OPUC on one foot
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- by Barry Simon PDF
- Bull. Amer. Math. Soc. 42 (2005), 431-460
Abstract:
We present an expository introduction to orthogonal polynomials on the unit circle (OPUC).References
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AkhB N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965; Russian original, 1961.
- A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 5, 490–503, 515 (Russian, with English and Armenian summaries). MR 931885
- A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Mat. Sb. (N.S.) 125(167) (1984), no. 2, 231–258 (Russian). MR 764479
- Jinho Baik, Percy Deift, and Kurt Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178. MR 1682248, DOI 10.1090/S0894-0347-99-00307-0
- J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the second row of a Young diagram under Plancherel measure, Geom. Funct. Anal. 10 (2000), no. 4, 702–731. MR 1791137, DOI 10.1007/PL00001635
- Jinho Baik, Percy Deift, Ken T.-R. McLaughlin, Peter Miller, and Xin Zhou, Optimal tail estimates for directed last passage site percolation with geometric random variables, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1207–1250. MR 1926668, DOI 10.4310/ATMP.2001.v5.n6.a7
- Jinho Baik and Eric M. Rains, Algebraic aspects of increasing subsequences, Duke Math. J. 109 (2001), no. 1, 1–65. MR 1844203, DOI 10.1215/S0012-7094-01-10911-3
- Glen Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471–487. MR 126126, DOI 10.1090/S0002-9947-1961-0126126-8
- Glen Baxter, A norm inequality for a “finite-section” Wiener-Hopf equation, Illinois J. Math. 7 (1963), 97–103. MR 145285
- M. Bello Hernández and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998), no. 2, 216–244. MR 1604927, DOI 10.1006/jath.1997.3126 Bern2 S. Bernstein, Sur une classe de polynomes orthogonaux, Commun. Kharkow 4 (1930), 79–93. BS A. Borodin and E. Strahov, Averages of characteristic polynomials in random matrix theory, preprint.
- Olivier Bourget, James S. Howland, and Alain Joye, Spectral analysis of unitary band matrices, Comm. Math. Phys. 234 (2003), no. 2, 191–227. MR 1962460, DOI 10.1007/s00220-002-0751-y
- M. J. Cantero, L. Moral, and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29–56. MR 1955452, DOI 10.1016/S0024-3795(02)00457-3 Ca07 C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95–115. Chris E. B. Christoffel, Über die Gaussische Quadratur und eine Verallgemeinerung derselben, J. Reine Angew. Math. 55 (1858), 61–82.
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338 DamKil D. Damanik and R. Killip, Half-line Schrödinger operators with no bound states, Acta Math. 193 (2004), 31–72. Dar G. Darboux, Mémoire sur l’approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série, Liouville J. (3) 4 (1878), 5–56; 377–416. DO P. Deift and J. Ostensson, A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, in preparation.
- Sergey A. Denisov, On Rakhmanov’s theorem for Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), no. 3, 847–852. MR 2019964, DOI 10.1090/S0002-9939-03-07157-0
- J. Dombrowski, Quasitriangular matrices, Proc. Amer. Math. Soc. 69 (1978), no. 1, 95–96. MR 467373, DOI 10.1090/S0002-9939-1978-0467373-3
- B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Uspehi Mat. Nauk 31 (1976), no. 1(187), 55–136 (Russian). MR 0427869
- Tamás Erdélyi, Paul Nevai, John Zhang, and Jeffrey S. Geronimo, A simple proof of “Favard’s theorem” on the unit circle, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 2, 551–556. MR 1150798 Fej L. Fejér, Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Math. Ann. 85 (1922), 41–48.
- H. Flaschka and D. W. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys. 55 (1976), no. 2, 438–456. MR 403368, DOI 10.1143/PTP.55.438 FrB G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford-New York, 1971.
