ABSTRACT
One of the benefits of working on equispaced data is the ability of a fast and easy analysis of the frequency contents of data, transforms and es- timators. Chapter 6 presents the Fourier framework for the design of wavelet transform from the frequency perspective. Whereas classical wavelet theory has a heavy Fourier component, readers not interested in this as- pect can easily skip this chapter. The chapter develops the two-scale equation and the corresponding wavelet transform in the Fourier domain. It then explores the interpretation of a Fourier analysis, and formulates the Heisenberg uncertainty principle. This principle puts a bound on how ac- curately a basis function can combine localisation in temporal or spatial position with localisation in frequency. The Fourier transform also carries information on the smoothness of the function in the original domain. More in particular, the decay of a Fourier transform can be linked to uniform Lip- schitz continuity. Another important operation studied from the frequency point of view is sampling, with the Nyquist-Shannon theorem. Finally, the orthogonal Shannon (or sinc) and Meyer wavelets are constructed from conditions imposed in terms of the Fourier analysis.
