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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