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

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:

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.

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