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An Introduction to Finite Tight Frames

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An Introduction to Finite Tight Frames Shayne F. D. Waldron An Introduction to Finite Tight Frames – Monograph – July 26, 2017 Springer To my wife Catherine Ann Etheredge. Preface This book gives a unified introduction to the rapidly developing area of finite tight frames. Fifteen years ago, the existence of equal–norm tight frames of n > d vectors for Rd and Cd was not widely known. Now equal–norm tight frames are known to be common, and those with optimal cross–correlation and symmetry properties are being constructed and classified. The impetus behind these rapid developments are applications to areas as diverse as signal processing, quantum information theory, multivariate orthogonal polynomials and splines, and compressed sensing. It can be thought of as an extension of the first chapter of Ole Christensen’s book: An introduction to Frames and Riesz Bases (in this series), which deals mostly with the infinite dimensional case. For finite dimensional Hilbert spaces the technicalities of Riesz bases disappear (though infinite frames are still of interest), and, with some work, usually a nice tight frame can be constructed explicitly. Hence the focus is on finite tight frames, which are the most intuitive generalisation of orthonormal bases. In addition to analogues of familiar ideas from the infinite dimensional setting such as group frames, there is a special geometry, e.g., the variational characterisation and the frame force. The book is structured into chapters, with the first paragraph intended to give a feeling for its content. These give a logical development, while being as independent as possible. For example, one could jump to those giving the important examples of harmonic frames, equiangular frames and SICs, referring back to the motivating chapters on symmetries and group frames as desired. Similarly, in the text we give forward references to such nice examples. Grey boxes are used to emphasize or paraphrase some key ideas, and may help readers to navigate this book. Each chapter has Notes, which primarily give a brief description of my source material and suggestions for further reading. The interdisciplinary nature of the sub- ject makes it difficult (and often senseless) to give exact attribution for many results, e.g., the discrete form of Na˘ımark’s theorem, and so I have attempted to do so only in some cases. This introductory book is deliberately as short as possible, and so its scope and list of references is far from definitive. If only for this reason, some contributions may not be noted explicitly. vii viii Preface The chapters conclude with Exercises, which contain more details and examples. Brief solutions are given, since some are parts of proofs, and in the interests of being both introductory and self contained. Those marked with a m are suitable for using a computer algebra package such as matlab, maple and magma. To get a real sense for finite tight frames the reader is encouraged construct the various examples given in the book numerically or algebraically. In particular, the harmonic frames, MUBs and Weyl–Heisenberg SICs for d 100 (or higher), which really are quite remarkable. ≤ This book can be used for an introductory (graduate or undergraduate) course in finite tight frames: chapters 1, 2, 5, 6, 7, 8, 11, together with some examples of interest, e.g., group frames (chapter 10), equiangular tight frames (chapter 12) and SICs (chapter 14). The latter chapters rely heavily on parts of classical algebra, such as graph theory (real equiangular tight frames), character theory (harmonic frames), representation theory (group frames), and Galois theory (SICs). Examples from these could be included in a graduate algebra course, e.g., Theorem 10.8 as an orthogonal decomposition of FG–modules. It could also act as the theoretical background for a course involving applications of tight frames, e.g., some of the topics treated in Finite frames Theory and Applications (in this series). It is also suitable for self study, because of the complete proofs and solutions for exercises. How does one come to write such a book? My initial interest in finite tight frames was accidental. While investigating the eigenstructure of the Bernstein operator, I encountered deficiencies in the existing orthogonal and biorthogonal expansions for the Jacobi polynomials on a triangle. Having heard of frames while a graduate student in Madison, I supposed that a finite tight frame might just be what was required. This eventually proved to be so (see 15), and along the way I found that finite frames are a fascinating and very active area§ of research. When I started this book, there was just enough material for a short book on finite tight frames, a small part of frame theory, at the time. Since