Refinable function

In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function ${\displaystyle \varphi }$ is called refinable with respect to the mask ${\displaystyle h}$ if

${\displaystyle \varphi (x)=2\cdot \sum _{k=0}^{N-1}h_{k}\cdot \varphi (2\cdot x-k)}$

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

Using the convolution (denoted by a star, *) of a function with a discrete mask and the dilation operator ${\displaystyle D}$ one can write more concisely:

${\displaystyle \varphi =2\cdot D_{1/2}(h*\varphi )}$

It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back. There is a similarity to iterated function systems and de Rham curves.

The operator ${\displaystyle \varphi \mapsto 2\cdot D_{1/2}(h*\varphi )}$ is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not uniquely defined. That is, if ${\displaystyle \varphi }$ is a refinable function, then for every ${\displaystyle c}$ the function ${\displaystyle c\cdot \varphi }$ is refinable, too.

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