## Two-scale difference equations. II: Local regularity, infinite products of matrices, and fractals.(English)Zbl 0788.42013

The authors represent the solution to the functional equation $$f(x)= \sum^ N_{n=0} c_ n f(kx- n)$$, where $$k\geq 2$$ is an integer and $$\sum^ N_{n=0} c_ n= k$$, in the time-domain, in terms of infinite products of matrices that vary with $$x$$. They give sufficient conditions on $$\{c_ n\}$$ for a continuous $$L^ 1$$-solution to exist, and additional sufficient conditions to have $$f\in C^ r$$. This representation is used to bound from below the degree of regularity of such an $$L^ 1$$-solution and to estimate the Hölder exponent of continuity of the highest-order well-defined derivative of $$f$$. In Part I [same journal 22, No. 5, 1388-1410 (1991; Zbl 0763.42018)] the authors used a Fourier transform approach to show that equations of the above type have at most one $$L^ 1$$-solution, up to a multiplicative constant, which necessarily has compact support in $$[0,N/(k-1)]$$.

### MSC:

 42C15 General harmonic expansions, frames 28A80 Fractals 39A10 Additive difference equations

Zbl 0763.42018
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