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AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 34

Finding What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other

Ulrich Daepp Pamela Gorkin Andrew Shaffer Karl Voss 10.1090/car/034

Finding Ellipses What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other

AMS/MAA THE CARUS MATHEMATICAL MONOGRAPHS

VOL 34

Finding Ellipses What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other

Ulrich Daepp Pamela Gorkin Andrew Shaffer Karl Voss Committee on Books Jennifer J. Quinn, Chair

2017–2018 Editorial Committee Fernando Gouvea, Editor

Francis Bonahon Alex Iosevich Gizem Karaali Kristin Estella Lauter Steven J. Miller David P. Roberts 2010 Subject Classification. Primary 47A05, 47A12, 30J10, 15-02, 15A60, 51-02, 51M04, 51N35.

For additional information and updates on this book, visit www.ams.org/bookpages/car-34

Library of Congress Cataloging-in-Publication Data Names: Daepp, Ulrich, author. Title: Finding ellipses: What Blaschke products, Poncelet’s theorem, and the numerical range know about each other / Ulrich Daepp [and three others]. Description: Providence, Rhode Island: MAA Press, An imprint of the American Math- ematical Society, [2018] | : The Carus mathematical monographs; volume 34 | Includes bibliographical references and index. Identifiers: LCCN 2018021655 | ISBN 9781470443832 (alk. paper) Subjects: LCSH: . | Conic sections. | Geometry, Projective. | AMS: Operator theory – General theory of linear operators – General (adjoints, conjugates, products, inverses, domains, ranges, etc.). msc | Operator theory – General theory of linear operators – Numerical range, numerical radius. msc | Functions of a complex variable – Function theory on the disc – Blaschke products. msc | Linear and multilinear ; theory – Research exposition (monographs, survey articles). msc | Linear and multi- linear algebra; matrix theory – Basic linear algebra – Norms of matrices, numerical range, applications of functional analysis to matrix theory. msc | Geometry – Research exposition (monographs, survey articles). msc | Geometry – Real and complex geom- etry – Elementary problems in Euclidean geometries. msc | Geometry – Analytic and descriptive geometry – Questions of classical . msc Classification: LCC QA559 .F56 2018 | DDC 516/.152–dc23 LC record available at https://lccn.loc.gov/2018021655

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© 2018 by the author. All rights reserved. Printed in the United States of America. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18 Contents

Preface vii

Part 1 1

Chapter 1 The Surprising Ellipse 3

Chapter 2 The Ellipse Three Ways 13

Chapter 3 Blaschke Products 23

Chapter 4 Blaschke Products and Ellipses 35

Chapter 5 Poncelet’s Theorem for 47

Chapter 6 The Numerical Range 61

Chapter 7 The Connection Revealed 75

Intermezzo 85

Chapter 8 And Now for Something Completely Different... Benford’s Law 87

Part 2 101

Chapter 9 Compressions of the Shift Operator: The Basics 103

Chapter 10 Higher : Not Your Poncelet Ellipse 121

Chapter 11 Interpolation with Blaschke Products 133

Chapter 12 Poncelet’s Theorem for 푛-Gons 147

v vi Contents

Chapter 13 Kippenhahn’s and Blaschke’s Products 159

Chapter 14 Iteration, Ellipses, and Blaschke Products 177

On Surprising Connections 195

Part 3 199

Chapter 15 Fourteen Projects for Fourteen Chapters 201 15.1 Constructing Great Ellipses 201 15.2 What’s in the Envelope? 201 15.3 Sendov’s Conjecture 206 15.4 Generalizing Steiner 210 15.5 Steiner’s Porism and Inversion 213 15.6 The Numerical Range and Radius 222 15.7 Pedal and Foci 224 15.8 The Power of Positivity 228 15.9 and the Numerical Range 231 15.10 The Importance of Being Zero 234 15.11 Building a Better Interpolant 237 15.12 Foci of Algebraic Curves 241 15.13 Companion Matrices and Kippenhahn 245 15.14 Denjoy–Wolff Points and Blaschke Products 251

Bibliography 255

Index 263 Preface

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glo- rious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing.... They are intimately interconnected, and they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That’s what mathematics is to me. –Paul Halmos, Celebrating 50 Years of Mathematics Discovering surprising connections between one of mathematics and another area is not only exciting, it is useful. Laplace transforms al- low us to switch between differential and algebraic equations, the fundamental theorem of algebra can be proved using complex analy- sis, and group theory plays a fundamental role in the study of public-key cryptography. If we can think of an object from different angles, these different perspectives aid our understanding. It is in that vein thatwe have written this book: We look at three seemingly different stories and a hidden connection between them. The first of our stories is complex—that is, it is about complex func- tion theory. In complex analysis, linear fractional transformations are among the first functions that we study. We understand their geometric and function-theoretic properties, and we know that they play an im- portant role in the study of analytic functions. In the class of linear frac- tional transformations, the automorphisms of the unit disk stand out for the geometric and analytic insight they provide. The functions that we on are a natural generalization of the disk automorphisms; they are

