CBMS. Toeplitz Approach to Problems of the Uncertainty Principle. Alexei Poltoratski. Regional Conference Series in Mathematics

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1 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 121 Toeplitz Approach to Problems of the Uncertainty Principle Alexei Poltoratski American Mathematical Society with support from the National Science Foundation

2 Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 121 Toeplitz Approach to Problems of the Uncertainty Principle Alexei Poltoratski Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation

3 NSF-CBMS Regional Conference in the Mathematical Sciences: Uncertainty Principles in Harmonic Analysis: Gap and Type Problems, held at Clemson University, Clemson, South Carolina, August 12 16, Partially supported by the National Science Foundation 2010 Mathematics Subject Classification. Primary 30-XX, 33-XX, 34-XX, 42-XX. For additional information and updates on this book, visit Library of Congress Cataloging-in-Publication Data Poltoratski, Alexei, 1966 Toeplitz approach to problems of the uncertainty principle / Alexei Poltoratski. pages cm. (CBMS regional conference series in mathematics ; number 121) Partially supported by the National Science Foundation. Based on the NSF-CBMS Regional Conference in the Mathematical Sciences on Uncertainty Principles in Harmonic Analysis: Gap and Type Problems, held at Clemson University, Clemson, South Carolina, August 12 16, Includes bibliographical references. ISBN (alk. paper) 1. Heisenberg uncertainty principle Congresses. 2. Functional analysis Congresses. 3. Toeplitz, Otto, I. National Science Foundation (U.S.) II. Title. QC H4P dc Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center s RightsLink R service. For more information, please visit: Send requests for translation rights and licensed reprints to reprint-permission@ams.org. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. 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

4 We demand rigidly defined areas of doubt and uncertainty! Douglas Adams iii

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6 Contents Thanks vii Chapter 1. Mathematical Shapes of Uncertainty 1 1. Basic notations 3 2. Variety of mathematical forms of UP 4 3. Function theoretic background 10 Chapter 2. Gap Theorems Classical Gap Theorems Spectral gap as a property of the support Toeplitz kernels and uniform approximation A formula for the gap characteristic of a set Examples and applications Appendix: Proof of the gap formula Appendix: Technical lemmas Appendix: De Branges theorem 66 in Toeplitz form 61 Chapter 3. A Problem by Pólya and Levinson Pólya sequences A theorem on existence of a de Branges space in L Beurling Malliavin densities Two theorems on Toeplitz kernels A description of Pólya sequences Technical lemmas Proofs of theorems 74 Chapter 4. Determinacy of Measures and Oscillations of High-pass Signals Sign changes of a measure with a spectral gap Entire functions and densities A lemma on d-uniform sequences M. Riesz-type criterion and its consequences Extreme measures in the indeterminate case Measures annihilating Paley-Wiener spaces Sign changes of measures with spectral gap 93 Chapter 5. Beurling Malliavin and Bernstein s Problems A problem on completeness of exponentials Structure of proof of BM Theorem: BM theory Bernstein s problem Semi-continuous weights Characteristic sequences 103 v

7 vi CONTENTS 6. Equivalence between weighted uniform and L p -approximation A criterion for completeness of polynomials in C W Lemmas and proofs Examples and corollaries 114 Chapter 6. The Type Problem General case p Known examples Polynomial decay Completeness of exponentials in L p and C W Main results Classical results and further corollaries Auxiliary statements Proofs of main results 139 Chapter 7. Toeplitz Approach to UP Spaces of entire functions and their zero sets Model spaces and Toeplitz operators Spectral theory Outer, inner and Herglotz functions Model spaces Weyl inner functions Modified Fourier transform Entire functions Square root transformation Dimension and triviality of Toeplitz kernels General form of Levinson s completeness theorem Applications to UP 164 Chapter 8. Toeplitz Version of the Beurling Malliavin Theory The language of Toeplitz kernels Super-exponential case Sub-exponential case The structure of BM theory One-sided Lipschitz condition for the Hilbert transform Triviality of Toeplitz kernels Non-triviality of Toeplitz kernels in Smirnov Nevanlinna class Multiplier theorem Non-triviality of Toeplitz kernels in Hardy spaces 204 Bibliography 211

