How Smooth Is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism

A popular wavelet reference [W] states that “in theoretical and practical studies, the notion of (wavelet) regularity has been increasing in importance.” Not surprisingly, the study of wavelet regularity is currently a major topic of investigation. Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. We present an algorithm for computing the Sobolev regularity of wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of wavelets that are smoother than the Daubechies wavelets and have the same support. We establish smooth dependence of the regularity for wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional wavelets. 0. Introduction While the Fourier transform is useful for analyzing stationary functions, it is much less useful for analyzing non-stationary cases, where the frequency content evolves over time. In many applications one needs to estimate the frequency content of a nonstationary function locally in time, for example, to determine when a transient event occurred. This might arise from a sudden computer fan failure or from a pop on a music compact disk. The usual Fourier transform does not provide simultaneous time and frequency localization of a function. The windowed or short-time Fourier transform does provide simultaneous time and frequency localization. However, since it uses a fixed time window width, the same window width is used over the entire frequency domain. In applications, a fixed window width is frequently unnecessarily large for a signal having strong high frequency components and unnecessarily small for a signal having strong low frequency components. In contrast, the wavelet transform provides a decomposition of a function into components from different scales whose degree of localization is connected to the size of the scale Typeset by AMS-TEX 1 2 M. POLLICOTT AND H. WEISS window. This is achieved by integer translations and dyadic dilations of a single function: the wavelet. The special class of orthogonal wavelets are those for which the translations and dilations of a fixed function, say, ψ form an orthonormal basis of L(R). Examples include Haar wavelet, Shannon wavelet, Meyer wavelet, Battle-Lemari wavelets, Daubechies wavelets, and Coiffman wavelets. A standard way to construct an orthogonal wavelet is to solve the dilation equation φ(x) = √ 2 ∞ ∑ k=−∞ ckφ(2x− k), (0.1) with the normalization ∑ k ck = 1. Very few solutions of the dilation equation for wavelets are known to have closed form expressions. This is one reason why it is difficult to determine the regularity of wavelets. Provided the solution φ satisfies some additional conditions, the function ψ defined by