# Contents

## Idea

Multiresolution analysis is an approach to construct wavelets.

Let’s say person A chooses a function ($f: [0, 1] \to [0, 1]$, $f$ sufficiently nice) and would like to tell person B about it. According to the philosophy of multiresolution analysis, A could, for starters, tell B the overall average of the function: $\int_0^1 f(x) \; d x$. In a second step, A tells B how much the function deviates from this average on the interval $[0, \frac{1}{2}]$ and on the interval $[ \frac{1}{2}, 1]$. Both players could continue this game until B can reconstruct the average of the function on any prescribed subinterval, so that she knows all values of the integral of f over every element of the Borel algebra of $[0, 1]$.

This is a multiresolution analysis of $f$ in the sense that one starts with information of averages of $f$ on a big scale, and adds information on smaller scales to it, until $f$ is completely specified. (It is a multiresolution in the precise sense of the word if one uses the Haar wavelet on $[0, 1]$.

## Details

### Definition

A multiresolution analysis is usually defined via 5 conditions, although we will see that these conditions are not completely independent. But for simplicity we will stick to the tradition.

###### Multiresolution analysis

A two-sided sequence of closed subspaces $V_j$ of $L^2(\mathbb{R})$ is a multiresolution (analysis), if

(1) $V_{j+1} \subset V_j$, that is the higher the index, the “coarser” is the approximation of a given function in it.

(2) for every $f \in V_j$ we have $f(\frac{t}{2}) \in V_{j+1}$, that is if we “spread” $f$ by a factor of 2, then the “next coarser” subspace contains it.

(3) $\lim_{j \to \infty} V_j = \{ 0 \}$

(4) $\overline{ \lim_{j \to -\infty} V_j} = L^2(\mathbb{R})$

(5) there is a function $\phi(t)$ in $V_0$ such that the set consisting of $\phi(t - k), k \in \mathbb{Z}$ is an orthonormal basis of $V_0$. This function is called a scaling function of the multiresolution.

The conditions are not independent:

###### Dependency of Conditions

Conditions (1), (2) and (5) imply (3).

For a proof see e.g. theorem 1.6 in

• Eugenio Hernández and Guido Weiss: A first course on wavelets. (ZMATH, first edition 1996)

### Wavelet Construction

In this paragraph we will sketch how one can construct a wavelet starting from a multiresolution analysis.

Let us first note that every $V_{j+1}$ is contained in $V_{j}$ and therefore has an orthogonal complement $W_{j+1}$, such that:

$V_j = V_{j+1} \oplus W_{j+1}$

Continuing this by induction letting the index on the right side go to $\infty$, we see that

$V_j = \cup_{k = j+1}^{\infty} W_{k}$

Now, when we let $j \to -\infty$, we see that

$L^2(\mathbb{R}) = \cup_{- \infty}^{\infty} W_{k}$

To find an orthonormal wavelet, therefore, all we need to do is to find a function $\psi$ such that the set $\{ \psi(t - k), k \in \mathbb{Z} \}$ is an orthonormal basis for $W_0$. If we find one, then we will get a basis of every $W_{-j}$ from $\{ 2^{\frac{j}{2}} \psi (2^j t - k), k \in \mathbb{Z} \}$, which shows that $\psi$ is a wavelet.

We will simply state the construction: From the assumptions we get that the following expansion has to exist:

$\frac{1}{2} \phi(\frac{1}{2} t) = \sum_{k = - \infty}^{\infty} a_k \phi(t -k)$

We can therefore a function that is usually called the low pass filter associated to a given multiresolution analysis:

$m_0(\chi) := \sum_{k = - \infty}^{\infty} a_k \exp(i k \chi)$
###### Wavelet Construction

Suppose $\phi$ is a scaling function for a given multiresolution analysis, and $m_0$ is the associated low-pass filter; then a function $\psi \in W_0$ is an orthonormal wavelet if and only if

$\hat{\psi} (2 \chi) = \exp{(i \chi)} \; \eta(2 \chi) \; \overline{m_0 (\chi + \pi)} \; \hat{\phi}(\chi)$

with a function $\eta$ that is measurable, $2 \pi$-periodic with $\| \eta \| = 1$ almost everywhere. (The hat denotes the Fourier transform.)

See wavelet.