vii viii Preface finite products of these automorphisms and are called Blaschke prod- ucts, in honor of the German geometer Wilhelm Blaschke. While we know, for example, that automorphisms of the disk map to cir- cles (if we agree that lines are circles), much less is known about the geometry of Blaschke products. The second story comes from . Here we focus on Poncelet’s closure theorem, a beautiful geometric result about conics in- scribed in polygons that are themselves inscribed in conics. We devote some attention to precursors of Poncelet’s theorem, including Chapple’s formula and Fuss’s theorem. In the final part of this book we will also consider variations on this theme, including Steiner’s porism. The final story is about the numerical range of a matrix. The numer- ical range is the range of a associated with a matrix but restricted to the unit of ℂ푛. The numerical range always con- tains the eigenvalues of the matrix, and it often can tell you more about a matrix than the eigenvalues can. While questions about the numerical ranges of 2 × 2 matrices are (relatively) easily analyzed, consideration of 푛 × 푛 matrices for 푛 > 2 is deeper. Even 3 × 3 matrices lead to deep and interesting questions! What is the hidden connection between these three stories? Curiously, the connection is an object we have not discussed much yet: It is the ellipse. In fact, this book is really a story about the ellipse, an object studied by the Greek mathematicians , Euclid, Ap- polonius of , and Pappus, among others. Later, Kepler developed a description of planetary motion, leading to his first law: The ofa is elliptical with one focus at the Sun. In spite of this long history, ellipses never cease to surprise us. It turns out that ellipses provide a remarkable amount of information about Blaschke products. Ellipses can tell us when a Blaschke product has a nontrivial factorization with respect to composition. And when we study the dynamics of Blaschke products, ellipses can tell us what to expect. The ellipse even establishes a connection between Blaschke products and a useful tool for detecting tax fraud, known as Benford’s law. But why does an ellipse know anything about these particular ra- tional functions? That is the story we wish to tell, and it is a story that Preface ix relies heavily on properties of the numerical range of a class of operators and that story, in turn, relies on projective geometry. The first part of this book (Chapter 1 through Chapter 7) focuses on Blaschke products that are products of three disk automorphisms, el- lipses that are inscribed in triangles, and 2 × 2 matrices. In this case, ellipses suggest directions to study. Each Blaschke product of degree 3 is naturally associated with an ellipse, as is each 2×2 matrix. The relation- ship between Blaschke products, 2 × 2 matrices, and Poncelet’s theorem will be revealed in Chapter 7. This part of the book also introduces the basic ideas of two-dimensional projective geometry and provides a proof of Poncelet’s theorem in the event that the circumscribing polygons are triangles. After concluding Part 1, the reader will be treated to an inter- mezzo (Chapter 8), as we focus on the surprising connection between Poncelet’s theorem and Benford’s law. In the second part of the book (Chapter 9 through Chapter 14) we consider the connection between ellipses and Blaschke products that are products of more than three analytic disk automorphisms. We will see that though the boundaries of the numerical ranges of the matrices we study need not be elliptical, the boundaries always satisfy a Poncelet-like property. In addition, when the boundary is elliptical, it provides insight into the function-theoretic behavior of the corresponding Blaschke prod- uct. Because Poncelet’s ideas are intimately connected to our presenta- tion, we provide a recent and beautiful proof of the general theorem. This part of the book is also a bit more challenging than the first part, mathematically speaking, but we strive for a self-contained treatment, providing appropriate proofs and references. Finally, the third part of the book provides a range of exercises and projects, from straightforward exercises for someone first entering the field to potential research problems. Each chapter has a corresponding project that is usually introduced with new material. To be prepared to work through a project, active reading is required. Projects include things we find particularly interesting; for example, we discuss Sendov’s conjecture, interpolation, inversion, and the spectral radius. In addi- tion, we provide suggestions for the creation of several algorithms that will enhance one’s understanding. We hope that the reader will pick one x Preface

(or more) of these projects and use it (or them) to develop a deeper ap- preciation of the subject, to write an honors thesis, or to begin a research project. The necessary background and a starter bibliography for each project is provided, but we encourage readers to do a thorough search of the literature.

Using this book. The first two parts of the book are meant to tellthe story we have just described; a story that is meant to be read. However, this book can be used in various ways. It can serve as a reference for independent study, as the text for a capstone course in mathematics, or as a reference for a researcher. We have written this book for an active reader—paper and in hand—and though there are a few exercises embedded in this reading, it is Part 3 of the text that includes a significant number of exercises as well as projects. These projects provide a range of both material and depth by including exercises well within the reach of every reader, suggested research papers, and research problems that have been open for many years.

Applets. There are several interactive applets designed to go along with the book, and we encourage using the applets while reading. If we feel that the time is ripe for experimentation, we will direct the reader to the applet using the symbol ,.1 Our applets illustrate various results in the text, such as the main result on the connection of Blaschke prod- ucts to ellipses. The proof of Poncelet’s theorem that we present depends on Pascal’s and Brianchon’s theorem, and we provide applets associated with each of these results. In addition, we have applets that illustrate the mapping behavior of Blaschke products (including locating fixed points in the closed unit disk), composition with Blaschke products, and these applets also produce the numerical range of particular matrices. It has been our experience that these tools are not only fun to work with, they also enhance one’s understanding and are great for testing conjectures.

Acknowledgments. The authors wish to thank the MAA Editorial Board of the Carus Series for many suggestions that have improved the manuscript. We also thank the editors at the AMS for their invaluable help preparing this book. We are grateful to Bucknell University for its

1http://pubapps.bucknell.edu/static/aeshaffer/v1/ Preface xi support and to the many students we have had who have played an im- portant role in the creation of this book. In particular, we wish to thank Robert C. Rhoades, Benjamin Sokolowsky, Nathan Wagner, and Beth Skubak Wolf. We are also grateful to Kelly Bickel, David Farmer, Gunter Semmler, Elias Wegert, and Jonathan Partington for valuable discus- sions and to Bucknell L&IT for technical support. Pamela Gorkin’s work was supported by Simons Foundation Collaboration Grant 243653.

Bibliography

[1] S. Adlaj, An eloquent formula for the of an ellipse, Notices Amer. Math. Soc. 59 (2012), no. 8, 1094–1099, DOI 10.1090/noti879. MR2985810 [2] M. Agarwal, J. Clifford, and M. Lachance, Duality and inscribed ellipses, Comput. Methods Funct. Theory 15 (2015), no. 4, 635–644, DOI 10.1007/s40315-015-0124-0. MR3428821 [3] J. Agler, Geometric and topological properties of the numerical range, Indiana Univ. Math. J. 31 (1982), no. 6, 767–777, DOI 10.1512/iumj.1982.31.31053. MR674866 [4] G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic-, ellipses, 휋, and the Ladies diary, Amer. Math. Monthly 95 (1988), no. 7, 585– 608, DOI 10.2307/2323302. MR966232 ′ [5] V. I. Arnol d and A. Avez, Ergodic problems of , W. A. Ben- jamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR0232910 [6] A. Avez, Ergodic theory of dynamical systems, vol. 1, University of Minnesota, Insti- tute of Technology, 1966. [7] S. Axler, Linear algebra done right, 3rd ed., Undergraduate Texts in Mathematics, Springer, Cham, 2015. MR3308468 [8] E. Badertscher, A simple direct proof of Marden’s theorem, Amer. Math. Monthly 121 (2014), no. 6, 547–548, DOI 10.4169/amer.math.monthly.121.06.547. MR3225469 [9] E. J. Barbeau, , Problem Books in Mathematics, Springer-Verlag, New York, 1989. MR987938 [10] J. Barnes, Gems of geometry, 2nd ed., Springer, Heidelberg, 2012. MR2963305 [11] N. Bebiano, J. da Providência, A. Nata, and J. P. da Providência, Revisiting the inverse field of values problem, Electron. Trans. Numer. Anal. 42 (2014), 1–12. MR3183614 [12] A. Berger and T. P. Hill, A basic theory of Benford’s law, Probab. Surv. 8 (2011), 1–126, DOI 10.1214/11-PS175. MR2846899 [13] A. Berger and T. P. Hill, An introduction to Benford’s law, Princeton University Press, Princeton, NJ, 2015. MR3242822 [14] A. Beutelspacher and U. Rosenbaum, Projective geometry: from foundations to appli- cations, Cambridge University Press, Cambridge, 1998. MR1629468 [15] H. J. M. Bos, C. Kers, F. Oort, and D. W. Raven, Poncelet’s closure theorem, Exposition. Math. 5 (1987), no. 4, 289–364. MR917349 [16] P. S. Bourdon and J. H. Shapiro, When is zero in the numerical range of a composi- tion operator?, Equations Operator Theory 44 (2002), no. 4, 410–441, DOI 10.1007/BF01193669. MR1942033 [17] C. B. Boyer and U. C. Merzbach, A , John Wiley & Sons, 2011.

255 256 Bibliography

[18] R. E. Bradley and E. Sandifer, Leonhard Euler: Life, work and legacy, vol. 5, Elsevier, 2007. [19] D. A. Brannan, On a conjecture of Ilieff, Proc. Cambridge Philos. Soc. 64 (1968), 83– 85. MR0220906 [20] M. A. Brilleslyper and B. Schaubroeck, Explorations of the Gauss–Lucas theorem, PRIMUS 27 (2017), no. 8-9, 766–777. [21] J. E. Brown and G. Xiang, Proof of the Sendov conjecture for polynomials of degree at most eight, J. Math. Anal. Appl. 232 (1999), no. 2, 272–292, DOI 10.1006/jmaa.1999.6267. MR1683144 [22] R. B. Burckel, Iterating analytic self-maps of discs, Amer. Math. Monthly 88 (1981), no. 6, 396–407, DOI 10.2307/2321822. MR622955 [23] W. Calbeck, Elliptic numerical ranges of 3 × 3 companion matrices, Linear Algebra Appl. 428 (2008), no. 11-12, 2715–2722, DOI 10.1016/j.laa.2007.12.018. MR2416583 [24] G. Cassier and I. Chalendar, The group of the invariants of a finite Blaschke product, Complex Variables Theory Appl. 42 (2000), no. 3, 193–206, DOI 10.1080/17476930008815283. MR1788126 [25] W. Chapple, An essay on the properties of triangles inscribed in and circumscribed about two given circles, Miscellanea Curiosa Mathematica 4 (1746), 117–124. [26] M.-D. Choi and C.-K. Li, Constrained unitary dilations and numerical ranges, J. Op- erator Theory 46 (2001), no. 2, 435–447. MR1870416 [27] M. Chuaqui and C. Pommerenke, On Schwarz–Christoffel mappings, Pacific J. Math. 270 (2014), no. 2, 319–334, DOI 10.2140/pjm.2014.270.319. MR3253684 [28] W. Cieślak and E. Szczygielska, On Poncelet’s porism, Ann. Univ. Mariae Curie- Skłodowska Sect. A 64 (2010), no. 2, 21–28, DOI 10.2478/v10062-010-0011-0. MR2771117 [29] J. B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathe- matics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR503901 [30] J. B. Conway, The theory of subnormal operators, Mathematical Surveys and Mono- graphs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR1112128 [31] R. Courant, Differential and integral calculus. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Translated from the German by E. J. McShane; Reprint of the 1936 original; A Wiley-Interscience Publication. MR1009559 [32] D. Courtney and D. Sarason, A mini-max problem for self-adjoint Toeplitz ma- trices, Math. Scand. 110 (2012), no. 1, 82–98, DOI 10.7146/math.scand.a-15198. MR2900072 [33] C. C. Cowen, Finite Blaschke products as compositions of other finite Blaschke prod- ucts, arXiv:1207.4010, 2012. [34] C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR1397026 [35] U. Daepp, P. Gorkin, and R. Mortini, Ellipses and finite Blaschke products, Amer. Math. Monthly 109 (2002), no. 9, 785–795, DOI 10.2307/3072367. MR1933701 [36] U. Daepp, P. Gorkin, A. Shaffer, B. Sokolowsky, and K. Voss, Decomposing fi- nite Blaschke products, J. Math. Anal. Appl. 426 (2015), no. 2, 1201–1216, DOI 10.1016/j.jmaa.2015.01.039. MR3314888 Bibliography 257

