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\begin{document}
\section*{Fourier data analysis}
Wavelets are tools mainly for time series and image analysis. It
has a very short history (1983 to present). Wavelets are small
waves (sinusoids are big waves). One object of wavelet
is to recover the mean response function $\mu$ from observed data
$$Y(t)= \mu(t) + \epsilon$$, where $\mathbb{E}\epsilon =0$. In a
classical fourier series expansion, we try to approximate $\mu$ as
a sum of linear sine or cosine functions, i.e.
$$\mu(t)= a_0 + a_1 \cos(\pi t) + a_2\cos(2\pi t) + \cdots.$$
It can be used to estimate a smooth function. It can be shown that
$$a_j=\sqrt{2}\int_0^1 \mu(t)\cos({\pi}jt) \; dt.$$

In general, any collection of functions $\psi_j$ that satisfies
$\int_0^1 \psi_i(t) \psi_j(t) \; dt = \delta_{ij}$ can be used as
building block for approximating functions in the following form
$$\mu(t)=\sum_{j=0}^{\infty} a_j\psi_j(t), \text{ where } a_j=\int_0^1 \mu(t)\psi_j(t)\; dt.$$
Further it is possible to make $\int_0^1 \psi_j(t) \; dt =0$.
$\psi_0(t)=1$ and $\psi_j(t)=\sqrt{2}\cos(\pi j t)$ would satisfy
the above conditions. Then for $N$ observations $Y_1,\cdots,Y_N$,
$a_j = \frac{1}{N}\sum_{i=1}^N \psi_j(\frac{i}{N})Y_i$. In this
example $a_0=\bar Y$ and $a_j=\frac{\sqrt{2}}{N}\sum_{i=1}^{N}
\cos(\frac{\pi ij}{N})Y_i$. Let $I=(1,2,\cdots,N)$ and
$J=(1,2,\cdots,M)'$, where $M$ is the terms in the approximation
we wish to use. $J \otimes I = (ji)$





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