Wavelet and Frame Constructions, with Applications

Dear Colleagues,

We envision a collection of papers in applied and computational harmonic analysis with an emphasis on frame analysis. In technical terms, Hilbert space-frames generalize more standard notions of bases. They are configurations of vectors in Hilbert space, which form overcomplete spanning families; and, hence, they generalize more standard notions of bases. In frame analysis, we give up the orthogonality requirements in ONBs. The introduction and use of frames are dictated by such applications as decomposition of signals, such as images, sound, e.g., music, speech, big data in finance, astrophysics, etc. The advantage of frames over the orthonormal bases is due their flexibility, and their use in reconstruction, multi-band analysis, in sampling theory, and to their ability to account for redundancy and aliasing. In detail, referring to the analysis/synthesis problem: with the use of suitable frames, each Hilbert space vector (signal) can be represented in many different ways in terms of frame elements. This makes the frames very useful in algorithms for recovering of data, for example data lost in Internet coding (so called erasure), in error correction in coding and decoding theory, in minimizing of noise in signal and image processing, and in speech recognition, signal processing, quantum computing, coding theory, and sparse representations, and in electrical engineering. Riesz bases and orthonormal bases are examples of frames.

Frames are constructed on various settings such as Euclidean spaces, manifolds, Lie groups, finite groups, finite fields, graphs and fractals. The structure and role of a frame varies by its application to a given explicit problem, and its adaptation adapts well to the desired setting. Most well known frames are Gabor frames (in connection to signal processing), Grassmannian frames (in connection to the area of spherical code, with application to wireless communication and to multiple description coding), and probabilistic frames as probabilistic measure in relation to statistics, geometry of convex bodies, quantum computing, and compress sensing.

A special kind of Hilbert space frames is wavelets. The construction of wavelet frames takes advantage of similarity up to scale. The functions making up a wavelet frame are obtained by action of a pair of operator families, dilations and translations, and acting on a single function a so-called ``mother function". Wavelets are usually obtained with the use of suitable multiscale analyses; and are popular in applications and decomposition problems due to their efficiency, especially in time-frequency localization problems. The wavelets, as analytic tools, have found a great number of applications in astrophysics, and harmonic analysis, such as in geometric discrepancy theory and in the study of irregularities of distributions.

Prof. Palle E.T. Jorgensen
Dr. Azita Mayeli
Guest Editors

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