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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
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