[37] U. Daepp, P. Gorkin, and K. Voss, Poncelet’s theorem, Sendov’s conjecture, and Blaschke products, J. Math. Anal. Appl. 365 (2010), no. 1, 93–102, DOI 10.1016/j.jmaa.2009.09.058. MR2585079 [38] C. Davis, The Toeplitz–Hausdorff theorem explained, Canad. Math. Bull. 14 (1971), 245–246, DOI 10.4153/CMB-1971-042-7. MR0312288 [39] J. Dégot, Sendov conjecture for high degree polynomials, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1337–1349, DOI 10.1090/S0002-9939-2014-11888-0. MR3162254 [40] J. Ding and T. H. Fay, The Perron–Frobenius theorem and limits in geometry, Amer. Math. Monthly 112 (2005), no. 2, 171–175, DOI 10.2307/30037416. MR2121328 [41] W. F. Donoghue Jr., On the numerical range of a bounded operator, Michigan Math. J. 4 (1957), 261–263. MR0096127 [42] R. G. Douglas, S. Sun, and D. Zheng, Multiplication operators on the Bergman space via analytic continuation, Adv. Math. 226 (2011), no. 1, 541–583, DOI 10.1016/j.aim.2010.07.001. MR2735768 [43] V. Dragović and M. Radnović, Poncelet porisms and beyond, Frontiers in Mathemat- ics, Birkhäuser/Springer Basel AG, Basel, 2011. Integrable billiards, hyperelliptic Ja- cobians and pencils of . MR2798784 [44] S. W. Drury, Symbolic calculus of operators with unit numerical radius, Lin- ear Algebra Appl. 428 (2008), no. 8-9, 2061–2069, DOI 10.1016/j.laa.2007.11.007. MR2401640 [45] J. Eising, D. Radcliffe, and J. Top, A simple answer to Gelfand’s question, Amer. Math. Monthly 122 (2015), no. 3, 234–245, DOI 10.4169/amer.math.monthly.122.03.234. MR3327713 [46] P. A. Fillmore, On similarity and the diagonal of a matrix, Amer. Math. Monthly 76 (1969), 167–169, DOI 10.2307/2317264. MR0237526 [47] L. Flatto, Poncelet’s theorem, American Mathematical Society, Providence, RI, 2009. Chapter 15 by S. Tabachnikov. MR2465164 [48] A. Fletcher, Unicritical Blaschke products and domains of ellipticity, Qual. Theory Dyn. Syst. 14 (2015), no. 1, 25–38, DOI 10.1007/s12346-015-0133-4. MR3326210 [49] M. Frantz, How conics govern Möbius transformations, Amer. Math. Monthly 111 (2004), no. 9, 779–790, DOI 10.2307/4145189. MR2104049 [50] E. Fricain and J. Mashreghi, The theory of ℋ(푏) spaces. Vol. 2, New Mathematical Monographs, vol. 21, Cambridge University Press, Cambridge, 2016. MR3617311 [51] M. Fujimura, Inscribed ellipses and Blaschke products, Comput. Methods Funct. The- ory 13 (2013), no. 4, 557–573, DOI 10.1007/s40315-013-0037-8. MR3138353 [52] N. Fuss, De quadrilateris quibus circulum tam inscribere quam circumscribere licet, Nova Acta Acad. Sci. Petrop. 10 (1797), 103–125. [53] M. Gardner, New mathematical diversions, revised edition, MAA Spectrum, Mathe- matical Association of America, Washington, DC, 1995. MR1335231 [54] J. B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR2261424 [55] H.-L. Gau and P. Y. Wu, Numerical range of 푆(휙), Linear Multilinear Algebra 45 (1998), no. 1, 49–73. [56] H.-L. Gau and P. Y. Wu, Lucas’ theorem refined, Linear and Multilinear Algebra 45 (1999), no. 4, 359–373, DOI 10.1080/03081089908818600. MR1684719 258 Bibliography

[57] H.-L. Gau and P. Y. Wu, Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl. 364 (2003), 213–222, DOI 10.1016/S0024-3795(02)00548- 7. MR1971096 [58] H.-L. Gau and P. Y. Wu, Numerical range and Poncelet property, Taiwanese J. Math. 7 (2003), no. 2, 173–193, DOI 10.11650/twjm/1500575056. MR1978008 [59] H.-L. Gau and P. Y. Wu, Companion matrices: reducibility, numerical ranges and similarity to contractions, Linear Algebra Appl. 383 (2004), 127–142, DOI 10.1016/j.laa.2003.11.027. MR2073899 [60] H.-L. Gau and P. Y. Wu, Numerical ranges of companion matrices, Linear Algebra Appl. 421 (2007), no. 2-3, 202–218, DOI 10.1016/j.laa.2006.03.037. MR2294336 [61] C. Glader, Minimal degree rational unimodular interpolation on the unit , Elec- tron. Trans. Numer. Anal. 30 (2008), 88–106. MR2480071 [62] A. W. Goodman, Q. I. Rahman, and J. S. Ratti, On the zeros of a and its derivative, Proc. Amer. Math. Soc. 21 (1969), 273–274, DOI 10.2307/2036982. MR0239062 [63] P. Gorkin, L. Laroco, R. Mortini, and R. Rupp, Composition of inner functions, Results Math. 25 (1994), no. 3-4, 252–269, DOI 10.1007/BF03323410. MR1273115 [64] P. Gorkin and R. C. Rhoades, Boundary interpolation by finite Blaschke prod- ucts, Constr. Approx. 27 (2008), no. 1, 75–98, DOI 10.1007/s00365-006-0646-3. MR2336418 [65] P. Gorkin and E. Skubak, Polynomials, ellipses, and matrices: two ques- tions, one answer, Amer. Math. Monthly 118 (2011), no. 6, 522–533, DOI 10.4169/amer.math.monthly.118.06.522. MR2812283 [66] P. Gorkin and N. Wagner, Ellipses and compositions of finite Blaschke products, J. Math. Anal. Appl. 445 (2017), no. 2, 1354–1366, DOI 10.1016/j.jmaa.2016.01.067. MR3545246 [67] T. Gosset, “The Kiss Precise (Generalized)”, in Strange attractors: Poems of love and mathematics (S. Glaz and J. Growney, editors), A K Peters, Ltd., Wellesley, MA, 2008, 189. MR 2490399 [68] J. W. Green, Classroom notes: On the envelope of curves given in parametric form, Amer. Math. Monthly 59 (1952), no. 9, 626–628, DOI 10.2307/2306769. MR1528268 [69] P. Griffiths and J. Harris, A Poncelet theorem in space, Comment. Math. Helv. 52 (1977), no. 2, 145–160, DOI 10.1007/BF02567361. MR0498606 [70] K. Gustafson, The Toeplitz–Hausdorff theorem for linear operators, Proc. Amer. Math. Soc. 25 (1970), 203–204, DOI 10.2307/2036559. MR0262849 [71] E. Gutkin, The Toeplitz–Hausdorff theorem revisited: relating linear algebra and geometry, Math. Intelligencer 26 (2004), no. 1, 8–14, DOI 10.1007/BF02985393. MR2034035 [72] U. Haagerup and P. de la Harpe, The numerical radius of a nilpotent opera- tor on a Hilbert space, Proc. Amer. Math. Soc. 115 (1992), no. 2, 371–379, DOI 10.2307/2159255. MR1072339 [73] A. J. Hahn, Mathematical excursions to the world’s great buildings, Princeton Uni- versity Press, Princeton, NJ, 2012. MR2962336 [74] L. Halbeisen and N. Hungerbühler, A simple proof of Poncelet’s theorem (on the oc- casion of its bicentennial), Amer. Math. Monthly 122 (2015), no. 6, 537–551, DOI 10.4169/amer.math.monthly.122.6.537. MR3361732 Bibliography 259