8 Thanks These notes are based on a Conference Board of the Mathematical Sciences mini-course given in August of 2013 at Clemson, South Carolina. I am deeply grateful to Constanze Liaw, Mishko Mitkovski and Brett Wick for inviting me to be a CBMS speaker and for all the hard work they put into organizing the conference. More than 40 researchers attended the conference, several of whom contributed outstanding lectures. The lectures were given by Michael Lacey, Doron Lubinsky, Shahaf Nitzan, Bill Ross, Carl Sundberg and Sergei Treil. On behalf of the organizers and participants of the conference, I wish to thank the National Science Foundation and Conference Board of the Mathematical Sciences for their generous support. Much credit is due to Nikolai Makarov, whose ideas initiated the Toeplitz approach in the area of the Uncertainty Principle. The last two chapters of the notes are devoted to our joint work, where these methods were developed. I also would like to thank Misha Sodin for convincing me to apply this approach to the Gap and Type problems and for numerous invaluable discussions. My interest in the Pólya-Levinson problem and Bernstein s weighted uniform approximation also stems from communications with Nikolai and Misha. I am grateful to Alexander Eremenko for bringing the problem on oscillations of Fourier integrals to my attention. Thought-provoking collaboration with my former student Mishko Mitkovski produced chapters 3 and 4. Original results discussed in these notes were obtained with support of the NSF through grants DMS , DMS and DMS I am especially indebted to my wife Svetlana for helping me find the classical texts mentioned in the notes and for constant moral support. vii

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11 Bibliography [1] Alexandrov, A. Isometric embeddings of coinvariant subspaces of the shift operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 232 (1996), Issled. po Linein Oper. i Teor. Funktsii. 24, 5 15, 213; translation in J. Math. Sci. (New York) 92 (1998), no. 1, [2] Aleksandrov, A. B. On A-integrability of the boundary values of harmonic functions, Mat. Zametki 39 (1982), (Russian) [3] Akhiezer, N. I. On the weighted approximation of continuous functions by polynomials on the real axis, Uspekhi Mat. Nauk 11(56), 3-43, AMS Transl. (ser 2), 22 (1962), [4] Arnold s Problems, Fazis Moscow, (in Russian), [5] Amrein, W.O., Berthier, A.M. On support properties of L p -functions and their Fourier transforms. J. Punct. Anal. 24 N3 (1977) [6] Avdonin, S.A., Ivanov, S.A. Families of exponentials. Cambridge Univ. Press, Cambridge, 1995 [7] Baranov, A. Completeness and Riesz bases of reproducing kernels in model subspaces, Int. Math. Res. Notices, Vol. 2006, 34 pages. [8] Bakan, A. Representation of measures with polynomial denseness in Lp(R,dμ), 0 <p<1, and its application to determinate moment problems, Proc. Amer. Math. Soc., 136 (2008), no. 10, [9] Beckner, W. Inequalities in Fourier analysis. Annals of Mathematics, Vol. 102, No. 6 (1975) pp [10] Benedetto, J.J.Uncertainty Principle Inequalities and Spectrum Estimation, Recent Advances in Fourier Analysis and Its Applications NATO ASI Series Volume 315, 1990, pp [11] Benedetto, J., Heil, C. and Walnut, D. Uncertainty Principles for time-frequency operators, in: Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations, Oper. Theory Adv. Appl. 58, I. Gohberg, ed., Birkhaüser, Basel (1992), 1 25 [12] Benedicks, M. Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106 Nl (1985) [13] Bernstein, S. N. Le probleme de l approximation des fonctions continues sur tout l axe reel et l une de ses applications, Bull. Math. Soc. France, 52(1924), [14] Beurling, A. On quasianalyticity and general distributions, Mimeographed lecture notes, Summer institute, Stanford University (1961) [15] Beurling, A.Collected works, vols. I.-II. Birkhauser, Boston [16] Beurling, A., Malliavin, P. On Fourier transforms of measures with compact support, Acta Math. 107 (1962), [17] Beurling, A., Malliavin, P. On the closure of characters and the zeros of entire functions, Acta Math. 118 (1967), [18] Boas,R.P.Entire Functions, Academic Press, New York, 1954 [19] Borg, G. Uniqueness theorems in the spectral theory of y +(λ q(x))y =0, Proc. 11- th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, [20] Borichev, A.A. Boundary uniqueness theorems for almost analytic functions and asymmetric algebras, Math. Sbornik, 136 N3 (1988) [21] Borichev, A., Sodin, M. Weighted exponential approximation and non-classical orthogonal spectral measures, Adv. in Math., 226, (2011) [22] Böttcher A., Silbermann B. Analysis of Toeplitz operators. Academie-Verlag and Springer- Verlag, Berlin,

