Using the convolution * of a function with a discrete mask and the dilation operator you can write more concisely:
The operator is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not defined. That is, if is a refinable function, then for every the function is refinable, too.
These functions play a fundamental role in wavelet theory as scaling functions.
A refinable function is defined only implicitly. It may also be that there are several functions which are refinable with respect to the same mask. If shall have finite support and the function values at integer arguments are wanted, then the two scale equation becomes a system of simultaneous linear equations.
Let be the minimum index and be the maximum index of non-zero elements of , then one obtains
This is again a fixed-point equation. But this one can now be considered as an eigenvector-eigenvalue problem. That is, a finitely supported refinable function exists only (but not necessarily), if has the eigenvalue 1.
From the values at integral points you can derive the values at dyadic points, i.e. points of the form , with and .
If is refinable with respect to , and is refinable with respect to , then is refinable with respect to .
Computing the scalar products of two refinable functions and their translates can be broken down to the two above properties. Let be the translation operator. It holds
Because of the above property, is refinable with respect to , and its values at integral arguments can be computed as eigenvectors of the transfer matrix.
A refinable function usually has a fractal shape. The design of continuous or smooth refinable functions is not obvious. Before dealing with forcing smoothness it is necessary to measure smoothness of refinable functions. Using the Villemoes machine one can compute the smoothness of refinable functions in terms of Sobolev exponents.
In a first step the refinement mask is divided into a filter , which is a power of the smoothness factor (this is a binomial mask) and a rest . Roughly spoken, the binomial mask makes smoothness and represents a fractal component, which reduces smoothness again. Now the Sobolev exponent is roughly the order of minus logarithm of the spectral radius of .
The concept of refinable functions can be generalized to functions of more than one variable, that is functions from . The most simple generalization is about tensor products. If and are refinable with respect to and , respectively, then is refinable with respect to .
The scheme can be generalized even more to different scaling factors with respect to different dimensions or even to mixing data between dimensions. Instead of scaling by scalar factor like 2 the signal the coordinates are transformed by a matrix of integers. In order to let the scheme work, the absolute values of all eigenvalues of must be larger than one. (Maybe it also suffices that .)
Formally the two-scale equation does not change very much: