CHAPTER 9 Approximations in Bases

It is time to wonder why we are constructing so many different orthonormal bases. In signal processing, orthogonal bases are of interest because they can provide sparse representations of certain types of signals with few vectors. Compression and denoising are applications studied in Chapters 10 and 11.

Approximation theory studies the error produced by different approximation schemes. Classic sampling theorems are linear approximations that project the analog signal over low-frequency vectors chosen a priori in a basis. The discrete signal representation may be further reduced with a linear projection over the first few vectors of an orthonormal basis. However, better nonlinear approximations are obtained by ...

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