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Sparse Representation of Signals in Hardy Space

Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Mathematically, signals can be seen as functions in certain spaces. And processing is more efficient in a sparse representation where few coefficients reveal the information. Such representations are constructed by decomposing signals into elementary waveforms. A set of all elementary waveforms is called a dictionary. In this chapter, we introduce a new kind of sparse representation of signals in Hardy space \(\it{H}^2(\mathbb{D})\) via the compressed sensing (CS) technique with the dictionary
$$\begin{array}{lll}\mathfrak{D} = \begin{array}{llll}\big\{e_{a}\;:\;e_{a}(z)\;=\; \frac{{\sqrt {1\, - \,\left| a \right|^2 } }} {{1\, - \,\bar az}},\;a\;\in\;\mathbb{D}\big\}\end{array}.\end{array}$$
where ⅅ denotes the unit disk. In addition, we give examples exhibiting the algorithm.

Keywords

Hardy space compressed sensing analytic signals reproducing kernels sparse representation redundant dictionary l1minimization. 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MacauMacauPeople’s Republic of China

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