5. Elongations of MRA-Based Wavelets

In the previous chapter, we have introduced the concept of an MRA by which we have constructed several types of orthonormal wavelets in \(L^{2}(\mathbb{R})\). However, the only example we have seen so far of a compactly supported wavelet has been the Haar wavelet. Recall that the Haar space V ₀ was generated by the Haar scaling function ϕ(t) = χ _[0,1](t) (see Example 4.​2.​2). Although this scaling function has many desirable properties such as short support, symmetry about the line t = 1∕2, and orthogonal to its translates, it is not continuous and its derivative is zero almost everywhere. Moreover, we saw that the analytic expression for the scaling function and wavelet is, in general, not available. Therefore, it is desirable to construct wavelets with greater degrees of smoothness and having compact support. In Section 5.2, we construct wavelets that are smooth and piecewise polynomial. Specifically, we construct a wavelet that is C ⁿ⁻¹ on \(\mathbb{R}\) and that is piecewise polynomial of degree n. These wavelets are called spline wavelets and the well-known Franklin and Battle-Lemarié wavelets are the special cases of these wavelets. Although B-splines are continuous and compactly supported, they fail to form an orthonormal basis. In Section 5.3, we develop the tools to construct orthonormal wavelets whose scaling functions are both differentiable and compactly supported. These wavelets were first constructed by Daubechies (1988b) that created a lot of excitement in the study of wavelets. Compactly supported wavelets possess certain desirable properties such as compact support, orthogonality, symmetry, smoothness, high order of vanishing moments, and so on. In Section 5.4, we construct another intersecting class of orthonormal wavelets called harmonic wavelets. Harmonic wavelets are complex functions and band-limited in the frequency domain, so that they can be used to analyze frequency changes as well as oscillations in a small interval of time. They are closely related to Shannon wavelets: their real part is an even function which is identical to the Shannon wavelet and their imaginary part is a kin but an odd function. Harmonic wavelets are also referred to as physical family of wavelets because they were proposed for the analysis of physical problems, particularly in the fields of vibration and acoustic analysis (see Newland 1993a). In the end, we present a novel and simple procedure for the construction of nonuniform wavelets associated with nonuniform MRA. In this nonstandard setting, the associated translation set is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair, and the associated dilation is an even positive integer related to the given spectral pair.