Frames, Riesz bases, and discrete Gabor/wavelet expansions

By Ole W. Christensen. 2001

146 citations recorded by the source corpus.

This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in L 2 (R). The concept of an orthonormal basis {fn}, allowing every function f ∈ L 2 (R) to be written f = cnfn for suitable coefficients {cn}, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients {cn} are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geophysics

View Paper: https://www.semanticscholar.org/paper/4342da8969b48277da7933ce03080f537bdc1747

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Citations in the corpus

Approximation of the Fourier Transform and the Dual Gabor Window (2005, methodology): “For details on Gabor systems we refer to [24, 25, 28], see also [8, 13, 31, 33, 40], [9, Sections 8-9], and [13, Chapters 3-4].”
Characterizations of g-frames and g-Riesz bases in Hilbert spaces (2008, methodology): “{Λj ∈ L (U ,V j )}j∈J is a g-frame for U with respect to C with bounds A and B if and only if {fj}j∈J is a frame for U with bounds A and B. From Theorem 2.4, we may obtain Theorem 2.5 in [ 4 ], Theorem 2.5 in [16], Theorem 2.1 in [17] and Theorem 2.1 in [18], where we also obtain the bounds of the g”
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator (2004, methodology): “We assume that the reader is familiar with the basic elements of frame theory and Riesz bases [18, 22, 14 ].”
A Survey on Multicarrier Communications: Prototype Filters, Lattice Structures, and Implementation Aspects (2012, methodology): “From the mathematical point of view, lattice staggering corresponds to utilizing Wilson bases on each real and imaginary parts of the scheme, which is equivalent to the linear combinations of two Gabor systems where δ(Λ) = 2 [7], [9], [17], [18].”