Acta Appl Math (2009) 107: 1–3 DOI 10.1007/s10440-009-9457-x

Preface Dorin Dutkay · Deguang Han · Palle Jorgensen · Qiyu Sun

Received: 16 January 2009 / Accepted: 16 January 2009 / Published online: 24 January 2009 © Springer Science+Business Media B.V. 2009

The broad areas of mathematics involved in this special issue are harmonic analysis (more precisely computational harmonic analysis), but understood broadly to include representations of non-Abelian groups; functional analysis; the theory of linear operators in Hilbert space (now including the analysis of bases and frames in Hilbert space); symbolic dynamics; digital quantization; numerical analysis; approximation theory; and sampling. The coming together of these broad fields in the past two decades was inspired to a large degree by the study of wavelets. The reference to wavelets is to be understood in a wide sense, so including the explicit construction of bases in function spaces (typically Hilbert spaces). In fact these variety of constructions divide up roughly in two classes: The first begins with a specific system of operators representing a choice of scaling and of translation, in the sense of S. Mallat, Y. Meyer, and I. Daubechies. The second method again arrives at practical and computable bases, including generalized bases (called frames) but is now instead based on a different duality system of operators, usually referred to as a time-frequency duality in the sense of D. Gabor. The time-frequency wavelets are continuous-time (analog) signals, and so they related more directly to classical harmonic analysis. Although, historically the time-frequency theory began with the continuous case, by now, almost all practically useful discrete wavelet transforms use discrete-time filter-banks (details below). D. Dutkay · D. Han · Q. Sun () Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA e-mail: [email protected] url: http://math.ucf.edu/~qsun D. Dutkay e-mail: [email protected] D. Han e-mail: [email protected] P. Jorgensen Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA e-mail: [email protected]

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Dennis Gabor is known for electron optics, and the invention of holography: a tool for perfecting optical imaging, dividing image information into amplitude and phase. Specifically, the use of amplitude in optical imaging, as well as image phase; and resulting in a complete holo-spatial picture. In the 1950s, Gabor further researched how human beings communicate and hear; the result of his investigations was the theory of granular synthesis; again a synthesis technique, albeit of a different kind. Starting in the 1980s, wavelets emerged on the scene as a mathematical theory. Here the starting point is a function, or a system of functions. Wavelet and wavelet transforms are used in the analysis and synthesis problems arising naturally in a host of mathematical areas, as well as in applications. With the use of wavelets, one is able to achieve a recursive and computable subdivision of a given function, or a continuous-time signal, into different frequency components, and then to study and process each component with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets in turn are scaled and translated copies of a single function (the resulting doubleindexed system of functions is known as “daughter wavelets”). The generating function is of finite-length and/or fast-decaying oscillating waveform (known as the “mother wavelet”). Wavelet transforms have numerous advantages over traditional Fourier transforms for representing functions with discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals. This list of advantages includes the following two: Wavelets are localized and recursive by their nature (referring to some chosen resolution system which possesses a specific scaling similarity). These two traits make analysis and synthesis with wavelets easy to program; and with digitized data that compresses well. In fact a typical wavelet representation has much fewer nonzero terms, and fewer significant terms than does a Fourier representation. Hence wavelets are used in the digitizing of fingerprints. The computation of wavelet coefficients may be obtained as a result of fast and efficient matrix operations, perfectly suited for numerical analysis. And in fact the matrices involved in this scheme are slanted in a way the makes the necessary matrix multiplications faster; even if the matrices involved are large. The resulting matrices become more and more sparse in each step in a numerical wavelet algorithm. So typically, the complexity in matrix-based wavelet algorithms is of the order n log n, as opposed to n2 , or n!, which is typical for traditional matrix operations. More importantly, the wavelet matrix operations are repeated and sorted with the use of so called “filter-banks.” To understand the subject as it evolved over the last two decades, one must keep in mind this symbiotic relationship between function theory and such applications as signal processing (quadrature-mirror filters, high-pass, low-pass, frequency bands), image processing (systems of resolution, level of detail in a digital image), vision (zooming and focus within a scale of resolutions), data-mining, and artificial intelligence. This interplay between the mathematical theory on the one hand, and engineering developments on the other has fuelled both, with each side benefiting from ideas derived on the other side. The engineering tools might at their inception have been invented for one purpose, but they have subsequently found exciting uses in mathematics; even uses that could not have been predicted initially. The special issue contains the latest developments on the following five topics: (1) (2) (3) (4) (5)

General frame theory Construction of wavelets Topological properties of wavelet frames Fractals, self-similarity and applications Sampling and signal processing

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The papers in this special issue are selected in order to present a subject which on the one hand is rich in diversity of strands, and at the same time forms an attractive unity; with the different strands linked together by a common core of theoretical tools. While each of the contributions in the issue presents new research, the authors have written for a wide audience, covering a spectrum of areas in both pure and applied mathematics. And the individual papers contain both motivation and applications. Some authors present tutorials, and all aim to communicate across boundaries between fields of specialization. Here is the short introduction for each paper which should help readers who might have an interest in taking a look at some of the papers, but perhaps don’t know where to begin. The topic of frame theory includes the paper on coorbit theory by Christensen and Olafsson, on perturbation of frame expansion by Li and Yan, and on Gabor analysis on discrete sets by Li and Lian. The topic of wavelet constructions contains the paper on smoothing Parseval frame wavelet sets by Benedetto and King, on multivariate interpolating refinable function vectors by Han and Zhuang, and on supersets of wavelet sets by Viriyapong and Sumetkijakan. The topic of topological properties of wavelet frames includes the paper on closure of the set of tight frame wavelets by Bownik, on path-connectivity of s-elementary frame wavelets by Dai, Diao and Li, on density and connectivity of wavelet frames by Han and Larson, and on connectivity of multivariate filters by Li. The topic of fractals, self-similarity and applications includes the paper on automata and graphs by Cho and Jorgensen, on affine iterated function systems by Dutkay and Jorgensen, and on encoding for robust analog/digital conversion by Jimenez and Wang. The topic of sampling and signal processing includes the paper on least squares for nonlinear signal model by Aldroubi and Zaringhalam, on robust reconstruction for signals with finite rate of innovation from noisy samples by Bi, Nashed and Sun, on sampling in shift-invariant spaces by Grochenig, Hogan and Lakey, and on localization and sampling of multiband signals by Izu and Lakey. The idea of this special issue arose from the Special Session on Wavelets and Sampling of the 32nd SIAM Southeastern-Atlantic Section Conference (SIAM SEAS 2008) held on March 14–15, 2008 at University of Central Florida. We thank all the participants at the special session of the conference, the authors who submitted manuscripts to this special issue, and all the referees for their hard work and conscientious refereeing. We also thank Lenny Gumapac in the editorial office of the journal for her constant help.