- I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. MR 0073805, DOI 10.1090/trans2/001/11
- J. S. Geronimo, Polynomials orthogonal on the unit circle with random recurrence coefficients, Methods of approximation theory in complex analysis and mathematical physics (Leningrad, 1991) Lecture Notes in Math., vol. 1550, Springer, Berlin, 1993, pp. 43–61. MR 1322290, DOI 10.1007/BFb0117473
- J. S. Geronimo and R. A. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations 132 (1996), no. 1, 140–178. MR 1418504, DOI 10.1006/jdeq.1996.0175
- J. S. Geronimo and R. Johnson, An inverse problem associated with polynomials orthogonal on the unit circle, Comm. Math. Phys. 193 (1998), no. 1, 125–150. MR 1620309, DOI 10.1007/s002200050321
- J. Geronimus, On polynomials orthogonal on the circle, on trigonometric moment-problem and on allied Carathéodory and Schur functions, Rec. Math. [Mat. Sbornik] N. S. 15(57) (1944), 99–130 (Russian., with English summary). MR 0012715
- J. Geronimus, On the trigonometric moment problem, Ann. of Math. (2) 47 (1946), 742–761. MR 18265, DOI 10.2307/1969232
- Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79. MR 0061706 Ger57 Ya. L. Geronimus, On some equations in finite differences and the corresponding system of orthogonal polynomials, Zap. Mat. Otdel Fiz.-Mat. Fak. i Khar’kov. Mat. Obšč 25 (1957), 87-100 [Russian].
- L. Ya. Geronimus, Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval, Consultants Bureau, New York, 1961. Authorized translation from the Russian. MR 0133643 GZ1 F. Gesztesy and M. Zinchenko, A Borg-type theorem associated with orthogonal polynomials on the unit circle, preprint, 2004. GZ2 F. Gesztesy and M. Zinchenko, Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, preprint, 2004. GoIb B. L. Golinskii and I. A. Ibragimov, On Szegő’s limit theorem, Math. USSR Izv. 5 (1971), 421–444.
- L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), no. 3, 169–186. MR 1919160, DOI 10.1023/A:1019765002077 Golppt L. Golinskii, Orthogonal polynomials on the unit circle, Szegő difference equations and spectral theory of unitary matrices, second Doctoral thesis, Kharkov, 2003.
- Leonid Golinskii and Paul Nevai, Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle, Comm. Math. Phys. 223 (2001), no. 2, 223–259. MR 1864433, DOI 10.1007/s002200100525 GolSim L. Golinskii and B. Simon, Results on spectral theorem using CMV matrices in Section 4.3 of B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.
- I. A. Ibragimov, A theorem of Gabor Szegő, Mat. Zametki 3 (1968), 693–702 (Russian). MR 231114
- V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2–3, 246–251 (Russian, with Armenian and English summaries). MR 0301565
- Kurt Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476. MR 1737991, DOI 10.1007/s002200050027
- William B. Jones, Olav Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), no. 2, 113–152. MR 976057, DOI 10.1112/blms/21.2.113
- Thomas Kailath, A view of three decades of linear filtering theory, IEEE Trans. Inform. Theory IT-20 (1974), 146–181. MR 465437, DOI 10.1109/tit.1974.1055174
- Thomas Kailath, Signal processing applications of some moment problems, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 71–109. MR 921085, DOI 10.1090/psapm/037/921085
- Thomas Kailath, Norbert Wiener and the development of mathematical engineering, The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 60, Amer. Math. Soc., Providence, RI, 1997, pp. 93–116. MR 1460278, DOI 10.1090/pspum/060/1460278
- Sergei Khrushchev, Schur’s algorithm, orthogonal polynomials, and convergence of Wall’s continued fractions in $L^2({\Bbb T})$, J. Approx. Theory 108 (2001), no. 2, 161–248. MR 1815919, DOI 10.1006/jath.2000.3500
- S. V. Khrushchev, Classification theorems for general orthogonal polynomials on the unit circle, J. Approx. Theory 116 (2002), no. 2, 268–342. MR 1911083, DOI 10.1006/jath.2002.3674 KilNen R. Killip and I. Nenciu, Matrix models for circular ensembles, Internat. Math. Res. Notices, (2004) no. 50, 2665–2701.
