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Multiresolution analysis of functions defined on the dyadic topological group. (English) Zbl 0894.42011

Let G be the dyadic group of all sequences x \(= (x_1,x_2,\ldots)\), \(x_i\in\{0,1\}\), \(i=1,2,\ldots\), equipped with the group operation x + y \(= (x_1\dot{+} y_1, x_2\dot{+} y_2, \ldots)\) (\(\dot{+}\) denotes addition modulo 2) and metrized with the distance \(\beta(\)x,y) \(= \sum_{i=1}^\infty 2^{-i}| x_i - y_i| = \sum_{i=1}^\infty 2^{-i}(x_i \dot{+} y_i)\). The author considers the Hilbert space \({\mathcal C}\) of all continuous real functions \(f\): G \(\to \mathbb R\) with the usual integral scalar product and the corresponding \(L_2\)-norm. The continuity here is defined with respect to the topology in G induced by the distance \(\beta(\cdot,\cdot)\). The author defines: (i) dyadic exponential functions \(\Lambda\in\mathcal C\), some of which are pixel functions of rank \(s\); (ii) Rademacher-type transformations \(r_s : {\mathcal C} \to {\mathcal C}\), \(s=1,2,\ldots\;\).
Let \(V_0\) denote the one dimensional linear subspace of \({\mathcal C}\) spanned over a given dyadic exponential function \(w_0\) having its \(L_2\)-norm equal to \(1\). The following theorem is proved.
Theorem. For every natural \(s\) define \(W_s = r_s(V_{s-1}), V_s = V_{s-1}\oplus W_s,\quad s=1,2,3,\ldots\), and \[ w_{2^{s-1} + p}(\mathbf{x}) = r_s(w_p;\text\textbf{x}), p=0,1,2,\ldots,2^{s-1}-1. \] Then \(V_{s-1} \bot W_s\), (1) is an orthonormal basis in \(W_s\) and \(\bigcup_{s=0}^\infty V_s\) is uniformly dense in \({\mathcal C}\).
The multiresolution analysis based on this theorem can be adapted to any individual function \(f\), choosing appropriately the dyadic exponential function \(w_0\) as a pixel function. For the corresponding optimization the author uses a criterion close to that one given by Coifman and Wickerhauser [see, e.g., R. R. Coifman, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 879-887 (1991; Zbl 0754.43001)].
Reviewer: T.Boyanov (Sofia)

MSC:

42C15 General harmonic expansions, frames
20F38 Other groups related to topology or analysis
68U10 Computing methodologies for image processing
22A10 Analysis on general topological groups

Citations:

Zbl 0754.43001
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