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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Comprehensive textbook examining meaningful connections between the subjects of Lie theory, differential geometry, and signal analysis
A Bridge Between Lie Theory and Frame Theory serves as a bridge between the areas of Lie theory, differential geometry, and frame theory, illustrating applications in the context of signal analysis with concrete examples and images.
The first part of the book gives an in-depth, comprehensive, and self-contained exposition of differential geometry, Lie theory, representation theory, and frame theory. The second part of the book uses the theories established in the early part of the text to characterize a class of representations of Lie groups, which can be discretized to construct frames and other basis-like systems. For instance, Lie groups with frames of translates, sampling, and interpolation spaces on Lie groups are characterized.
A Bridge Between Lie Theory and Frame Theory includes discussion on:
- Novel constructions of frames possessing additional desired features such as boundedness, compact support, continuity, fast decay, and smoothness, motivated by applications in signal analysis
- Necessary technical tools required to study the discretization problem of representations at a deep level
- Ongoing dynamic research problems in frame theory, wavelet theory, time frequency analysis, and other related branches of harmonic analysis
A Bridge Between Lie Theory and Frame Theory is an essential learning resource for graduate students, applied mathematicians, and scientists who are looking for a rigorous and complete introduction to the covered subjects.
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Seitenzahl: 414
Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
Acknowledgments
1 Introduction
1.1 Organization of the Book
1.2 Proficiency Expectations
1.3 Aims
1.4 Scope and Material Selection
1.5 Catering to Diverse Learning Approaches and Expertise Levels
References
2 Differentiable Manifolds
2.1 Calculus on Euclidean Space
2.2 Topological Manifolds
References
Note
3 Lie Theory
3.1 Lie Derivatives
3.2 Lie Groups and Lie Algebras
3.3 Exponential Map
3.4 Invariant Measure on Lie Groups
3.5 Homogeneous Spaces
3.6 Matrix Lie Theory
3.7 Construction of Spline‐Type Partitions of Unity
References
4 Representation Theory
4.1 Representations of Lie Groups and Lie Algebras
4.2 A Survey on the Theory of Direct Integrals
4.3 Induced Representations
4.4 Integrability of Induced Characters
References
5 Frame Theory
5.1 Series Expansions in Hilbert Spaces
5.2 Riesz Bases
5.3 Frames
References
6 Frames on Euclidean Spaces
6.1 Wavelets and the
ax
+
b
Group
6.2 Gabor Systems and the Heisenberg Group
References
7 Frames on Lie Groups
7.1 Discretization of Induced Characters
7.2 Localized Frames on Matrix Lie Groups
7.3 A Generalization
References
8 Frames on Homogeneous Spaces
8.1 Localized Frames on Homogeneous Spaces
8.2 Frames on Spheres
8.3 Frames on the Klein Bottle
References
9 Groups with Frames of Translates
9.1 Frames and Bases of Translates on the
ax
+
b
Lie Group
References
10 Sampling and Interpolation on Unimodular Lie Groups
10.1 Admissible Representations
10.2 Gröchenig–Führ's Method of Oscillations
10.3 Sampling on Locally Compact Groups
10.4 Bandlimitation for Extensions of
References
Note
11 Finite Frames Maximally Robust to Erasures
11.1 Inductive Construction of All Complex ‐Frames
11.2 Infinite Singly Generated Subgroups of
11.3 Random Sampling
References
Index
End User License Agreement
List of Illustrations
Chapter 6
Figure 6.1 A window function generating a frame.
Figure 6.2 A sampling set.
Figure 6.3 Some elements of a frame generated by the
ax
+
b
Lie group.
Figure 6.4 A generator of a frame induced by an irreducible action of the He...
Chapter 7
Figure 7.1 Showing a sample of orbit points in the orbit of the Heisenberg g...
Figure 7.2 Showing a sample of orbit points in the orbit of the
ax
+
b
group...
Figure 7.3 Orbit and sampling set on the
ax
+
b
group.
Figure 7.4 Orbit and sampling set on the Heisenberg group.
Figure 7.5 A graph of .
