# Regularity Analysis

The Fourier transform analyses the global
regularity of a function.

The wavelet transform makes it possible to analyze
the pointwise regularity of a function.

A signal is regular if it can be locally
approximated by a polynomial. The definition of the Lipschitz
regularity is

Naturally, this a **global** regularity
condition.

To get conditions on the local or even pointwise
regularity of a signal, it is necessary to use a transform which is
localized in time.

Assume that the wavelet has n vanishing
moments:

and has n continuous derivatives with a fast
decay.

A fast decaying wavelet has n vanishing moments
if and only if its is the n^{th} derivative of a fast
decaying function.

If f is a function which is a little bit more than
n times differentiable at point v, then it can be approximated by a
polynomial of degree n. The wavelet transform of this polynomial is
zero; around v, its order is that of the error between the polynomial
and the function. If this error can be uniformly estimated on an
interval, this yields a tool for regularity analysis on an
interval.

This condition relates the pointwise regularity of
a signal to the decay of its wavelet transform's modulus.

It can be extended to an interval and, of course,
to the whole real axis.

### Example

A signal and its wavelet transform, computed with the
derivative of a Gaussian.

Finer scales are at top.

Zero coefficients are represented by a medium gray.

Hence, the regular parts are medium gray.

Notice the cones below the singularities.

Detection of
Singularities