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Two-scale difference equations. I: Existence and global regularity of solutions. (English) Zbl 0763.42018
The authors use a Fourier transform approach to describe the existence and uniqueness of \(L^ 1\)-solutions to two-scale difference equations 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. In particular, \(L^ 1\)- solutions with compact support are characterized. Finally, compactly supported continuous solutions to lattice two-scale difference equations are studied. Global regularity of compactly supported solutions can be bounded above purely in terms of their support width. Two examples are considered: the de Rham continuous nowhere-differentiable function [communicated by Y. Meyer (1987)] and the Lagrange interpolation functions of Deslauriers and Dubuc [G. Deslauriers and S. Dubuc, Constructive Approximation 5, No. 1, 49-68 (1989; Zbl 0659.65004)].

MSC:
42C15 General harmonic expansions, frames
39A10 Additive difference equations
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
28A80 Fractals
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