Refinement Equations
There is a non-trivial integrable solution to a two scale difference equation with non-negative
coefficients involving f if and only if the corresponding random expansion is absolutely continuous. Utilizing this
characterization a simple description in terms of the coefficients is given of all refinement equations
with no more than six coefficients possessing an integrable solution.
The functional equation (D) is variously known as a dilation equation, a refinement
equation,
or a two scale difference equation. Its solutions are important in wavelets and in subdivision schemes.
Our interest is simply in the existence of a solution to (D) without regard to its potential use. We show that there
is a solution to (D) if and only if the measure describing Y, whose cumulative
distribution function F is absolutely continuous with respect to Lebesgue measure, and in that case, f can
be taken as the probability density of Y. The proof of the main theorem involves a
forest of trees related
to points in the complex plane.
See also the article by Deliu and Spruill "Existence results for refinement equations" pp 20-37 in volume 59 (2000) of the
journal
Aequationes Mathematicae
published
by Birkhauser Verlag<