[75] P. R. Halmos, Numerical ranges and dilations, Acta Sci. Math. (Szeged) 25 (1964), 1–5. MR0171168 [76] P. R. Halmos, A Hilbert space problem book, 2nd ed., Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR675952 [77] T. R. Harris, M. Mazzella, L. J. Patton, D. Renfrew, and I. M. Spitkovsky, Numerical ranges of cube roots of the identity, Linear Algebra Appl. 435 (2011), no. 11, 2639– 2657, DOI 10.1016/j.laa.2011.03.020. MR2825272 [78] F. Hausdorff, Der Wertvorrat einer Bilinearform (German), Math. Z. 3 (1919), no. 1, 314–316, DOI 10.1007/BF01292610. MR1544350 [79] W. K. Hayman, Research problems in function theory, The Athlone Press University of London, London, 1967. MR0217268 [80] W. M. Higdon, On the numerical ranges of composition operators induced by map- pings with the Denjoy–Wolff point on the boundary, Integral Equations Operator The- ory 85 (2016), no. 1, 127–135, DOI 10.1007/s00020-016-2287-0. MR3503182 [81] T. P. Hill, A statistical derivation of the significant-digit law, Statist. Sci. 10 (1995), no. 4, 354–363. MR1421567 [82] J. Hilmar and C. Smyth, Euclid meets Bézout: intersecting algebraic curves with the Euclidean algorithm, Amer. Math. Monthly 117 (2010), no. 3, 250–260, DOI 10.4169/000298910X480090. MR2640851 [83] H. Hilton, Plane algebraic curves, Oxford University Press, 1920. [84] K. Hoffman and R. Kunze, Linear algebra, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR0276251 [85] J. Holbrook and J.-P. Schoch, Theory vs. experiment: multiplicative inequalities for the numerical radius of commuting matrices, Topics in operator theory. Volume 1. Oper- ators, matrices and analytic functions, Oper. Theory Adv. Appl., vol. 202, Birkhäuser Verlag, Basel, 2010, pp. 273–284, DOI 10.1007/978-3-0346-0158-0_14. MR2723281 [86] C. G. J. Jacobi, Gesammelte Werke. Bände I–VIII (German), Herausgegeben auf Ve- ranlassung der Königlich Preussischen Akademie der Wissenschaften. Zweite Aus- gabe, Chelsea Publishing Co., New York, 1969. MR0260557 [87] A. Jamain, Benford’s law, Master’s thesis, Imperial College of London (2001). [88] D. Kalman, Solving the ladder problem on the back of an envelope, Math. Mag. 80 (2007), no. 3, 163–182. MR2322082 [89] Y. Katznelson, An introduction to harmonic analysis, 3rd ed., Cambridge Mathemat- ical Library, Cambridge University Press, Cambridge, 2004. MR2039503 [90] D. S. Keeler, L. Rodman, and I. M. Spitkovsky, The numerical range of 3 × 3 matri- ces, Linear Algebra Appl. 252 (1997), 115–139, DOI 10.1016/0024-3795(95)00674-5. MR1428632 [91] S. M. Kerawala, Poncelet porism in two circles, Bull. Calcutta Math. Soc. 39 (1947), 85–105. MR0026339 [92] D. Khavinson, R. Pereira, M. Putinar, E. B. Saff, and S. Shimorin, Borcea’s variance conjectures on the critical points of polynomials, Notions of positivity and the ge- ometry of polynomials, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2011, pp. 283–309, DOI 10.1007/978-3-0348-0142-3_16. MR3051172 [93] C. Kimberling, The and history of the ellipse in Washing- ton, D.C., Department of Mathematics, University of Evansville, http://faculty.evansville.edu/ck6/ellipse.pdf. (Accessed October 2016). 260 Bibliography

[94] J. L. King, Three problems in search of a measure, Amer. Math. Monthly 101 (1994), no. 7, 609–628, DOI 10.2307/2974690. MR1289271 [95] R. Kippenhahn, Über den Wertevorrat einer Matrix (German), Math. Nachr. 6 (1951), 193–228, DOI 10.1002/mana.19510060306. MR0059242 [96] R. Kippenhahn, On the numerical range of a matrix, Linear Multilinear Algebra 56 (2008), no. 1-2, 185–225, DOI 10.1080/03081080701553768. Translated from the Ger- man by Paul F. Zachlin and Michiel E. Hochstenbach [MR0059242]. MR2378310 [97] H. Klaja, J. Mashreghi, and T. Ransford, On mapping theorems for numerical range, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3009–3018, DOI 10.1090/proc/12955. MR3487232 [98] A. Kock, Envelopes—notion and definiteness, Beiträge Algebra Geom. 48 (2007), no. 2, 345–350. MR2364794 [99] J. C. Lagarias, C. L. Mallows, and A. R. Wilks, Beyond the Descartes circle theorem, Amer. Math. Monthly 109 (2002), no. 4, 338–361, DOI 10.2307/2695498. MR1903421 [100] J. S. Lancaster, The boundary of the numerical range, Proc. Amer. Math. Soc. 49 (1975), 393–398, DOI 10.2307/2040652. MR0372644 [101] J. C. Langer and D. A. Singer, Foci and foliations of real algebraic curves, Milan J. Math. 75 (2007), 225–271, DOI 10.1007/s00032-007-0078-4. MR2371544 [102] C.-K. Li, A simple proof of the elliptical range theorem, Proc. Amer. Math. Soc. 124 (1996), no. 7, 1985–1986, DOI 10.1090/S0002-9939-96-03307-2. MR1322932 [103] H. Licks, Recreations in mathematics, D. Van Nostrand Company, New York, 1917. Available online in public domain. [104] E. H. Lockwood, A book of curves, Cambridge University Press, New York, 1961. MR0126191 [105] C. Maclaurin, A treatise on fluxions, Ruddimans, Edinburgh, 1742. [106] F. Malmquist, Sur la détermination d’une classe de fonctions analytiques par leurs valeurs dans un ensemble donné de poits, in C.R. 6ième Cong. Math. Scand. Kopen- hagen, 1925. Gjellerups, Copenhagen, 1926, pp. 253–259. [107] M. Marcus and B. N. Shure, The numerical range of certain 0, 1-matrices, Linear and Multilinear Algebra 7 (1979), no. 2, 111–120, DOI 10.1080/03081087908817266. MR529878 [108] M. Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, Amer- ican Mathematical Society, Providence, R.I., 1966. MR0225972 [109] M. Marden, The search for a Rolle’s theorem in the complex domain, Amer. Math. Monthly 92 (1985), no. 9, 643–650, DOI 10.2307/2323710. MR810661 [110] J. Mashreghi, Derivatives of inner functions, Fields Institute Monographs, vol. 31, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013. MR2986324 [111] V. J. Matsko, Generic ellipses as envelopes, Math. Mag. 86 (2013), no. 5, 358–365, DOI 10.4169/math.mag.86.5.358. MR3141737 [112] S. J. Miller, A quick introduction to Benford’s law, Benford’s law: Theory and applications, Princeton Univ. Press, Princeton, NJ, 2015, pp. 3–22, DOI 10.1515/9781400866595. MR3411056 [113] S. Mills, Note on the Braikenridge–Maclaurin theorem, Notes and Records Roy. Soc. London 38 (1984), no. 2, 235–240, DOI 10.1098/rsnr.1984.0014. MR783589 [114] B. Mirman, Numerical ranges and Poncelet curves, Linear Algebra Appl. 281 (1998), no. 1-3, 59–85, DOI 10.1016/S0024-3795(98)10037-X. MR1645335 Bibliography 261

