15.13 Approximation and projections in hilbert space

$x\in \mathbb{R}^3$ , $V=\mathrm{span}(\{\begin{pmatrix}1\\ 0\\ 0\\ \end{pmatrix}, \begin{pmatrix}0\\ 1\\ 0\\ \end{pmatrix}\})$ , $x=\begin{pmatrix}a\\ b\\ c\\ \end{pmatrix}$ . So, $$y=\sum_{i=1}^2 (x\cdot b_i)b_i=a\begin{pmatrix}1\\ 0\\ 0\\ \end{pmatrix}+b\begin{pmatrix}0\\ 1\\ 0\\ \end{pmatrix}=\begin{pmatrix}a\\ b\\ 0\\ \end{pmatrix}$$

V = {space of periodic signals with frequency no greater than $3w_0$ }. Given periodic f(t), what is the signal in V that best approximates f?

1. { $\frac{1}{\sqrt{T}}e^{i{w}_{0}kt}$ , k = -3, -2, ..., 2, 3} is an ONB for V
2. $g(t)=\frac{1}{T}\sum_{k=-3}^3 (f(t)\cdot e^{i{w}_{0}kt})e^{i{w}_{0}kt}$ is the closest signal in V to f(t) ⇒ reconstruct f(t) using only 7 termsof its Fourier series .

Let V = {functions piecewise constant between the integers}

1. ONB for V.

$$b_i=\begin{cases}1 & \text{if $i-1\le t< i$}\\ 0 & \text{otherwise}\end{cases}$$ where { $b_i$ } is an ONB.

Best piecewise constant approximation? $$g(t)=\sum_{i=()}$$∞ ∞ f b i b i $$f\cdot b_i=\int_{()} \,d t$$∞ ∞ f t b i t t i 1 i f t

This demonstration explores approximation using a Fourier basis and a Haar Wavelet basis.See here for instructions on how to use the demo.