Chapter 10
Figure 10.1 Graph of the oscillation of a Gaussian.
Figure 10.2 Graphs of and .
Chapter 11
Figure 11.1 Sampling along a unit circle.
Figure 11.2 Sampling over the finite subgroups of the unit circle, , where
Figure 11.3 A dense subset of .
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
End User License Agreement
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A Bridge Between Lie Theory and Frame Theory
Applications of Lie Theory to Harmonic Analysis
Vignon Oussa
Bridgewater State UniversityBridgewater, MA, USA
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Library of Congress Cataloging‐in‐Publication Data
Names: Oussa, Vignon, author.
Title: A bridge between Lie theory and frame theory : applications of Lie
theory to harmonic analysis / Vignon Oussa.
Description: Hoboken, New Jersey : Wiley, [2025] | Includes bibliographical
references and index.
Identifiers: LCCN 2024057768 (print) | LCCN 2024057769 (ebook) | ISBN
9781119712138 (hardback) | ISBN 9781119712145 (adobe pdf) | ISBN
9781119712152 (epub)
Subjects: LCSH: Frames (Vector analysis) | Lie groups. | Geometry,
Differential. | Harmonic analysis.
Classification: LCC QA433 .O87 2023 (print) | LCC QA433 (ebook) | DDC
512/.482–dc23/eng/20241226
LC record available at https://lccn.loc.gov/2024057768
LC ebook record available at https://lccn.loc.gov/2024057769
Cover Design: Wiley
Cover Image: © Flavio Coelho/Getty Images
Preface
Duffin and Schaeffer developed frame theory in the 1950s to solve problems in nonharmonic Fourier series. The quest for redundant and flexible, basis‐like reproducing systems for signal analysis led to the rediscovery of frames in the early 1980s. The seminal work of Daubechies, Meyer, Grossman, and others underscored the influential role of frames in the study of signal analysis via wavelet theory and time‐frequency analysis. Frame theory, a branch of harmonic analysis, has now matured into a dynamic and active field, drawing its strengths from diverse areas like representation theory and Lie theory. The proposed book centers around the discretization problem of representations of Lie groups, which can be articulated as follows: Given a representation of a Lie group, under what conditions is it possible to sample one of its orbits for constructing frames with prescribed properties?
The book aims to offer a differential geometric treatment of the mathematics involved in finding a satisfactory solution to the discretization problem for a class of representations relevant to current research in wavelet, Gabor wavelet, shearlet theories, and their various generalizations.
The first part of the book delivers a self‐contained exposition of differential geometry, Lie theory, representation theory, and frame theory. This material will arm the reader with the technical tools necessary to probe the discretization problem of a class of representations at a deep level. The second part of the book applies the theories laid down in the initial section to characterize a type of representations of Lie groups that can be discretized to construct frames and other basis‐like systems. For example, Lie groups with frames of translates, sampling, and interpolation spaces on Lie groups are examined. Inspired by applications in signal analysis, the book introduces novel constructions of frames possessing additional sought‐after features like boundedness, compact support, continuity, fast decay, and smoothness. The book features an extensive list of meticulously composed and concrete examples. Moreover, it draws relevant connections with ongoing dynamic research problems in frame theory, wavelet theory, time‐frequency analysis, and other related branches of harmonic analysis.
Vignon Oussa
Massachusetts 2023
Acknowledgments
The evolution of novel approaches to solving problems in mathematics is often a nonlinear process. Various individuals significantly shaped my growth throughout the journey of writing this book. I want to specifically acknowledge the profound influence Hartmut Führ's book, “Abstract Harmonic Analysis of Continuous Wavelet Transforms,” has had on my mathematical development. As a Ph.D. student, Hartmut's book was not just a gateway to my own mathematical journey but also exemplified excellent mathematical writing to my novice self. Additionally, in his monograph, Hartmut introduced original results that sparked my interest in further expanding these ideas.