- A. N. Kolmogoroff, Stationary sequences in Hilbert’s space, Bolletin Moskovskogo Gosudarstvenogo Universiteta. Matematika 2 (1941), 40pp (Russian). MR 0009098
- M. Krein, On a generalization of some investigations of G. Szegö, V. Smirnoff and A. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N.S.) 46 (1945), 91–94. MR 0013457
- M. Krein, On a problem of extrapolation of A. N. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N. S.) 46 (1945), 306–309. MR 0012700
- Igor Moiseevich Krichever, Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 215–216 (Russian). MR 510681
- B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
- H. J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 47–77. MR 866018, DOI 10.1090/S0273-0979-1987-15464-4
- Norman Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. Phys. Mass. Inst. Tech. 25 (1947), 261–278. MR 19257, DOI 10.1002/sapm1946251261 MFMS A. Martinez-Finkelshtein, K. T.-R. McLaughlin, and E. B. Saff, Strong asymptotics of Szegő orthogonal polynomials with respect to an analytic weight, preprint.
- Attila Máté, Paul Nevai, and Vilmos Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985), no. 1, 63–69. MR 766095, DOI 10.1007/BF01890022
- Attila Máté, Paul Nevai, and Vilmos Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), no. 2, 239–281. MR 882423, DOI 10.2307/2374574
- H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217–274. MR 397076, DOI 10.1007/BF01425567
- Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
- H. N. Mhaskar and E. B. Saff, On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory 63 (1990), no. 1, 30–38. MR 1074079, DOI 10.1016/0021-9045(90)90111-3 Nen I. Nenciu, Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle, to appear in Internat. Math. Res. Notices.
- Paul Nevai, Characterization of measures associated with orthogonal polynomials on the unit circle, Rocky Mountain J. Math. 19 (1989), no. 1, 293–302. Constructive Function Theory—86 Conference (Edmonton, AB, 1986). MR 1016182, DOI 10.1216/RMJ-1989-19-1-293
- Paul Nevai, Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991), no. 3, 322–340. MR 1109411, DOI 10.1016/0021-9045(91)90095-R
- Paul Nevai and Vilmos Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53 (1989), no. 1-2, 99–104. MR 1018677
- Franz Peherstorfer, Orthogonal and extremal polynomials on several intervals, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 187–205. MR 1246858, DOI 10.1016/0377-0427(93)90322-3
- F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), no. 2, 161–185. MR 1393285, DOI 10.1007/s003659900008
- Franz Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), no. 2, 180–195. MR 1849545, DOI 10.1006/jath.2001.3571
- Franz Peherstorfer, Inverse images of polynomial mappings and polynomials orthogonal on them, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 371–385. MR 1985708, DOI 10.1016/S0377-0427(02)00628-3
- Franz Peherstorfer and Robert Steinbauer, Perturbation of orthogonal polynomials on the unit circle—a survey, Orthogonal polynomials on the unit circle: theory and applications (Madrid, 1994) Univ. Carlos III Madrid, Leganés, 1994, pp. 97–119. MR 1317108
- Franz Peherstorfer and Robert Steinbauer, Orthogonal polynomials on arcs of the unit circle. I, J. Approx. Theory 85 (1996), no. 2, 140–184. MR 1385813, DOI 10.1006/jath.1996.0035
- Franz Peherstorfer and Robert Steinbauer, Orthogonal polynomials on arcs of the unit circle. II. Orthogonal polynomials with periodic reflection coefficients, J. Approx. Theory 87 (1996), no. 1, 60–102. MR 1410612, DOI 10.1006/jath.1996.0092
- Franz Peherstorfer and Robert Steinbauer, Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients, J. Approx. Theory 88 (1997), no. 3, 316–353. MR 1432577, DOI 10.1006/jath.1996.3026
- Franz Peherstorfer and Robert Steinbauer, Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. II. Weak asymptotics, J. Approx. Theory 105 (2000), no. 1, 102–128. MR 1768526, DOI 10.1006/jath.2000.3450
- Franz Peherstorfer and Robert Steinbauer, Orthogonal polynomials on the circumference and arcs of the circumference, J. Approx. Theory 102 (2000), no. 1, 96–119. MR 1736047, DOI 10.1006/jath.1999.3383
- Franz Peherstorfer and Robert Steinbauer, Strong asymptotics of orthonormal polynomials with the aid of Green’s function, SIAM J. Math. Anal. 32 (2000), no. 2, 385–402. MR 1781222, DOI 10.1137/S0036141098343045 PrS00 M. Praehofer and H. Spohn, Universal distributions for growth processes in $1+1$ dimensions and random matrices, Phys. Rev. Lett. 84 (2000), 4882–4885. Rakh77 E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977), 199–213. Rakh83 E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105–117.
