Scaling functions ‘or connect the dots’

Fix filter no restrictions yet:

FUND. DEFN: Scaling Function

relates at two levels of resolution.

Basic condition:

Examples so far:

Box:

Tent centered at :

Daubechies D4: does there exist ?

Fractal example:

Dyadic rationals:

determined at dyadic rationals :

Convolution on integers? Powers of 2?

KNOW ALL , THEN KNOW ALL

Construct on all as limiting fixed point!

Iterative process: with limit

Construct sequence functions such that

Then

What about convergence? Pointwise, in Energy?

Pointwise: start with Tent function

In energy: start with Box function

Getting started:

Tent function centered at origin:

for suitable

Basic idea: set

Filter conditions:

Need in

so that

Conditions on :

Solve using Fourier Transforms as usual.

Fourier Transforms:

Set

Then

So:

Up-sampling again!

Recall

Crucial results:

where in z-transform notation:

Use these to compute .

Connect the dots! Daub-4

Depths: 1, 2, 4, 6

Cascade Algorithm: convergence in energy

Start with box function: can exploit orthonormality.

with

as before, but no Vetterli condition yet. So

Orthonormality:

Case: k = 0

Can we recognize sequence:

?

Finally Vetterli!

Consider first:

Crucial identification:

Fourier transform:

Finally Vetterli!

When

we deduce that

So , hence

ORTHONORMAL FAMILY for each k.

,

In the limit!

When

in energy, then

so Vetterli ensures

orthonormal family in .

Finally wavelets:

Assume convergence in energy and Vetterli.

Fix FIR filter

Set define wavelet by

compactly supported if compactly supported.

By same argument as for :

So need to identify .

More results for wavelets:

Recall

so,

By Vetterli yet again:so

Thus orthonormal family.

,

Still more results:

By same argument yet again:

But by Fourier transforms yet again:

where remember,

Thus: all .

.

Main Theorem Part 1:

FIR filter

If

then

is a continuous function with derivatives.

Main Theorem Part 2:

Suppose also satisfies Vetterli condition.

Define wavelet:

Then:

,orthonormal families,

1.

2.

3.

complete orthonormal family in .

,

.

so

hence Daub-4

continuous,

not quite differentiable

Main Theorem applied to Daub-4: