Refinable function

A refinable function is a function which fulfills the following self-similarity condition:

$\phi (x)=\sum _{k=0}^{N-1}a_{k}\phi (2x-k)$

This condition is called refinement equation or two-scale equation.

Using the convolution * of a function with a discrete mask and the dilation operator $D$ you can write more concisely:

$\phi =D_{1/2}(a*\phi )$

It means that the you obtain the function, again, if you convolve the function with a discrete mask and then scale it back. There is an obvious similarity to iterated_function_systems.

The operator Failed to parse (unknown function "\mapto"): {\displaystyle \phi\mapto D_{1/2} (a * \phi)} is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not defined. That is, if $\phi$ is a refinable function, then for every $c$ the function $c\cdot \phi$ is refinable, too.

These functions play a fundamental role in wavelet theory as scaling functions.

Properties

Values at integral points

Solve an eigenvector-eigenvalue problem.

Scalar products

Integral points of the autocorrelation of the function.

Smoothness

Villemoes machine