Two-Scale Difference Equations. I. Existence and Global Regularity of Solutions

A two-scale difference equation is a functional equation of the form $f(x) = \sum _{n = 0}^N c_n f(\alpha x - \beta _n )$, where $\alpha > 1$ and $\beta _0 < \beta _1 <\cdots <\beta _n $, are real constants, and $c_n $ are complex constants. Solutions of such equations arise in spline theory, in interpolation schemes for constructing curves, in constructing wavelets of compact support, in constructing fractals, and in probability theory. This paper studies the existence and uniqueness of $L^1 $-solutions to such equations. In particular, it characterizes $L^1 $-solutions having compact support. A time-domain method is introduced for studying the special case of such equations where $\{ {\alpha ,\beta _0 , \cdots ,\beta _n } \}$ are integers, which are called lattice two-scale difference equations. It is shown that if a lattice two-scale difference equation has a compactly supported solution in $C^m (\mathbb{R})$, then $m < {{(\beta _n - \beta _0 )} / {(\alpha - 1)}} - 1$.