12 212 BIBLIOGRAPHY [23] Borichev, A.A., Volberg, A.L. Uniqueness theorems for almost analytic functions. Algebra and Analysis, (1989) [24] Bourgain, J. A remark on the uncertainty principle for Hilbertian basis, J. FUNCT. ANAL. 01/1988; 79(1): [25] Bourgain, J. A problem of Douglas and Rudin on factorization, Pacific J. Math. 121 (1986), [26] De Branges, L. Hilbert spaces of entire functions. Prentice-Hall, Englewood Cliffs, NJ, 1968 [27] de Branges, L., The Stone-Weierstrass theorem, Proc. Amer. Math. Soc., 10, 1959, [28] de Branges, L., Some applications of spaces of entire functions, Canad. J. Math., 15, 1963, [29] De Branges, L. The Bernstein problem, Proc. Amer. Math. Soc., 10(1959), [30] De Branges, L. Some Hilbert spaces of entire functions II, Trans. Amer. Math. Soc. 99 (1961), [31] Bruna J., Olevskii, A., Ulanovskii, A. Completeness in L 1 (R) of discrete translates and related questions for quasi-analytic classes, Rev. Mat. Iberoamericana 22 (2005), 1 16 [32] J.-F. Burnol Two complete and minimal systems associated with the zeros of the Riemann zeta function, Journal de Théorie des Nombres de Bordeaux 16 (2004), [33] Carleson, L. Bernstein s approximation problem, Proc. Amer. Math. Soc., 2(1951), [34] Calderón,A.,Zygmund,A.On the existence of certain singular integrals, Acta Math. 88 (1952), [35] Clark, D. One dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972), [36] Coifman, R.R., Weiss, G. Extensions of Hardy spaces and their use in analysis, Bull. AMS 83 (1977), [37] Cornu, F. and Jancovici, B. On the two-dimensional Coulomb gas, Journal of statistical physics, 49:1-21-2, 33 56, Springer, [38] Duffin, R., Schaeffer, A. Power series with bounded coefficients, American Journal of Mathematics, 67 (1945), [39] Daubechies, I. Ten Lectures of Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1992 [40] Dyakonov, K. Zero sets and multiplier theorems for star-invariant subspaces, J. Anal. Math. 86 (2002), [41] Dym, H. On the span of trigonometric sums in weighted L 2 spaces, Linear and Complex Analysis Problem Book 3, Part II, Lecture Notes in Math., Springer, 1994, [42] Dym, H. An introduction to de Branges spaces of entire functions with applications to differential equations of the Sturm-Lioville type, Advances in Math. 5 (1971), [43] Dym H., McKean H.P. Gaussian processes, function theory and the inverse spectral problem Academic Press, New York, 1976 [44] Dzhrbashyan, M.M. Uniquencss theorems for Fourier transforms and infinitely differentiable func- tions. Izv. AN ArmSSR, ser. Hz-mat, 10, N6-7 (1957) (Russian). [45] Eremenko, A., Novikov, D., Oscillation of Fourier integrals with a spectral gap, J. demath. Pures Appl. 83, 2004, [46] Eremenko, A., Novikov, D., Oscillation of Fourier integrals with a spectral gap, Proc. Acad. Nat. Sci., 101, 2004, [47] Everett III, H.The Many-Worlds Interpretation of Quantum Mechanics: the theory of the universal wave function, Ph.D. thesis, Princeton, 1957 [48] Everitt, W. N. On a property of the m-coefficient of a second order linear differential equation, J. London Math. Soc. 4 (1972), [49] Folland, G. A course in abstract Harmonic Analysis, CRC press, Boca Raton, Florida, 1995 [50] Folland, G. B. and Sitaram, A. The Uncertainty Principle: A Mathematical Survey, J. of Fourier Analysis and Applications, Vol. 3, No. 3, 1997 [51] Gabor, D. Theory of communication, J. Inst. Elec. Engr., 1946, 93, [52] Garnett, J. Bounded analytic functions. Academic Press, New York, 1981 [53] Gelfand, I, M., Levitan, B. M. On the determination of a differential equation from its spectral function (Russian), Izvestiya Akad. Nauk SSSR, Ser. Mat., 15 (1951), ; English translation in Amer. Math. Soc. Translation, Ser. 2, 1 (1955),

