Editing Wavelet (section)

== Mother wavelet ==
For practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT) and in general for theoretical reasons, one chooses the wavelet functions from a subspace of the [[Lp space|space]] <math>L^1(\R)\cap L^2(\R).</math> This is the space of [[Lebesgue measure|Lebesgue measurable]] functions that are both [[Absolutely integrable function|absolutely integrable]] and [[Square-integrable function|square integrable]] in the sense that
<math display="block">\int_{-\infty}^{\infty} |\psi (t)| \, dt <\infty</math> and <math display="block">\int_{-\infty}^{\infty} |\psi (t)|^2 \, dt < \infty.</math>

Being in this space ensures that one can formulate the conditions of zero mean and square norm one:
<math display="block">\int_{-\infty}^{\infty} \psi (t) \, dt = 0</math> is the condition for zero mean, and
<math display="block">\int_{-\infty}^{\infty} |\psi (t)|^2\, dt = 1</math> is the condition for square norm one.

For ''ψ'' to be a wavelet for the [[continuous wavelet transform]] (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform.

For the [[discrete wavelet transform]], one needs at least the condition that the [[wavelet series]] is a representation of the identity in the [[Lp space|space]] ''L''<sup>2</sup>('''R'''). Most constructions of discrete WT make use of the [[multiresolution analysis]], which defines the wavelet by a scaling function. This scaling function itself is a solution to a functional equation.

In most situations it is useful to restrict ψ to be a continuous function with a higher number ''M'' of vanishing moments, i.e. for all integer ''m'' < ''M''
<math display="block">\int_{-\infty}^{\infty} t^m\,\psi (t)\, dt = 0.</math>

The mother wavelet is scaled (or dilated) by a factor of ''a'' and translated (or shifted) by a factor of ''b'' to give (under Morlet's original formulation):

<math display="block">\psi _{a,b} (t) = {1 \over \sqrt a}\psi \left( {t - b \over a} \right).</math>

For the continuous WT, the pair (''a'',''b'') varies over the full half-plane '''R'''<sub>+</sub> × '''R'''; for the discrete WT this pair varies over a discrete subset of it, which is also called ''affine group''.

These functions are often incorrectly referred to as the basis functions of the (continuous) transform. In fact, as in the continuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation uses a subtly different formulation (after Delprat).

Restriction：
# <math> \frac{1}{\sqrt{a}} \int_{-\infty}^\infty \varphi_{a1,b1}(t)\varphi\left(\frac{t-b}{a}\right) \, dt</math> when {{math|1=''a''<sub>1</sub> = ''a''}} and {{math|1=''b''<sub>1</sub> = ''b''}},
# <math>\Psi (t)</math> has a finite time interval