1
Introduction

As a doctoral candidate at Saint-Louis University, I have had the opportunity to study Frame Theory, taught by Professor Darrin Speegle, Representation Theory, taught by Professor Bradley Currey, and Differential Geometry, a course conducted by Professor James Hebda. In my pursuit of a research direction, I found myself captivated by the field of Abstract Harmonic Analysis, specifically continuous wavelets on noncommutative domains [3]. Through my explorations, I have come to appreciate the immense utility of the tools provided by Abstract Harmonic Analysis across other subjects within Harmonic Analysis, including Frame Theory, Wavelet Theory, and Shearlet Theory [1, 4, 6].

Recognizing the broad significance of Frame Theory within the Harmonic Analysis community, my aim was to enhance the accessibility and clarity of the connections between Abstract Harmonic Analysis and Frame Theory in academic literature. However, my previous approach, which heavily leaned on Representation Theory, potentially obscured these connections. This was compounded by the fact that only a minority of researchers in Harmonic Analysis possess expertise in Representation Theory.

In response, I endeavored to minimize reliance on Representation Theory in framing constructions of frames, wavelets, and their broader generalizations. Instead, I introduced a series of constructions characterized by a more pronounced emphasis on Lie Theory and differential geometry.

Despite dispensing with stringent Representation-theoretic assumptions, the Lie theoretic/differential geometric approach still necessitated a decent understanding of Differential Geometry and Lie Theory. Thus, I decided to pen this book in order to clarify these connections and make them more comprehensible to researchers and graduate students interested in studying Frame Theory within the context of wavelet, time-frequency, and shearlet theories.

A pertinent question arises for any vector space, whether finite- or infinite-dimensional, structured as a Hilbert space: how can we systematically construct basis-like objects such that any vector within the given vector space can be expressed as a series expansion by appropriately scaling the elements of the chosen building block? The term "basis-like" underscores that we do not necessarily aim for an outcome where every vector has a unique series expansion in terms of the provided building blocks. Randomly selecting vectors is not an efficient approach for the systematic construction of these building blocks, particularly if we are seeking blocks with prescribed properties. One approach, established throughout the literature, begins with a unitary representation. A unitary representation is a continuous map between a group and a group of unitary operators acting on the given Hilbert space. For example, if the group is connected, a particular orbit associated with the representation may be excessively redundant to be of use. Thus, we might consider sampling an orbit of a representation along a countable set, with the hope that the countable set of vectors retained can serve as building blocks for the space. However, the following question remains: how can we achieve this?

In this book, I delve into the fundamental question of what the primary building blocks of any data collection are and how one can systematically generate such a collection. Any data, whether it be a recorded piece of music, a movie, or a signal from distant galaxies, can be regarded as a function defined over a set. This set, generally has the structure of a (smooth) manifold and can be equipped with a measure with respect to which a satisfactory theory of integration can be established. This allows us to view the space of all possible data on a fixed manifold as a Hilbert space. The study of frames, wavelets and shearlets in this context, will serve as the central theme of this book.

As a first example, let us consider the picture below:

Color intensity map represented on a grid.

This data could be naturally viewed as a function defined over a finite grid (a zero-dimensional manifold). Different real numbers represent varying intensities of the same color in each cell, and we may for instance code this image as the following square matrix of order two:

Thus, there is a one-to-one correspondence between the set of all pictures described in the format described above and the set of all real square matrices of order two. Indeed, let

be the vector space of objects representing pictures modeled as square matrices of order two, with basis , where

Every picture is represented by the linear combination

for some unique sequence of real numbers Moreover, the collection is a set of basic building blocks for Lastly, consider the isomorphism that transforms every square matrix of order two to its flatten version. Applying to a matrix with elements and results in a vertical vector with elements and from top to bottom.

and letting be the cyclic group of order four generated by matrix

That is

The first matrix is an identity matrix of order four. The second matrix has its first element in the fourth position, its second element in the first position, its third element in the second position, and its fourth element in the third position, with all other elements as zeros. The third matrix has its first element in the third position, its second element in the fourth position, its third element in the first position, and its fourth element in the second position. The fourth matrix has its first element in the second position, its second element in the third position, its third element in the fourth position, and its fourth element in the first position. Letting (the vector obtained by extracting the -column of ) it is easy to verify that

In other words, the set of basic building blocks of is generated by allowing a finite cyclic group of order 4 to act on a single element of this set. This process provides us with a systematic method for constructing a basis for the vector space of pictures made of pixels with different intensities of the same color.

The interested reader is invited to interact with a graphical user interface available through Mathematica for further exploration of these concepts. [Click here]

More generally, let be an invertible matrix of order 4 commuting with and therefore every element of To this end, it suffices to allow to be an invertible circulant matrix of the type

such that

In other words, is a polynomial function of and for any integer As a result,

is a basis for as well. This procedure describes a very convenient method for constructing in a systematic fashion, a large class of bases for

For a second example, we consider a thin rod modeled as the open unit interval (a bounded one-dimensional smooth manifold). Let us imagine that the temperature of the rod is naturally regarded as a function defined over the open set . Assume furthermore that the space of all possible temperatures on this rod is modeled as the Hilbert space Thanks to Fourier analysis [7], it is known that the collection

of complex exponentials parametrized by the integers forms an orthonormal basis for . Moreover, any square-integrable function (in this case, temperature) on a unit interval can be written as a series expansion of complex exponentials suitably scaled by some complex coefficients. Precisely, for any vector , there exists a sequence such that

with convergence in the norm of Next, the sequence of coefficients obtained in such a series expansion are scalars given by projecting the fixed vector on each element of this building block. That is, each complex number is uniquely determined as follows:

Moreover, the series obtained from this procedure (1.2) is stable, since convergence is unconditional when one is concerned with orthonormal bases. To construct this orthonormal basis from a group acting on a single vector, one may proceed as follows. First, let be the additive group acting unitarily in by multiplication by complex exponentials as follows. Given real numbers and a vector

and it turns out that is an orthonormal basis obtained by sampling the orbit along the integers [7]. Furthermore, let be a bounded invertible operator such that for every integer Then is a Riesz basis [1] for For instance, letting be a uniformly continuous function on with an empty zero set and letting be a bounded invertible multiplicative operator acting as follows: then one obtains that is a Riesz basis for

If we were to replace the rod discussed in the previous example by a rod of infinite length modeled as the real line (an unbounded one-dimensional manifold), we would be interested in constructing an orthonormal basis for which is obtained by sampling the group orbit of some vector. A viable strategy would be to view the real line as an infinite union of finite intervals. By constructing an orthonormal basis for each finite interval, the union of such bases would therefore form a basis for the entire space itself. To accomplish this, let us consider the Heisenberg group

acting unitarily on by translation, modulation (frequency-translation) and multiplication by...