13 BIBLIOGRAPHY 213 [54] Gesztezy F., Simon B. Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, Trans. AMS 352 (2000), [55] Gesztezy F., Simon B. m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. d Analyse Math. 73 (1997), [56] Gohberg, I., Krein, M. Theory and applications of Volterra operators in Hilbert space. AMS, Providence, RI, 1970 [57] Hall, T. Sur l approximation polynômiale des fonctions continues d une variable réelle, Neuvième Congrès des Mathématiciens Scandinaves 1938, Helsingfors (1939), [58] Havin, V.P. On the Uncertainty Principle in Harmonic Analysis, lecture notes. [59] Hardy, G.H. A theorem concerning Fourier transforms, Journal of the London Mathematical Society 8 (3), 1933, , [60] Havin, V. P., Jöricke, B. The uncertainty principle in harmonic analysis. Springer-Verlag, Berlin, [61] Havin, V., Mashreghi, J. Admissible majorants for model subspaces of H 2 ;I.Slowwinding of the generating inner function, II. Fast winding of the generating inner function, Canad. J. Math. 55 (2003), , [62] Hedenmalm, H. Heisenberg s uncertainty principle in the sense of Beurling, J. Anal. Math., 118, 2012, [63] Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 1927, Volume 43, Issue 3 4, pp [64] Higgins, J.R. Completeness and basis properties of sets of special functions. Cambridge Univ. Press, Cambridge, 1977 [65] Hochstadt, H., Lieberman, B. An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), [66] Horváth M. Inverse spectral problems and closed exponential systems, Preprint (2004) [67] I.I. Hirschman, Jr. A note on entropy, American Journal of Mathematics (1957) pp [68] Hruschev S., Nikolskii, N., Pavlov, B. Unconditional bases of exponentials and of reproducing kernels, Lecture Notes in Math., Vol. 864, [69] Hunt R., Muckenhoupt B., Wheeden R. Weighted norm inequalities for the conjugate functions and Hilbert transform, Trans. AMS 176 (1973), [70] Kakutani, S. Review: Nobert Wiener, Extrapolation, interpolation, and smoothing of stationary time series with engineering applications, Bull. Amer. Math. Soc. Volume 56, Number 4 (1950), [71] Izumi, S., Kawata, T. Quasi-analytic class and closure of tn in the interval (, ), Tohoku Math. J., 43(1937), [72] Kahane, J.-P. Travaux de Beurling et Malliavin, Seminaire Bourbaki. Exposés 223 á 228, 1962 [73] Kahane, J.-P. Sur la totalité des suites d exponentielles imaginaires, Ann. Inst. Fourier,(Grenoble), 8, 1959, [74] Kerov, S. V. Equilibrium and orthogonal polynomials, Algebra i Analiz, 12:6 (2000), [75] Kennard, E. H. Zur Quantenmechanik einfacher Bewegungstypen, Zeitschriftfür Physik 44 (4 5), (1927), 326 [76] Khodakovsky, A. M. Inverse spectral problem with partial information on the potential. PhD Thesis, Caltech, 1999 [77] Krein, M. G. On an extrapolation problem of A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), (Russian). [78] Krein, M. G. On a basic approximation problem of the theory of extrapolation and filtration of stationary random processes, Doklady Akad. Nauk SSSR (N.S.) 94, (1954), (Russian). [79] Krein, M. G. On the transfer function of a one- dimensional boundary problem of the second order (Russian), Doklady Akad. Nauk SSSR (N.S.) 88 (1953), [80] Krein, M. On the theory of entire functions of exponential type, Izv. AN SSSR 11 (1947), (Russian) [81] Kechris, A. Set theory and uniqueness for trigonometric series, unpublished lecture notes. [82] Khabibullin, B. Completeness of exponential systems and uniqueness sets. Bashkir State Univ. Press, Ufa, 2006 [83] Kislyakov, S.V. Classical Themes of Fourier Analysis, Commutative Harmonic Analysis I, Encyclopaedia of Mathematical Sciences Volume 15, 1991, pp