[115] B. Mirman, V. Borovikov, L. Ladyzhensky, and R. Vinograd, Numerical ranges, Pon- celet curves, invariant measures, Linear Algebra Appl. 329 (2001), no. 1-3, 61–75, DOI 10.1016/S0024-3795(01)00233-6. MR1822222 [116] H. F. Montague, Envelopes associated with a one-parameter family of straight Lines, Natl. Math. Mag. 13 (1938), no. 2, 73–75. MR1569600 [117] J. Munkres, Topology, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 2000. [118] R. Nevanlinna and V. Paatero, Introduction to complex analysis, Addison-Wesley Publishing Co., Reading, .-London-Don Mills, Ont., 1969. Translated from the German by T. Kövari and G. S. Goodman. MR0239056 [119] T. W. Ng and C. Y. Tsang, Chebyshev-Blaschke products: solutions to certain approxi- mation problems and differential equations, J. Comput. Appl. Math. 277 (2015), 106– 114, DOI 10.1016/j.cam.2014.08.028. MR3272168 [120] M. Nigrini, Benford’s law: Applications for forensic accounting, auditing and fraud detection, vol. 586, John Wiley & Sons, 2012. [121] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathemat- ical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. MR1892647 [122] J. J. O’Connor and E. F. Robertson, The MacTutor history of mathematics archive, School of Mathematics and , University of St. Andrews, Scotland, http://www-history.mcs.st-and.ac.uk/. (Accessed 12/15/2017). [123] C. S. Ogilvy, Excursions in geometry, Courier Corporation, 1990. [124] B. C. Patterson, The origins of the geometric principle of inversion, Isis 19 (1933), no. 1, 154–180. [125] J. V. Poncelet, Traité des propriétés projectives des figures: ouvrage utile à ceux qui s’ occupent des applications de la géométrie descriptive et d’opérations géométriques sur le terrain. T. 2, Gauthier-Villars, Imprimeur-Libraire, 1866. [126] T. E. Price, Products of Chord Lengths of an Ellipse, Math. Mag. 75 (2002), no. 4, 300– 307. MR1573631 [127] P. Psarrakos and M. Tsatsomeros, Numerical range: (in) a matrix nutshell, No. Bd. 1 in Mathematics Notes from Washington State University, Department of Mathematics, Washington State University, 2002. [128] H. Queffélec and K. Seip, Decay rates for approximation numbers of composition operators, J. Anal. Math. 125 (2015), 371–399, DOI 10.1007/s11854-015-0012-6. MR3317907 [129] G. Quenell, Envelopes and string art, Math. Mag. 82 (2009), no. 3, 174–185, DOI 10.4169/193009809X468779. MR2522910 [130] Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathemat- ical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford Univer- sity Press, Oxford, 2002. MR1954841 [131] J. Rickards, When is a polynomial a composition of other polynomials?, Amer. Math. Monthly 118 (2011), no. 4, 358–363, DOI 10.4169/amer.math.monthly.118.04.358. MR2800347 [132] J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), no. 1, 51–66, DOI 10.2307/1988911. MR1501189 [133] H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR1013117 262 Bibliography

[134] Z. Rubinstein, On the approximation by 퐶-polynomials, Bull. Amer. Math. Soc. 74 (1968), 1091–1093, DOI 10.1090/S0002-9904-1968-12057-9. MR0232003 [135] D. Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR1289670 [136] G. Schmeisser, Bemerkungen zu einer Vermutung von Ilieff (German), Math. Z. 111 (1969), 121–125, DOI 10.1007/BF01111192. MR0264040 [137] I. J. Schoenberg, A conjectured analogue of Rolle’s theorem for polynomials with real or complex coefficients, Amer. Math. Monthly 93 (1986), no. 1, 8–13, DOI 10.2307/2322536. MR824585 [138] G. Semmler and E. Wegert, Boundary interpolation with Blaschke products of minimal degree, Comput. Methods Funct. Theory 6 (2006), no. 2, 493–511, DOI 10.1007/BF03321626. MR2291147 [139] Bl. Sendov, Generalization of a conjecture in the geometry of polynomials, Serdica Math. J. 28 (2002), no. 4, 283–304. MR1965232 [140] J. H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR1237406 [141] T. Sheil-Small, Complex polynomials, Cambridge Studies in Advanced Mathematics, vol. 75, Cambridge University Press, Cambridge, 2002. MR1962935 [142] D. A. Singer, The location of critical points of finite Blaschke products, Conform. Geom. Dyn. 10 (2006), 117–124, DOI 10.1090/S1088-4173-06-00145-7. MR2223044 [143] F. Soddy, The Kiss Precise, Nature 137 (1936), 1021. [144] N. Stefanović and M. Milošević, A very simple proof of Pascal’s hexagon theorem and some applications, Proc. Indian Acad. Sci. Math. Sci. 120 (2010), no. 5, 619–629, DOI 10.1007/s12044-010-0047-7. MR2779392 [145] B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic analysis of operators on Hilbert space, revised and enlarged edition, Universitext, Springer, New York, 2010. MR2760647 [146] S. Takenaka, On the orthogonal functions and a new formula of interpolation, Jap. J. Math. 2 (1925), 129–145. [147] O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejér (German), Math. Z. 2 (1918), no. 1-2, 187–197, DOI 10.1007/BF01212904. MR1544315 [148] J. van Yzeren, A simple proof of Pascal’s hexagon theorem, Amer. Math. Monthly 100 (1993), no. 10, 930–931, DOI 10.2307/2324214. MR1252929 [149] J. L. Walsh, Interpolation and functions analytic interior to the unit circle, Trans. Amer. Math. Soc. 34 (1932), no. 3, 523–556, DOI 10.2307/1989366. MR1501650 [150] J. L. Walsh, Note on the location of zeros of the derivative of a rational function whose zeros and poles are symmetric in a circle, Bull. Amer. Math. Soc. 45 (1939), no. 6, 462–470, DOI 10.1090/S0002-9904-1939-07012-2. MR1564005 [151] E. Wegert, Visual complex functions, Birkhäuser/Springer Basel AG, Basel, 2012. An introduction with phase portraits. MR3024399 [152] E. W. Weisstein, Poncelet’s porism, from MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/PonceletsPorism.html. (Accessed July 2017). [153] J. W. Young, Projective geometry, The Carus Mathematical Monographs, The Math- ematical Association of America, Chicago, 1930. [154] N. Young, An introduction to Hilbert space, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1988. MR949693 Index

