Homework 5, Real Analysis 2

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.
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