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In this chapter, we give an overview on multiresolution analysis, wavelet series and wavelet estimators in the classical setting. By `classical' or `first-generation' wavelets, we mean waveletsthat were constructed initially to analyze signals observed at equispaced design points and having a sample size which is a power of two.The `second-generation' wavelet basis presented in the subsequent chapters will release these two constraints.
If one wants to analyze a function of time with a series expansion, the first idea that comes probably into one's mind is to use a Fourier series, i.e. decompose the function into sine and cosine at different frequencies. In this process, we hope that only a few coefficients in the series will carry mostof the information about the signal. Certain smooth functions have such an `economical' Fourier expansion. However, for most functions, a good Fourier seriesapproximation requires numerous sine and cosine basis functions. Indeed, the sine functions have a precise frequency but are not localized in time, hence a localizedinformation in the signal like a discontinuity will affect all the coefficients of the series. This drawback lead the researchers to look for more efficient bases, that is,bases which are localized both in time and in frequency. We will see in Multiresolution analysis and wavelets that a wavelet basis offers this property.
This chapter is structured as follows. We begin by recalling some notations in Function spaces: notion and notations . Next Multiresolution analysis and wavelets introduces the multiresolution analysis, the wavelet functions, and gives some simple examples of wavelet bases. Fast wavelet transform explains how to decompose a signal using the wavelet transform. Such wavelet transforms, also called `decimated', lack the property of translation-invariance. Non-decimated wavelet transform presents a widely used trick to make a wavelet transform translation-invariant. Since the main goal of a wavelet series is to providea good approximation of a function belonging to a given space, Approximation of Functions introduces some fundamental notions to measure the quality of such approximation. Finally, Nonparametric regression with wavelets presents how to construct a nonparametric regression estimator using wavelets. First, the classical case of equally spaced design is considered. In the last part of Nonparametric regression with wavelets , we review some existing methods that deal with irregular designs.
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