(푢퐻2)⟂, 112 ⟨⟨푥⟩⟩, 90 퐴, 117 ⟨푥, 푦⟩, 13 퐴⋆, 62 ⌊푥⌋, 98 퐵퐻2, 109 ℂ, 14 퐶(퐴), 163 ℂ∗, 134 푛 퐶휑, 236 ℂ , 13 2 퐶푀푎,휙 ∶ ℬ → ℬ, 192 ℙ (ℂ), 163 퐻2, 106 ℙ2(ℝ), 48 퐻2(픻), 105 ℝ∗, 134

퐻퐴 + 푖퐾퐴, 161 핋, 17 + 퐼ℓ, 148 퐇 , 134 − 퐼ℓ,푚, 148 퐇 , 134 퐽3, 164 ℬ, 192 퐽푛, 103 풮푛, 103 퐾ᵆ, 112 ‖퐴‖, 63, 76 퐿2(핋), 104 ‖푥‖, 13

푀푒, 162 휔+, 242 푀 , 177 휔−, 242 푎,휙 ←→ 푃+, 113 푢푣, 54, 179 푃−, 113 휓푐, 169, 177 푃ᵆ, 113 휎(퐸), 123, 232 ̃ 푆, 108 푘푎, 107 푆⟂, 71 tr(퐴), 16, 234 ⋆ 푆 , 111 휑푎, 114 2 푆ᵆ, 113 푒푗, standard basis 퐿 (핋), 104 푛 푈휆, 119 푒푗, standard basis ℂ , 63 푊(퐴), 14 푓(푛), 92

푋푝,푞, 148 푓푟, 105 푓(푛)̂ , 105 푘푎, 107 푤(퐴), 232

263 264 Index

푤(푇), 223 Braikenridge–MacLaurin theorem, 57 adjoint, 16 Brianchon’s theorem, 55 affine part, 59 Brianchon, Charles, 57 algebraic curve, 51 applets ,, x Cauchy kernel, 107 Blaschke product tools, 29–33, 121,Cauchy–Bunyakovsky–Schwarz in- 159, 170, 177, 209, 251 equality, 66, 79, 108

experimental tools, 138 chain of circles between 풞1 and 풞2, geometry tools, 8, 53, 56, 217 213 astroid, 204 Chapple, William, 45 Chapple–Euler formula, 43–46, 175, Bedford, Frank, 88 197 Bell, Eric Temple, 1 circular point, 242 Benford’s Law, 88 circumscribing a conic, 55 Beurling’s theorem, 111 class 풮푛, 103, 162 Blaschke Colosseum, 3 curve, 129, 174 companion matrix, 245 ellipse complete orthonormal basis, 105 (푛, 푝)-ellipse, 184 composition operator, 192, 236 3-ellipse, 33, 43, 60, 82 compressed shift operator, 113 product, 19, 23, 28 compression, 71 characterization, 19, 26 matrix, 117 composition, 159, 169, 171 conic, 51 decomposable, 169 associated matrix, 51 degree, 19 degenerate, 52 degree-2, 35 constructing an ellipse degree-3, 37–44, 82 envelope method, 201 degree-4, 169–174 folding, 7–10 finite, 19 pin and string method, 201 infinite, 111 contraction, 76 normalized, 23 curve properties, 28, 37–38 algebraic, 51 unicritical, 253 class, 163, 241 Blaschke, Wilhelm, 19 degree, 163, 241 Borcea, J., 209 dual, 51, 163, 241 boundary generating curve, 163 Index 265

