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Due March 9

Problem 1

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

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

Problem 2 (Riemann-Lebesgue Lemma)

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

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

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

Problem 3

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

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

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

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

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

Problem 4

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

$f*I = f.$

Problem 5

Let $f_n\to f$ in measure.

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

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