat f(\xi) \to 0$ as $latex |\xi|\to\infty$.

Problem 4

If $latex \Phi(x) = e^{-\pi |x|^2}$, the collection $latex \{\Phi_t(x)\}_{t>0}$ if its dilations is a collection of better kernels.

Problem 5

If $latex f,g\in L^1(\mathbb R^d)$, then $latex \widehat{f*g}(\xi) = \hat f(\xi) \hat g(\xi)$. (Hint: Apply Fubini's theorem to $latex F(x,y) = f(x-y) g(y) e^{-2\pi i x\cdot\xi}$ on $latex \mathbb R^d\times\mathbb R^d$.)