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Homework 5, Real Analysis 2

Due March 9

Problem 1

Let f be integrable, and for each $latex \alpha>0$ let $latex E_\alpha = \{x:|f(x)|>\alpha\}$. Then

$latex \displaystyle \int |f| = \int_0^\infty m(E_\alpha)d\alpha.$

Problem 2 (Riemann-Lebesgue Lemma)

For $latex f\in L^1(\R^d)$, let

$latex \displaystyle \hat{f}(\xi) = \int_{\R^d} f(x) e^{-2\pi i x\cdot\xi} dx$

be its Fourier transform. Then $latex \hat f(\xi) \to 0$ as $latex |\xi|\to0$.

Problem 3

Let $latex f,g\in L^1(\R^d)$.

  1. $latex (x,y) \mapsto f(x-y)g(y) \in L^1(\R^d\times\R^d)$.

  2. The convolution $latex \displaystyle f*g(x) = \int_{\R^d} f(x-y)g(y) dy$ is well defined for a.e. x.

  3. $latex ||f*g||_{L^1} \le ||f||_{L^1} ||g||_{L^1}$.

  4. $latex \widehat{f*g}(\xi) = \hat f(\xi) \hat g(\xi)$.

Problem 4

There does not exist $latex I\in L^1(\R^d)$ such that, for all $latex f\in L^1(\R^d)$,

$latex f*I = f.$

Problem 5

Let $latex f_n\to f$ in measure.

  1. If $latex f_n\ge 0$, then $latex \displaystyle \int f \le \liminf \int f_n$

  2. If there exists $latex g\in L^1$ such that $latex |f_n|\le g$ for all n, then $latex f_n\to f$ in $latex L^1$ and $latex \displaystyle \int f_n \to \int f$.