14 214 BIBLIOGRAPHY [84] Koosis, P. Introduction to H p spaces. Cambridge Univ. Press, Cambridge, 1980 [85] Koosis, P. The logarithmic integral, Vol. I & II, Cambridge Univ. Press, Cambridge, 1988 [86] Koosis, P. Lecons sur le Theorem de Beurling et Malliavin. Les Publications CRM, Montreal, 1996 [87] Koosis, P. A local estimate, involving the least superharmonic majorant, for entire functions of exponential type, Algebra i Analiz 10 (1998), 45 64; English translation in St. Petersburg Math. J. 10 (1999), no. 3, [88] Koosis, P. Sur la totalité des systèmes d exponentielles imaginaires C. R. Acad. Sci, 250, 1960, [89] Koosis, P. Kargaev s proof of Beurling s lemma (unpublished manuscript) [90] Körner, T.W. Uniqueness for trigonometric series, Ann. of Math, 126, N1 (1987), 1 34 [91] Lagarias, J. Zero Spacing Distributions for Differenced L-Functions, Acta Arithmetica 120, No. 2, (2005) [92] Lagarias, J. The Schrödinger Operator with Morse Potential on the Right Half Line, Communications in Number Theory and Physics 3 (2009), No.2, [93] Lax P. D., Phillips R.S. Scattering theory. Academic Press, New York, 1967 [94] Levin, B. Distribution of zeros of entire functions. AMS, Providence, RI, 1980 [95] Levin, B. Lectures on entire functions AMS, Providence, RI, 1996 [96] Levin, B. Completeness of systems of functions, quasi-analyticity and subharmonic majorants (Russian), Issled. Linein. Oper. Teorii Funktsii, 17, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 170 (1989), ; English translation in J. Soviet Math., 63 (1993), no. 2, [97] Levinson, N. Gap and density theorems, AMS Colloquium Publications, 26 (1940) [98] Levitan, B.M., Sargsjan, I.S. Sturm-Liouville and Dirac operators. Kluwer, Dordrecht, 1991 [99] Logvinenko, V.N., Sereda, Yu.F. Equivalent norms in spaces of entire functions of exponential type Teor. funk- tsii, funkt. analiz i ich prilozhenia 20 (1971) (Russian). [100] Lubinsky, D. S. A Survey of Weighted Polynomial Approximation with Exponential Weights, Surveys in Approximation Theory, 3, (2007) [101] Lyons, R. A Characterization of Measures Whose Fourier-Stieltjes Transforms Vanish at Infinity, Ph. D. thesis, Univ. of Michigan, (1983). [102] Lyons, R. Seventy Years of Rajchman Measures, J. Fourier Anal. Appl., Kahane Special Issue (1995), [103] Makarov, N., Poltoratski, A. Meromorphic inner functions, Toeplitz kernels, and the uncertainty principle, in Perspectives in Analysis, Springer Verlag, Berlin, 2005, [104] Makarov, N., Poltoratski, A. Beurling-Malliavin theory for Toeplitz kernels, Invent. Math., Vol. 180, Issue 3 (2010), [105] Makarov, N., Poltoratski, A., Sodin, M. Lectures on Linear Complex Analysis, inpreparation. [106] Marchenko, V. Some questions in the theory of one-dimensional linear differential operators of the second order, I, Trudy Mosk. Mat. Obsch. 1 (1952), [107] Marchenko, V. Sturm-Liouville operators and applications. Birkhauser, Basel, 1986 [108] Mashreghi, J., Nazarov, F., Havin, V. Beurling-Malliavin multiplier theorem: the seventh proof, St. Petersburg Math. J. 17 (2006), [109] Mitkovski, M. and Poltoratski, A. Polya sequences, Toeplitz kernels and gap theorems, Advances in Math., 224 (2010), pp [110] Morgan, C.W. A note on Fourier transforms. Journal of the London Math. Soc. 9 N3 (1934) [111] Menshov, D. E. Sur l unicité dudéveloppement trigonométrique, C. R. Acad. Sc. Paris, Sér. A-B 163, (1916), [112] Mergelyan, S. Weighted approximation by polynomials, Uspekhi mat. nauk, 11 (1956), , English translation in Amer. Math. Soc. Translations, Ser 2, 10 (1958), [113] Nienhuis, B. Coulomb gas formulation of two-dimensional phase transitions, Phase transitions and critical phenomena, vol. 11, C. Domb and J.L. Lebowitz, eds. (Academic, 1987.) [114] Nazarov, F. Local estimates of exponential polynomials and their applications to the uncertainty principle. Algebra i analiz, 5, N4, 1993, p (Russian; English translation: St.Petersburg Math.J., 5, N4, 1994, p ). [115] Nazarov, F. The Beurling lemma via the Bellman function (unpublished manuscript)

