Homework 15, Real Analysis

Due date: November 17

Problem 1

For each $n\in\Z$, let $e_n(x) = e^{2\pi i nx}$. Then

$\displaystyle \int_0^1 e_n(x) \overline{e_m(x)} dx = \begin{cases} 1 & n=m\\ 0 & n\not=m. \end{cases}$

Problem 2

For $f\in C([0,1])$, the sequence $\widehat f(n) \to 0$ as $|n|\to\infty$.

Problem 3

Let $f\in C^1([0,1])$ with $f(0)=f(1)$.

1. $\widehat{f'}(n) = 2\pi i n \widehat f(n)$
2. The Fourier series of f converges uniformly to f.

Problem 4

Let $E\subset \R$ and $U_n$ the open set

$U_n = \{x\in\R: d(x,E)<1/n\}.$
1. If E is compact, $|E| = \lim |U_n|$.
2. However, the previous conclusion may be false if either E is closed and unbounded, or bounded and open.

Problem 5

Let E be the subset of $[0,1]$ of numbers which do not have the digit 4 in their decimal expansion. Find $|E|$.

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