One-Dimensional Dyadic Wavelets
Authors: Luthy Peter , Šikić Hrvoje , Soria Fernando , Weiss Guido , Wilson Edward 
Published: October 2022

The theory of wavelets has been thoroughly studied by many authors; standard references include books by I. Daubechies, by Y. Meyer, by R. Coifman and Y. Meyer, by C.K. Chui, and by M.V. Wickerhauser. In addition, the development of wavelets influenced the study of various other reproducing function systems. Interestingly enough, some open questions remained unsolved or only partially solved for more than twenty years even in the most basic case of dyadic orthonormal wavelets in a single dimension. These include issues related to the MRA structure (for example, a complete understanding of filters), the structure of the space of negative dilates (in particular, with respect to what is known as the Baggett problem), and the variety of resolution structures that may occur. In this article we offer a comprehensive, yet technically fairly elementary approach to these questions. On this path, we present a multitude of new results, resolve some of the old questions, and provide new advances for some problems that remain open for the future.

In this study, we have been guided mostly by the philosophy presented some twenty years ago in a book by E. Hernandez and G. Weiss (one of us), in which the orthonormal wavelets are characterized by four basic equations, so that the interplay between wavelets and Fourier analysis provides a deeper insight into both fields of research. This book has influenced hundreds of researchers, and their effort has produced a variety of new techniques, many of them reaching far beyond the study of one-dimensional orthonormal wavelets. Here we are trying to close the circle in some sense by applying these new techniques to the original subject of one-dimensional wavelets. We are primarily interested in the quality of new results and their clear presentations; for this reason, we keep our study on the level of a single dimension, although we are aware that many of our results can be extended beyond that case.

Given

ψ
\psi

, a square integrable function on the real line, we want to address the following question: What sort of structures can one obtain from the affine wavelet family

{

2

j

/

2

ψ
(

2

j

x
−
k
)
:
j
,
k
∈

Z

}

\{2^{j/2} \psi ( 2^{j}x - k ) : j,k\in \mathbb Z\}

associated with

ψ
\psi

? It may be too difficult to directly attack this problem via the function

ψ
\psi

. We argue in this article that the appropriate object to study is the principal shift invariant space generated by

ψ
\psi

(these spaces were introduced by H.Helson decades ago and applied very successfully in the approximation theory by C. de Boor, R.A. DeVore, and A. Ron, with more recent applications to wavelets introduced by A. Ron and Z. Shen). With this goal in mind, in Chapter 1, we present a very detailed study of principal shift invariant spaces and their generating families. These include the relationships between principal shift invariant spaces, various basis-like and frame-like properties of their generating families, their classifications based on additional translation invariances, convergence properties of various reproducing families with emphasis on the case of unconditional convergence, and the special properties of maximal principal shift invariant spaces.

Given a principal shift invariant space

V
V

and the dyadic dilation

D
D

, our approach is that the entire theory can be developed by considering two basic relationships between

V
V

and

D
(
V
)

D(V)

. Chapter 2 is devoted to the first of these two cases, the one in which the space

V
V

is contained within

D
(
V
)

D(V)

. In this chapter, we completely resolve this case via an emphasis on generalized filter studies. We show that the entire generalized MRA theory is a natural consequence of this approach, with a detailed classification of all the special cases of what we term as Pre-GMRA structures. Special attention is devoted to the analysis of the general form of a filter associated with the space

V
V

. It is worth emphasizing two aspects of our approach. One is that the phase of the filter does not play a significant role in any of the main filter properties. This allows us to shift the analysis from complex valued filters into the analysis of their absolute values, which introduces significant simplifications of various convergence properties. The second aspect is related to the fact that most authors would base their approach to filters satisfying the Smith-Barnwell condition, which is natural since it is easy to construct functions that satisfy this important condition. However, we show that by lifting the theory to a more abstract level our view improves and several new features are revealed to us. The theory splits into two subcases, based on the filter properties with respect to dyadic orbits; we distinguish the “full-orbit” case and the “non full-orbit” case. In both cases we introduce new Tauberian conditions which provide complete characterizations of “usable” filters (Theorem 2.86 in Chapter 2 covers the “full-orbit” case and Theorem 2.101 covers the “non full-orbit” case). This approach further splits into the analysis of low frequencies versus high frequencies. There is a fundamental new result here which shows that, based on the “ergodic properties” of

ψ
^

\widehat {\psi }

, the two frequency regimes exhibit radically different behavior; low frequencies allow completely localized adjustments while high frequencies can only be treated in a global sense. Theorem 2.151 in Chapter 2 presents a new ergodic type condition on filters, previously unobserved, which is a far reaching generalization of standard filter properties. Various known results, like the Smith-Barnwell condition, the Cohen condition and its generalizations, the Lawton condition and its generalizations, are extracted naturally from our general approach. A multitude of new technical results are presented, with many examples and counter-examples exhibited to illustrate various subtle points of the theory. For example, the “full-orbit” filters are naturally connected with the “reversed” two-scale equation, the complete solution of which is given in Theorem 2.323 of Chapter 2.

The third and final chapter is devoted to the second case, i.e., when the space

V
V

is not contained in

D
(
V
)

D(V)

. This naturally leads to the space of negative dilates, and the theory again splits into two subcases, based on whether the original function

ψ
\psi

is contained within the space of its negative dilates or not. This is very much in the spirit of the Baggett problem, and we find it somewhat striking that the entire theory can be built based on such a simple property. Based on the classification of shift invariant spaces with respect to additional translation invariances (as introduced in Chapter 1) we can carefully build up various orthogonal-like properties starting from the MSF case and moving to the general case. The central result in Chapter 3 is Theorem 3.86, the main structural theorem for dyadic resolution levels of the affine family. It has several interesting consequences, as emphasized in the last section of the chapter. We end the article with a partial resolution of the Baggett problem, but the problem remains open when taken in its full scope.