15 BIBLIOGRAPHY 215 [116] Nikolskii, N. K. Treatise on the shift operator, Springer-Verlaag, Berlin (1986) [117] Nikolskii, N.K. Operators, functions, and systems: an easy reading, Vol. I & II. AMS, Providence, RI, 2002 [118] Ortega-Cedrá, J., Seip, K. Fourier frames, Annals of Math. 155 (2002), [119] Pavlov, B. The basis property of a system of exponentials and the condition of Muckenhoupt, Dokl. Acad. Nauk SSSR 247, (1979), [120] Paley, R., Wiener, N. Fourier transform in the complex domains. AMS, New York, 1934 [121] Paneah, B. P. On some theorems of Paly-Wiener type Doklady AN SSSR 138 N1 (19()1) (Russian). [122] Poltoratski, A., On the boundary behavior of pseudocontinuable functions, St. Petersburg Math. J.,5 (1994), [123] Poltoratski, A. Bernstein s problem on weighted polynomial approximation, preprint, to appear in proceedings of Abel Symposium, Oslo, [124] Poltoratski, A. Survival probability in rank-one perturbation problems, Comm. Math. Phys., 203 (2001), No. 1, [125] Poltoratski, A. Kreĭn s spectral shift and perturbations of spectra of rank one, Algebra i Analiz, 10 (1998), no. 5, ; translation in St. Petersburg Math. J. 10 (1999), no. 5, [126] Poltoratski, A. and Sarason, D. Aleksandrov-Clark measures, Recent advances in operator-related function theory, 1 14, Contemp. Math., 393, Amer. Math. Soc., Providence, RI, 2006 [127] Poltoratski, A. Spectral gaps for sets and measures, Acta Math., 2012, Volume 208, Number 1, pp [128] Poltoratski, A., Problem on completeness of exponentials, Ann. Math., 178, , 2013 [129] Pólya, G. Jahresbericht der Deutchen Mathematiker-Vereinigung, Vol. 40 (1931), Problem 105 [130] Redheffer, R. Completeness of sets of complex exponentials, Advances in Math. 24, 1977, 1 62 [131] Riesz, F. Über die Fourierkoffizienten einer stetigen Funktion von beschränkter Schwankung, Math. Zeit. 2, , (1918). [132] Seip, K. Interpolation and sampling in spaces of analytic functions, AMS, 2004 [133] Sarason, D. Sub-Hardy Hilbert spaces in the unit disk. Univ. of Arkansas Lecture Notes in the Math. Sciences, vol. 10, J. Wiley and Sons, New York, 1994 [134] Sarason, D. Kernels of Toeplitz operators, Oper. Theory Adv. Appl. 71 (1994), [135] Schwartz, L. Études des sommes d exponentielles réelles. Hermann, Paris, 1943 [136] Simon, B. Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schroedinger operators (Vancouver, BC, 1993), , CRM Proc. Lecture Notes, 8, Amer. Math. Soc., Providence, RI, 1995 [137] Samaj, L. The statistical mechanics of the classical two-dimensional Coulomb gas is exactly solved, J. Phys. A36: , 2003 [138] Sodin, M. Which perturbations of quasianalytic weights preserve quasianalyticity? How to use de Branges theorem, J. Anal. Math. 69 (1996), [139] Sodin, M., de Branges spaces, Unpublished lecture notes, CRM Institute, Barcelona, [140] Sodin, M. and Yuditskii, P. Another approach to de Branges theorem on weighted polynomial approximation, Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), , Israel Math. Conf. Proc., 11, Bar-Ilan Univ., Ramat Gan, [141] Shreĭder, Yu. A.On the Fourier-Stieltjes coefficients of functions of bounded variation, (in Russian), Dokl. Acad. Nauk SSSR 74, (1950), [142] Titchmarsh, E.C. Eigenfunction expansion associated with second order differential equations, Clarendon Press, Oxford, 1946 [143] Treil S., Volberg, A. Embedding theorems for invariant subspaces of inverse shift, Proceedings of LOMI Seminars, 149 (1986), [144] Valiron, G. Sur la formule d interpolation de Lagrange, Bull. Sci. Math. 49 (1925), , [145] Vinogradov, S. A. Properties of multipliers of Cauchy Stieltjes-type integrals and some factorization problems for analytic functions, Proceedings of 7th Winter School (Drogobych, 1974), Moscow, 1974, 5 39 (in Russian).