then, there has been an explosion of research on finite tight frames, which continues apace. I have called time on the project: the book contains results from last week (Lucas-Fibonnacci SICs), but not next week (maybe a proof of Zauner’s conjecture). There are chapters that could have been written, e.g., ones on erasures and data transmission. I maintain a strong interest in finite tight frames. On my home page, there are various links of relevance to the book (lists of SICs, harmonic frames, typos, etc). It is my hope that this book conveys the basic theory of finite tight frames, in a friendly way, being mindful of its connections with existing areas and continuing development. The applications given are just the tip of an iceberg: an invitation to use finite tight frames in any area where there is a natural inner product on a finite dimensional space. Shayne F. D. Waldron Honeymoon Valley, Far North Aotearoa (New Zealand) July 2017 Acknowledgements There are many people to thank. Catherine for her support. My mentors Carl de Boor (Orcas island) and Len Bos (Verona). My colleagues at home: Tom ter Elst, Sione Ma’u and Warren Moors, and in Italy: Marco Vianello and Stefano Di Marchi. My students: Irine Peng, Richard Vale, Simon Marshall, Nick Hay, Helen Broome, Tan Do, Jennifer Bramwell, Xiaoyang Li, Paul Harris, Tuan Chien and Daniel Hughes. Many in the frame community: John Benedetto and Pete Casazza for their ongoing encouragement with this book, Joe Renes for introducing me to SICs, and others for their careful and insightful reading of various chapters of the book. Algebraists that I have bothered: Charles Leedham-Green, Mark Lewis, Martin Liebeck, Gabriel Verret, and those that I will, now that I have time. ix Contents 1 Introduction ................................................... 1 1.1 Somehistory ..................................... ......... 2 1.2 Desirableproperties ............................. ........... 3 Notes .............................................. ........... 4 Exercises .......................................... ............ 4 2 Tight frames ................................................... 7 2.1 Normalisedtightframes ........................... .......... 7 2.2 Unitarily equivalent finite tight frames . ............ 10 2.3 Projective and complex conjugate equivalences. ........... 11 2.4 The analysis, synthesis and frame operators . ........... 12 2.5 TheGramian ...................................... ........ 13 2.6 Tight frames as orthogonal projections . ........... 15 2.7 The construction of tight frames from orthogonal projections...... 17 2.8 Complementarytightframes ........................ ......... 18 2.9 Partitionframes................................. ........... 20 2.10 Real and complex tight frames. .......... 21 2.11 SICsandMUBs.................................... ........ 22 Notes .............................................. ........... 23 Exercises .......................................... ............ 24 3 Frames ................................................... 31 3.1 Motivation ...................................... .......... 31 3.2 Aframeanditsdual ................................ ........ 32 3.3 Thecanonicaltightframe.......................... .......... 35 3.4 Unitarilyequivalentframes ....................... ........... 40 3.5 Similar frames and orthogonal frames . ......... 41 3.6 Frames as orthogonal projections . .......... 43 3.7 Condition numbers and the frame bounds . ....... 46 3.8 Normalising frames and the distances between them . ........ 47 3.9 Approximate inverses of the frame operator. .......... 53 xi xii Contents 3.10 Alternateduals................................. ............ 54 3.11 Obliqueduals................................... ........... 57 Notes .............................................. ........... 60 Exercises .......................................... ............ 60 4 Canonical coordinates for vector spaces and affine spaces ........... 71 4.1 The canonical Gramian of a spanning sequence . ....... 72 4.2 The canonical coordinates of a spanning sequence . ......... 74 4.3 A characterisation of the canonical coordinates . ............ 76 4.4 Properties of the canonical coordinates . ............ 78 4.5 The canonical inner product for a spanning sequence . ......... 80 4.6 Canonical coordinates for cyclotomic fields . ........... 81 4.7 Generalised barycentric coordinates for affine spaces . ............ 87 4.8 TheBernsteinframe ............................... ......... 92 4.9 PropertiesoftheBernsteinframe ................... .......... 93 4.10 The generalised Bernstein operator . ............ 94 Notes .............................................. ........... 98 Exercises .......................................... ............ 98 5 Combining and decomposing frames ............................. 99 5.1
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