Later, at the onset of my career as an Assistant Professor of Mathematics at Bridgewater State University, I was fortunate enough to visit Hartmut in Aachen, Germany. During these three weeks, I had the opportunity to work, learn, and collaborate with him directly. I distinctly remember presenting him with a rough draft containing ideas related to the central theme of the present book (the discretization problem of representations for constructing frames), and his encouragement and insightful feedback were invaluable in helping me refine and expand these concepts. His input significantly boosted my confidence and further fueled my drive for mathematical research. These experiences underscore the profound and lasting impact effective mentorship can have on a young mathematician's career trajectory.
Writing a book is a daunting task, necessitating patience, dedication, and perseverance. When I was first approached to draft a proposal for this book, I received unwavering support from many individuals, including Hartmut Führ, Karlheinz Gröchenig, Akram Aldroubi, and Jens Christensen. As I stepped back from research to devote myself to this book, I encountered several challenges that tested my resolve to complete this project. However, the sustained support of friends, colleagues, and my children: Sena, Kemi, and Donah, kept me steadfast. I am particularly indebted to Hartmut once again, who generously proofread significant portions of the book, providing invaluable feedback that has been integrated into subsequent revisions.
The completion of this book was supported by the National Science Foundation under Grant No. 2205852. I gratefully acknowledge the NSF's support, which made this work possible. I would also like to thank Professor Kasso Okoudjou for his invaluable support and collaboration throughout this project.
Vignon Oussa
1Introduction
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 complex exponentials as follows. Given real numbers
In addition, it is worth pointing out that the Heisenberg group is also isomorphic to a semidirect product group of the type endowed with the noncommutative group operation
where
In any case, one searches for a pair such that and is a countable subset of the Heisenberg group such that is a basis‐like set for For instance, it is easy to verify that
is an orthonormal basis for This construction also gives rise to a large class of orthonormal bases. Indeed, let be a unitary operator which leaves the set
invariant. Then
and is an orthonormal basis for For instance, if is the Fourier transform acting on then, ,
and as a result, must also be an orthonormal basis for
For a fourth example, let be the space of square‐integrable vectors on the real line. Next, let be the semidirect product group equipped with the following operation: and let acts unitarily in as follows:
The interested reader is invited to verify that the above is indeed a group action, which is also norm‐preserving. Part of the group acts by translation, while another part of the group acts with a modulation‐like action. Next, let be compactly supported functions on the real line, and let us assume that is supported on some closed interval of the form for some positive real number Then the change of variable in the second integral below gives
and as such,
Observing that the length of the interval is equal to , if arcsinh then is a unit interval, and as such, the trigonometric system must be an orthonormal basis for Observing additionally that the scalar
is the length of the projection of the vector onto thanks to Parsevalś equality, the following is immediate:
Returning to (1.3), since
it follows that
Suppose additionally that is the indicator function of the unit interval In other words, and
Changing variable again by allowing to be equal to we obtain
Furthermore, since arcsinh and since the collection of sets forms a perfect tiling of the real line, appealing to the fact that is a unitary action, we derive the following:
Furthermore, since
Finally, appealing to the fact that the set of all compactly supported functions on the real line is dense in we conclude that the mapping
is an isometric embedding of into the sequence Hilbert space In other words, whatever information is encoded in any square‐integrable vector can also be compressed into the countable set
Additionally, it is possible to fully recover from As such, we may view the countable set as a building block for
This approach for constructing building blocks of function spaces by sampling the group orbit of a fixed vector is a method that has successfully been tackled in a variety of subjects: wavelet, generalized wavelet, shearlet, and generalized shearlet theory. Generally, this approach is described as follows. Fix a group acting unitarily on a Hilbert space. Given any fixed vector in the space, the collection of vectors produced via this group action is called an orbit. Moreover, the collection of all possible orbits forms a partition of the space. If for instance, the given action is faithful (that is, only the identity element fixes a vector), the given group parametrizes every orbit, and one aims to sample a fixed orbit to construct a countable building block for the given space.