16 216 BIBLIOGRAPHY [146] Volberg, A.L. Thin and thick families of rational fractions. (Seminar, Leningrad ). Lecture Notes in Math. 864 Springer, Bcrlin Heidelberg New York (1981) [147] Volberg, A.L., Joricke, B. Summability of the logarithm of an almost analytic function and the Levinson- Cartwright theorem. Mat. Sbornik, 130 N2 (1986) (Russian). [148] Weyl, H. Gruppentheorie und Quantenmechanik, Leipzig, Hirzel, 1928; transl. by H. P. Robertson, The Theory of Groups and Quantum Mechanics, 1931, rept Dover. ISBN [149] Wiener, N. I am a mathematician, The M.I.T. Press, 1964 [150] Wiener, N. Extrapolation, Interpolation, and Smoothing of Stationary Time Series, M.I.T. press, [151] Wiener, N. The quadratic variation of a function and its Fourier coefficients, Massachusetts J. of Math. 3, 1924, [152] Young, R. M. An introduction to non-harmonic Fourier series. Academic Press, New York, 1980 [153] Zygmund, A. Trigonometric series, Vol. I & II. Cambridge University Press, Cambridge, 1959

17 The Uncertainty Principle in Harmonic Analysis (UP) is a classical, yet rapidly developing, area of modern mathematics. Its first significant results and open problems date back to the work of Norbert Wiener, Andrei Kolmogorov, Mark Krein and Arne Beurling. At present, it encompasses a large part of mathematics, from Fourier analysis, frames and completeness problems for various systems of functions to spectral problems for differential operators and canonical systems. These notes are devoted to the so-called Toeplitz approach to UP which recently brought solutions to some of the long-standing problems posed by the classics. After a short overview of the general area of UP the discussion turns to the outline of the new approach and its results. Among those are solutions to Beurling s Gap Problem in Fourier analysis, the Type Problem on completeness of exponential systems, a problem by Pólya and Levinson on sampling sets for entire functions, Bernstein s problem on uniform polynomial approximation, problems on asymptotics of Fourier integrals and a Toeplitz version of the Beurling Malliavin theory. One of the main goals of the book is to present new directions for future research opened by the new approach to the experts and young analysts. For additional information and updates on this book, visit CBMS/121 AMS on the Web