real part, 163 perimeter, 5–7 , 51 Ellipse, President’s Park South, 3 cyclic quadrilateral, 221 elliptical range theorem, 16, 62, 68– 71, 164, 173, 222 Dégot, J., 207 envelope, 202, 204 degenerate, 52 Euclidean norm, 13 Denjoy–Wolff point, 251 Euler, Leonhard, 45 Denjoy–Wolff theorem, 251 Descartes’s circle theorem, 221 Fatou’s radial limit theorem, 106 Descartes, René, 85 field of values, 61 diagonal, 55 first digit, 88 direct sum focus, 242 of matrices, 166 ellipse, 3 disk automorphism, 23, 134, 177 real, 242 elliptic, 178 singular, 243 canonical, 182 Fourier coefficient, 105 characterization, 178 Frantz’s theorem, 179 convex order, 182 frequency, 90 order, 182 Fuss’s theorem, 175, 197 hyperbolic, 178 Fuss, Nicolaus, 45 parabolic, 178 dual, 51, 241–242 Gauss–Lucas theorem, 206 dual curve, 163 Gelfand’s questions, 90, 98 dual , 241 general position duality, 47, 57 of lines, 55 of points, 53 eigenvalue, 14, 232 Gergonne, Joseph, 57 Hermitian matrix, 161 glissette, 224 eigenvector, 14 ellipse, 3 half-plane area enclosed, 5 lower, 134 construction, 8–10 upper, 134 , 3 Halmos, Paul, vii, 125 equation, 4 Hardy space, 105 focus, 3 Hermitian matrix major axis, 3 eigenvalue, 161 minor axis, 4 numerical range, 161 266 Index hexagon Kippenhahn, Rudolf, 123, 159–160 diagonal, 55 inscribed in conic, 53 Lagrange form, 140 opposite sides, 53 Lebesgue space, 104 opposite vertices, 55 Lebesgue, Henri, 104 principal diagonal, 55 , 48 side, 53 linear fractional transformation, 134 simple, 53 Möbius transformations, 134 hexagrammum mysticum theorem, Maclaurin expansion, 7 57 Maclaurin, C., 7 Hilbert space, 76 major axis, 3 homogeneous polynomial, 51 Mandart , 212 hyperbolic Marden, M., 210 convex hull, 209 matrix convex set, 209 adjoint, 16 hyperbolic geometry, 208 associated to conic, 51 Illiev, L., 206 Cartesian decomposition, 161 inner function, 109 companion, 245 inner product direct sum, 166 퐿2(핋), 104 Hermitian, 160, 222 standard, 13 imaginary part, 161 inscribed in a conic, 53 nilpotent, 222 interpolation, 134, 237–241 nonnegative, 228 invariant subspace, 109, 111 norm, 63, 76 inverse of a point, 216 normal, 62 inversion, 215–221 normal, numerical range, 68, 75 iterate, 92, 182, 251 positive, 228 positive definite, 222 Jensen’s theorem, 206 real part, 161 Jordan block, 78, 103, 131, 164, 168, reducible, 124, 246 222 self-adjoint, 62, 160 similar, 63 Kepler, Johannes, 6 trace, 16, 69, 234 King, Jonathan, 91 unitarily equivalent, 63 Kippenhahn curve, 163 unitary, 62, 75 Kippenhahn’s theorem, 163 unitary decomposition, 123 Index 267 maximum modulus theorem, 26 opposite vertices, 55 measure, 95–97 orthogonal complement, 71 Mersenne, Marin, 57 orthogonal projection, 71 minor axis, 4 , 165 model space, 112 parallel postulate, 208 negative , 227 Parseval’s identity, 105 Nelson, Wayne James, 87 Pascal line, 57 nephroid, 205 Pascal’s theorem, 53 nested form, 240 Pascal, Blaise, 57 Newcomb, Simon, 88 pedal curve, 227 Newton form, 238 pedal point, 227 nilpotent, 222 Perron–Frobenius theorem, 228 nonnegative matrix, 228 phase portrait, 30–32 norm Poincaré model, 208 Euclidean, 13 points at infinity, 48 matrix, 63 polygonal chain vector, 13 inscribed in conic, 53 normalized Blaschke product, 23 polynomial numerical radius homogenous, 51 matrix, 224, 232 Poncelet operator, 223 curve, 121, 130 numerical range, 14, 61 ellipse circular disk, 174 3-ellipse, 60, 82 elliptical disk, 159, 168, 171 4-ellipse, 175 Hermitian matrix, 161, 222 convex, 175 normal matrix, 75 property, 121 properties, 63, 68 Poncelet’s theorem, 92 alternate version, 157 operator false statements, 158 quasi-nilpotent, 223 for triangles, 18, 57 bounded, 76 general, 156 contraction, 77 Poncelet, Jean-Victor, 47 nilpotent, 222 porism, 213 norm, 77 positive matrix, 228 shift, 108 principal diagonal, 55 opposite sides, 53 268 Index projection, 71, 113 , 39, 211 projective geometry, 47 Steiner’s porism, 213 projective coordinates, 48 Steiner’s theorem, 210 , 48 Stone–Weierstrass theorem, 26 Ptolemy’s theorem, 221 strongly real of positive type, 135, 239 quasi-nilpotent, 223 support line, 124, 162–164 radial limit, 106 Takenaka–Malmquist basis, 114 radical axis, 219 tangent line, 51 rational function, 135 tangential equation, 241 degree, 135 Toeplitz, Otto, 61 reference circle, 216 Toeplitz–Hausdorff theorem, 61–62, reproducing kernel, 107 71–73 normalized, 107, 114 trace, 16, 234 Traité des propriété projectives des Saratov notebook, 47 figures, 48 Schmeisser, G., 207 Schur’s theorem, 64 unit circle, 17 Sendov’s conjecture, 206 unitary 1-dilation, 76–82, 119 shift unitary dilation, 76 backward, 111 forward, 109 vector, 49 shift operator, 108 cross product, 49 adjoint, 111 , 49 compressed, 113 stochastic, 229 side of a polygonal chain, 53 Siebeck’s theorem Walsh’s two-circle theorem, 206 general, 243 Weyl’s theorem, 98 triangles, 211 zero inclusion question, 236 Soddy’s formula, 221 spectral radius, 231 spectral theorem for normal matrices, 67 spectrum, 123, 232 standard form, 117 standard inner product, 13 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS

Mathematicians delight in finding surprising connections between seemingly disparate of mathematics. Whole domains of modern mathematics have arisen from exploration of such connections—consider analytic number theory or algebraic topology. Finding Ellipses is a delight-filled romp across a three-way unexpected connection between complex analysis, linear algebra, and projective geometry. The book begins with Blaschke products, complex-analytic functions that are generalizations of disk automorphisms. In the analysis of Blaschke products, we encounter, in a quite natural way, an ellipse inside the unit disk. The story continues by introducing the reader to Poncelet’s theorem—a beautiful result in projective geometry that ties together two conics and, in particular, two ellipses, one circum- scribed by a polygon that is inscribed in the second. The Blaschke ellipse and the Poncelet ellipse turn out to be the same ellipse, and the connection is illuminated by considering the numerical range of a 2 × 2 matrix. The numerical range is a convex subset of the complex plane that contains information about the geometry of the transformation represented by a matrix. Through the numerical range of n × n matrices, we learn more about the interplay between Poncelet’s theorem and Blaschke products. The story ranges widely over analysis, algebra, and geometry, and the exposi- tion of the deep and surprising connections is lucid and compelling. Written for advanced undergraduates or beginning graduate students, this book would be the perfect vehicle for an invigorating and enlightening capstone exploration. The exer- cises and collection of extensive projects could be used as an embarkation point for a satisfying and rich research project. You are invited to read actively using the accompanying interactive website, which allows you to visualize the concepts in the book, experiment, and develop original conjectures.

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