Similarly to the previous example, with wavelet theory [1, 4], the underlying group is also the affine group (that is, the group of affine transformations on the real line) acting by translation and dilation in (a difference choice of action). This group can also be realized as a matrix group of the type
Note that is isomorphic to a semidirect product group of the form with group operation given by
Furthermore acts unitarily on as follows:
and it is a useful exercise for the reader to verify that is an injective homomorphism (a monomorphism). Additionally, one seeks building blocks of the form for some countable set To this end, recall that the Fourier transform, formally defined as:
is a unitary operator which intertwines translation with pointwise multiplication by complex exponentials. In this context, it follows that
Next, let be the Fourier inverse of the indicator function of That is,
As a subset of the real line, since the collection of sets
forms a tiling of the real line, it follows that is an orthonormal basis for Appealing to the fact that unitary operators map orthonormal bases to orthonormal bases, one verifies that the collection is an orthonormal basis for the Hilbert subspace Next, for any nonzero real number
and since tiles the real line as well,
Noting additionally that
we derive that the system of vectors
is an orthonormal basis for
Shearlet theory on the other hand, uses a group obtained by extending the affine group by some shearing action. Precisely, let be a semidirect product group of the form with group operation given by where
Then acts unitarily on as follows:
This group operation includes shearing transformations, which are a kind of linear transformation that “slants” the space, besides translations and dilations. Moreover, one seeks a basis‐like set for of the form for some countable set and some vector To this end, we may proceed as follows. Let be a compact subset of with a nonempty interior. Next, let be a compactly supported and smooth function defined on (a test function). Also, let be a smooth vector‐valued function defined as follows:
Then letting be a continuous function defined on one verifies that
We will next prove that up to a change of variable, the formula above is the Fourier transform of a compactly supported function. Indeed, since is everywhere injective, letting , we obtain that
The factor in the integrand arises from the change of variables. This factor compensates for the way in which the transformation by stretches or compresses different regions of the space. Suppose next that is small enough to ensure that is contained in a fundamental domain of the integer lattice Then summation over gives
Noting that the sequence represents the Fourier coefficients of it follows that
Suppose now that is a continuous function satisfying Then
Furthermore, the change of variable allows us to proceed as follows:
Finally, suppose that is a fundamental domain for a full‐rank lattice Then
and since the set of all compactly supported functions on the real line is dense in we may conclude that
is an isometric embedding of into the sequence space
1.1 Organization of the Book
The underlying set (affine, Heisenberg, and shearlet groups) in each of the construction above has the structure of a semidirect product Lie group. Moreover, the operators arising from each group action are contained in the range of some unitary representation (called induced characters) of the groups at hand [2, 5]. Thus, to generalize the methods of wavelet, Gabor, and shearlet theories, a certain degree of fluency in the subjects of Lie theory seems useful.
The main aim of the present book is to (a) offer an introduction to both Lie and frame (basis‐like object) theories and (b) provide new constructions of bases‐like objects by only exploiting differential geometric properties of the group action of interest. Mainly motivated by the classical construction of wavelets, Gabor wavelets, shearlets, and their various generalizations, we will be particularly interested in a type of group action obtained from a class of representations known as induced characters. When concerned with the construction of basis‐like objects associated with induced characters, we will establish a novel theory based only on differential geometric properties of the group action of interest. We will aim to describe a general theory in which all examples described above fit. To this end, we will consider semidirect product groups of the type , where is a simply connected commutative Lie group and is a closed subgroup of the automorphism group of For any fixed character (a one‐dimensional representation of ), there is an associated action of on the space of square‐integrable vectors on (with respect to an ‐invariant measure ) such that given
where is determined by the action of as an automorphism map on By only considering specific differential geometric properties of the smooth map we will provide explicit construction of frames generated by a fixed vector exhibiting a number of desired prescribed properties (compact support and various level of smoothness). The theory developed in this book stands in stark contrast with the representation‐theoretic approach, which is far more common in the literature. Additionally, we hope that the interested reader will find the (differential geometric) approach proposed in this text relevant to ongoing research topics on the subjects of wavelet theory, time‐frequency analysis, and shearlet theory.
