Preface
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Preface Sergei Silvestrov · Palle Jorgensen · Ola Bratteli · David Kribs · Gestur Olafsson
Received: 7 June 2009 / Accepted: 7 June 2009 / Published online: 12 November 2009 © Springer Science+Business Media B.V. 2009
Fractals are everywhere in nature and in technology: Each portion forming a reduced-size copy of the whole, a fractal is forever fragmented, both chaotic and ordered, endlessly complex: By viewing and varying the zoom, you see scale-similarity, and you see new definite patterns emerging in great variety, some deterministic and some chaotic; —emerging from the fabric of an environment in nature; —in such diverse fields as physics (called “renormalization groups” there!), quantum theory and also quantum algorithms manipulating elusive “qubits” made up of polarized photons; and in signal processing, in images, in wavelets, and in digital filters, —etc. This special issue began with an international research workshop in December 2006 at the research center in Banff, Canada. The focus of this special issue is interdisciplinary: mathematical foundations for wavelet analysis, dynamical and iterated function systems, spectral and tiling duality, fractal iteration processes and non-commutative dynamical systems, including quantum information. The basic methods are a combination of operator theory, operator algebras, harmonic analysis and representation theory. From seemingly unpredictable measurements, under closer scrutiny, one often discerns surprising scale-similarities, appearing from the mathematics; —as we navigate an eerie chaos, lurking behind a facade of order; —where what looks like a river . . . could be a misty fog; —with the mathematics revealing fundamental attributes of nature. S. Silvestrov () Lund University, Lund, Sweden e-mail: [email protected] P. Jorgensen The University of Iowa, Iowa City, USA O. Bratteli University of Oslo, Oslo, Norway D. Kribs University of Guelph, Guelph, Canada G. Olafsson Louisiana State University, Baton Rouge, USA
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When you look at fractals in a microscope, or in a telescope, hidden patterns appear as similar and repeated structures and features, emerging in different scales. They may be well hidden though, for example in huge data sets from the Internet; in digital searches; in output from data mining. Fractal analysis, image processing, and data mining are among the applications that reveal these features; —repeated at varying scales of resolution; and making up the fundamental constituents in a yet new and relatively uncharted domain of science. Fitting into this wider framework, we have wavelet theory. It stands on the crossroads between three interrelated areas, signal processing, operator theory, and harmonic analysis. It is concerned with the mathematical tools involved in digitizing continuous data with a view to storage and compression, and with the synthesis process, recreating the desired picture (or time signal) from the stored data. The algorithms involved go under the name of filter banks, and their spectacula