- E. A. Rakhmanov, Asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szegő condition, Mat. Sb. (N.S.) 130(172) (1986), no. 2, 151–169, 284 (Russian); English transl., Math. USSR-Sb. 58 (1987), no. 1, 149–167. MR 854969, DOI 10.1070/SM1987v058n01ABEH003097
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- I. Gohberg (ed.), I. Schur methods in operator theory and signal processing, Operator Theory: Advances and Applications, vol. 18, Birkhäuser Verlag, Basel, 1986. MR 902600, DOI 10.1007/978-3-0348-5483-2
- Barry Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203. MR 1627806, DOI 10.1006/aima.1998.1728
- Barry Simon, The Golinskii-Ibragimov method and a theorem of Damanik and Killip, Int. Math. Res. Not. 36 (2003), 1973–1986. MR 1991180, DOI 10.1155/S107379280313084X
- Barry Simon, Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line, J. Approx. Theory 126 (2004), no. 2, 198–217. MR 2045539, DOI 10.1016/j.jat.2003.12.002 OPUC1 B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1 S297 B. Simon, Aizenman’s theorem for orthogonal polynomials on the unit circle, to appear in Const. Approx. Saff2 B. Simon, Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, to appear in J. Approx. Theory. MSF B. Simon, Meromorphic Szegő functions and asymptotic series for Verblunsky coefficients, preprint.
- Barry Simon and Thomas Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys. 125 (1989), no. 1, 113–125. MR 1017742, DOI 10.1007/BF01217772
- Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105 Sz19 G. Szegő, Über Orthogonalsysteme von Polynomen, Math. Z. 4 (1919), 139–151. Sz20-21 G. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, I, II, Math. Z. 6 (1920), 167–202; 9 (1921), 167–190. Sz22a G. Szegő, Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 86 (1922), 114–139. Szb G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, American Mathematical Society, Providence, R.I., 1939; 3rd edition, 1967.
- G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplémentaire, 228–238. MR 51961 Talmud Talmud Bavli, Tractate Shabbos, 31a; see, for example, Schottenstein Edition, Mesorah Publications, New York, 1996.
- Vilmos Totik, Orthogonal polynomials with ratio asymptotics, Proc. Amer. Math. Soc. 114 (1992), no. 2, 491–495. MR 1065095, DOI 10.1090/S0002-9939-1992-1065095-9
- Pierre van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), no. 1, 45–81. MR 650253, DOI 10.1007/BF01418827 V35 S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proc. London Math. Soc. (2) 38 (1935), 125–157. V36 S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc. (2) 40 (1936), 290–320.
- Franz Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), no. 1-2, 9–15. MR 639135, DOI 10.1007/BF01292646
- Burton Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 554–555. MR 131120, DOI 10.1090/S0002-9939-1961-0131120-2
- Harold Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997–1013. MR 0209448
- Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications, Technology Press of The Massachusetts Institute of Technology, Cambridge, Mass.; John Wiley & Sons, Inc., New York, N.Y.; Chapman & Hall, Ltd., London, 1949. MR 0031213, DOI 10.7551/mitpress/2946.001.0001 W Wikipedia entry on Rodney Dangerfeld: For those in the international community who don’t know of this reference, see http://en.wikipedia.org/wiki/Rodney_Dangerfield.
Additional Information
- Barry Simon
- Affiliation: Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 189013
- Email: bsimon@caltech.edu
- Received by editor(s): February 2, 2005
- Received by editor(s) in revised form: April 19, 2005
- Published electronically: June 23, 2005
- Additional Notes: Supported in part by NSF grant DMS-0140592.
- © Copyright 2005 Barry Simon
- Journal: Bull. Amer. Math. Soc. 42 (2005), 431-460
- MSC (2000): Primary 42C05, 30E05, 42A70
- DOI: https://doi.org/10.1090/S0273-0979-05-01075-X
- MathSciNet review: 2163705