Chapter 2 provides a comprehensive introduction to differentiable manifolds, drawing on Dr. James Hebda's Differentiable Geometry course at Saint Louis University, textbooks by Spivak and John Lee, and discussions on math.stackexchange.com. It revisits Euclidean calculus concepts like the Implicit Function Theorem before introducing topological manifolds as locally linear, Hausdorff, and second countable spaces. By layering a differentiable structure on topological manifolds, the chapter extends familiar notions from Euclidean calculus, such as diffeomorphism and derivatives, to these broader structures. It explores different classes of submanifolds, introduces the concept of a derivation in the context of smooth functions on a manifold, and discusses how derivations form a Lie algebra. Furthermore, it defines the pushforward of a derivation in the context of a differentiable function between two manifolds, establishes the concept of tangent vectors, and concludes by introducing forms and tensor fields to facilitate integration on a manifold.
Chapter 3 introduces the basic concepts and results of Lie theory, which studies the properties of differentiable manifolds with group structures. Key topics covered include vector fields, Lie derivatives, Lie brackets, Lie groups, Lie algebras, exponential maps, invariant measures, homogeneous spaces, matrix Lie groups, and solvable Lie groups. The chapter draws from various sources, including lecture notes and textbooks, with the goal of preparing the groundwork for further developments in frame theory on manifolds.
Chapter 4 explores the action of Lie groups on vector spaces, particularly function spaces over manifolds, from the perspective of representation theory. This branch of mathematics represents abstract groups as linear transformations on vector spaces. Initiated by Sophus Lie and later expanded by Hermann Weyl and Eugene Wigner, the chapter discusses concepts like unitary representations of Lie groups and induced representations, which allow for the creation of new representations from existing ones. Key topics such as invariant spaces, equivalent representations, irreducible representations, Schur's lemma, direct integral, and induced representation are introduced, with examples and applications to the integrability theory of representations. The chapter recommends texts by Folland and Kaniuth and Taylor [2, 5] for a more comprehensive understanding of these topics.
Chapter 5 explores the concept of frames, which are sets of vectors that can be used to represent any vector in a Hilbert space. Frames generalize the notion of bases and orthonormality, essential tools in linear algebra and functional analysis. The chapter also reviews the concepts of orthonormal bases, Riesz bases, and Parseval's equality, which are related to frames [1, 4].
Chapter 6 investigates frames in the Hilbert space of square‐integrable functions on a Euclidean manifold. The chapter focuses on the discretization of two types of induced representations of the affine Lie group and the Heisenberg Lie group. These examples pave the way for a more general class of frames in later chapters.
Chapter 7 develops a general and unified theory of induced representations for linear Lie groups, which can be used to construct frames for function spaces over any such group. The chapter also provides an algorithm for generating frames that have desirable properties for various applications in analysis and operator theory. The chapter builds on the frames obtained from the affine and Heisenberg Lie groups in Chapter 6.
Chapter 8 introduces a novel method for constructing frames on homogeneous spaces, which are spaces obtained by quotienting a Lie group by a closed subgroup. The chapter focuses on the unit sphere, which is a homogeneous space of the special orthogonal group, and the Klein bottle, viewed as a homogeneous space.
Chapter 9 addresses the discretization problem for both left and right regular representations of a Lie group acting on square‐integrable vectors in a given space. This is because the two representations are unitarily equivalent. Specifically, this chapter demonstrates that a Lie group can admit frames of translates if it contains a closed subgroup, which is isomorphic to the two‐dimensional affine group, which was introduced earlier.
Chapter 10 focuses on the search for analogs of the Whittaker–Kotel'nikov–Shannon sampling series for noncommutative groups. The chapter revisits the concept of admissible representations and presents a methodology based on oscillation estimates by Gröchenig and Führ for discretizing an admissible representation to construct frames. It broadens the concept of bandlimitation on the real line, represented by Paley–Wiener‐type spaces, to a wider class of noncommutative Lie groups. The chapter provides sufficient conditions that offer a favorable resolution to the discretization problem.
Finally, Chapter 11 distinguishes itself from previous chapters by focusing on a specific type of finite frames. These frames are induced by the action of a Lie group and possess an additional characteristic known as “full spark.” As elaborated in Chapter 11, a “full spark” in a finite‐dimensional vector space is a spanning set for the vector space that has the added feature of any subset, equal in size to the dimension of the space, being a basis. This means that any information encoded by a full spark frame is highly robust to erasures, a concept that is explained in greater detail within the chapter.
