# Wavelets and signal analysis

Chui, Charles. K. (1992) Wavelets and signal analysis. Accademia Peloritana dei Pericolanti, Classe di Scienze FF. MM. NN., LXX . pp. 95-146.

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atti_3_1992_95.pdf - Submitted Version

The notion of multiresolution analysis (MRA) is a familiar concept to the approximation theorist. In fact, the family of $m^t^h$ order cardinal spline spaces $V_j^m, j \in \ \mathbb{Z} \$, with knot sequences $2^-^j \mathbb{Z} \$ is the most typical example being used to demonstrate the structure of an MRA. What has been somewhat neglected in approximation theory in the past is the study of the structure and implications of the complementary subspaces of an MRA. Any function that belongs to a set of generators of such complementary subspaces may be called a wavelet. The objective of this contribution is to give a brief review of wavelet analysis, with special emphasis on how it fits in and enhances the field of approximation theory, and why it has recently created much excitement in the mathematical and engineering communities. From the mathematical point of view, we will use cardinal splines to demonstrate the importance of the subject of wavelet analysis to the approximation theorist; and for engineering applications, we will restrict our discussion to why wavelet analysis has revolutionized the field of signal processing. Since there is already a vast literature, including at least three monographs and several edited volumes devoted to wavelet analysis, our presentation will be very brief.