1.2 Proficiency Expectations
This textbook is primarily written for readers at the graduate student level or above. A fundamental proficiency in real analysis, including differential calculus, integration theory, and topology, is assumed. Our ideal reader has a good understanding of these foundational principles and is ready to delve deeper into more complex concepts. For those who wish to brush up on these fundamentals or explore them in greater detail, a list of basic references will be provided. These references, while not exhaustive, represent a starting point for the topics covered in this book. The aim of this book is not just to impart knowledge, but to guide you on a journey of discovery. Our starting point is a solid grounding in differential geometry. From there, we delve into the world of Lie groups, representation theory, and frames.
1.3 Aims
The primary goal of this book, as suggested by the title, is to present a unique approach to the construction of frames through the action of Lie groups. The structure of the book is largely dictated by the necessity to furnish readers with the necessary technical tools required to undertake various steps in this construction process. At its core, this book provides a problem‐oriented introduction to the use of Lie group methods in frame construction. Its main objective is to develop and present a cohesive treatment of this material, a significant portion of which I have personally developed. The early sections of the book are specifically designed to help readers understand, appreciate, and potentially contribute to this scientific pursuit. By the time readers reach the conclusion of this book, they should have acquired a comprehensive understanding of the principles and techniques involved in the construction of frames using Lie group methods. This understanding will provide the foundation for their further exploration of the subject.
1.4 Scope and Material Selection
Recognizing the diverse audience this book caters to, it is important to note that the material chosen has been purposefully selected to align with the book's objectives, rather than aiming to provide a comprehensive review of the current state of the art.
Frame theory is a significant field in its own right, with comprehensive works penned by authors like Ole Christensen and Chris Heil. This book does not seek to compete with such exhaustive works. Instead, the selection of general frame theory in this book is motivated by its practical relevance to the material that follows.
Similarly, the sections on Lie theory represent a specific choice of material designed to serve the book's broader objectives, rather than attempting to provide a comprehensive and self‐contained introduction. Certain aspects of Lie theory, such as the adjoint and coadjoint actions of Lie groups and Lie algebras, their interplay, coadjoint orbits, differentials of homomorphisms, and so on, are touched upon lightly. While these areas could constitute an entire book in themselves, they are not the primary focus of this work.
However, certain foundational results, such as those of the inverse function theorem and the implicit function theorem, are included in this book. These are considered basic, indispensable knowledge, essential for understanding the advanced topics that this book delves into.
1.5 Catering to Diverse Learning Approaches and Expertise Levels
This book serves as a problem‐driven introduction to the use of Lie‐theoretic methods in frame construction. Rather than offering an exhaustive account of all theoretical aspects, it centers on those which are instrumental to the content of subsequent sections. The extent of exploration into specific areas, such as Lie theory and frames, is primarily determined by their necessity for further interaction within the content. The book occupies a unique niche in the literature, catering to the varied preferences of potential readers. Some readers prefer a thorough, ground‐up approach to learning theory, taking the time to explore fundamentals in depth before turning to concrete examples or applications. In contrast, others thrive on having a clear motivation guiding them through the material. This book is designed with the latter type of reader in mind, leveraging its strengths, particularly its wealth of concrete examples.
In serving as an entry point to both frame theory and, more importantly, Lie theory, the book presents fundamental definitions and results in tandem with concrete questions and constructions. This pairing not only motivates the introduction of these notions but also provides a solid foundation for the reader. To complement the introductory material on Lie theory or frame theory, systematic suggestions for further reading are provided, ensuring readers understand that the material presented in the book is not exhaustive. This approach aligns well with the decision to provide references at the end of each chapter rather than a full list at the end of the book.
Finally, the book's structure takes into consideration the needs of readers who may be mathematicians with strong expertise in one of the main areas of the book and less expertise in others. These readers may wish to skip certain parts of the book and revisit others on multiple occasions, possibly after or while reading later parts of the book. Enhancing the book's accessibility to enable this kind of nonlinear reading can significantly boost its utility, making it an effective resource for a wide range of readers.
For instance, for a reader with a strong background in frame theory but less understanding of differential geometry, Lie theory, and representation theory, I would recommend the following pathway through the book:
Chapter 5
“Frame Theory”:
Start here since you are already comfortable with frame theory. This will give you a sense of the book's approach and style.
Chapter 2
“Differentiable Manifolds”:
Move to this chapter next. This chapter provides foundational knowledge in differential geometry, which is essential for understanding the later chapters on Lie theory and representation theory. Spend time on the sections that are new to you, and do not hesitate to revisit this chapter as you work through the book.
Chapter 3
“Lie Theory”:
This chapter is a good next step. While it may be challenging due to your lack of background in the area, it is crucial for understanding the applications of Lie theory to frame construction in later chapters. Take your time to ensure you understand the key concepts.
Chapter 4
“Representation Theory”:
After you have grasped the basics of Lie theory, move on to this chapter. The concepts introduced here are essential for understanding the later chapters that delve into more complex applications.
Chapters 6
–
10
:
These chapters apply the theories discussed in the earlier chapters to different settings. You should be able to follow the frame theory sections comfortably, and your newly acquired knowledge in differential geometry, Lie theory, and representation theory will help you understand the rest.
Chapter 11
“Finite Frames Maximally Robust to Erasures”:
Finally, this chapter provides a nice capstone to the book, pulling together many of the threads from earlier chapters.
For a reader with a strong background in Lie theory and differential geometry, but limited knowledge of frame theory, I would recommend the following approach to navigate through the book:
Chapter 5
“Frame Theory”
: Start with this chapter to build a foundation in frame theory. Although the concepts may be new, your strong background in Lie theory and differential geometry will likely make the material more accessible.
Chapter 1
“Introduction”
: After gaining some understanding of frame theory, visit this chapter to get a general overview of the book's organization and aims.
Chapters 2
“Differentiable Manifolds” and 3 “Lie Theory”
: Skim through these chapters. Your expertise in differential geometry and Lie theory should allow you to go through this material quickly. However, pay attention to how these concepts are applied in the context of frame theory.
Chapter 4
“Representation Theory”
: Although not directly related to frame theory, the concepts in this chapter will be useful in later chapters that deal with more complex applications.
Chapters 6
–
10
: These chapters apply the theories discussed in the earlier chapters to different settings. As you read, try to focus on how the frame theory concepts you learned in
Chapter 5
are applied in these settings.
Chapter 11
“Finite Frames Maximally Robust to Erasures”
: Lastly, this chapter synthesizes many of the concepts from the earlier chapters and applies them in a complex setting. The material in this chapter should be easier to understand now that you have a foundation in frame theory and have seen how it interacts with Lie theory and differential geometry.
References
1
Ole Christensen.
An introduction to frames and Riesz bases
. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA, 2003.
2
Gerald B. Folland.
A course in abstract harmonic analysis
, volume 29. CRC Press, 2016.
3
Hartmut Führ.
Abstract harmonic analysis of continuous wavelet transforms
, volume 1863 of
Lecture Notes in Mathematics
. Springer‐Verlag, Berlin, 2005.
4
Christopher Heil.
A basis theory primer: expanded edition
. Springer Science & Business Media, 2010.
5
Eberhard Kaniuth and Keith F. Taylor.
Induced representations of locally compact groups
, volume 197. Cambridge University Press, 2013.
6
Gitta Kutyniok and Demetrio Labate. Introduction to shearlets. In
Shearlets
, pages 1–38. Springer, 2012.
7
Elias M. Stein and Rami Shakarchi.
Fourier analysis: an introduction
, volume 1. Princeton University Press, 2011.
2Differentiable Manifolds
The main aim of the ensuing chapter is to provide the reader with an introduction to the concept of differentiable manifolds. The structure of this chapter has been inspired by the curriculum of a year‐long course in differentiable geometry offered by Dr. James Hebda at Saint‐Louis University in the academic year 2011–2012.
