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Published online by Cambridge University Press:  31 March 2022

Jean Bernard Lasserre
Affiliation:
LAAS-CNRS, Toulouse
Edouard Pauwels
Affiliation:
Institut de Recherche en Informatique, Toulouse
Mihai Putinar
Affiliation:
University of California, Santa Barbara
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Aamari, E. and Levrard, C. 2019. Nonasymptotic rates for manifold, tangent space and curvature estimation. Ann. Statist., 47(1), 177204.CrossRefGoogle Scholar
Adler, M. and van Moerbeke, P. 2001. Darboux transforms on band matrices, weights, and associated polynomials. Int. Math. Res. Not., 935984.Google Scholar
Ahiezer, N. I. 1965. Lektsii po teorii approksimatsii, 2nd edition. Izdat. “Nauka”, Moscow.Google Scholar
Akhiezer, N. I. 1965. The Classical Moment Problem and Some Related Questions in Analysis, trans. N. Kemmer. Hafner, New York.Google Scholar
Alaoui, A. and Mahoney, M. 2015. Fast randomized kernel ridge regression with statistical guarantees. In Advances in Neural Information Processing Systems, NIPS, pp. 775783.Google Scholar
Alpay, D. (ed.). 2003. Reproducing Kernel Spaces and Applications. Operator Theory: Advances and Applications, vol. 143. Birkhäuser, Basel.CrossRefGoogle Scholar
Anjos, M. and Lasserre, J. B. (eds.). 2011. Handbook of Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research and Management Science, vol. 166. Springer, New York, NY.Google Scholar
Appell, P. 1890. Sur une classe de polynômes à deux variables et le calcul approché des intégrales doubles. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 4(2), H1H20.Google Scholar
Aronszajn, N. 1950. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68, 337– 404.CrossRefGoogle Scholar
Askari, A., Yang, F. and El Ghaoui, L. 2018. Kernel-based outlier detection using the inverse Christoffel function. arXiv preprint arXiv:1806.06775.Google Scholar
Bach, F. 2013. Sharp analysis of low-rank kernel matrix approximations. In Proceedings of the 26th Annual Conference on Learning Theory, PMLR, pp. 185209.Google Scholar
Bach, F. 2017. On the equivalence between kernel quadrature rules and random feature expansions. J. Mach. Learn. Res., 18(21), 138.Google Scholar
Baran, M. 1988. Siciak’s extremal function of convex sets in CN . Ann. Polon. Math., 48(3), 275280.Google Scholar
Baran, M. 1992. Bernstein type theorems for compact sets in Rn. J. Approx. Theory, 69(2), 156166.CrossRefGoogle Scholar
Baran, M., Białas-Cież, L. and Milówka, B. 2013. On the best exponent in Markov inequality. Potential Anal., 38(2), 635651.Google Scholar
Barvinok, A. 2002. A Course on Convexity. American Mathematical Society, Providence, R.I.Google Scholar
Batschelet, E. 1981. Circular Statistics in Biology. Academic Press, Cambridge, MA.Google Scholar
Bayer, C. and Teichmann, J. 2006. The proof of Tchakaloff’s theorem. Proc. Amer. Math. Soc., 134(10), 30353040.CrossRefGoogle Scholar
Beckermann, B., Putinar, M., Saff, E. B. and Stylianopoulos, N. 2021. Perturbations of Christoffel–Darboux kernels: detection of outliers. Found. Comp. Math., 21(1), 71124.CrossRefGoogle Scholar
Bedford, E. and Taylor, B. A. 1986. The complex equilibrium measure of a symmetric convex set in Rn. Trans. Amer. Math. Soc., 294(2), 705717.Google Scholar
Belkin, M. and Niyogi, P. 2008. Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. System Sci., 74(8), 12891308.CrossRefGoogle Scholar
Berman, R. J., Boucksom, S. and Witt Nyström, D. 2011. Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math., 207(1), 127.Google Scholar
Berman, R. J. 2009. Bergman kernels and equilibrium measures for line bundles over projective manifolds. Amer. J. Math., 131(5), 14851524.CrossRefGoogle Scholar
Bleher, P., Lyubich, M. and Roeder, R. 2020. Lee–Yang–Fisher zeros for the DHL and 2D rational dynamics, II. Global pluripotential interpretation. J. Geom. Anal., 30(1), 777833.Google Scholar
Bloom, T. 1997. Orthogonal polynomials in Cn. Indiana Univ. Math. J., 46(2), 427452.CrossRefGoogle Scholar
Bloom, T. and Levenberg, N. 2003. Weighted pluripotential theory in CN . Amer. J. Math., 125(1), 57103.CrossRefGoogle Scholar
Bloom, T. and Shiffman, B. 2007. Zeros of random polynomials on Cm. Math. Res. Lett., 14(3), 469479.Google Scholar
Bloom, T., Levenberg, N., Piazzon, F. and Wielonsky, F. 2015. Bernstein–Markov: a survey. Dolomites Res. Notes Approx., 8(Special Issue), 7591.Google Scholar
Bochnak, J., Coste, M. and Roy, M.-F. 1998. Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36. Springer, Berlin. Translated from the 1987 French original, revised by the authors.Google Scholar
Bos, L. P. 1989. Asymptotics for the Christoffel function for the equilibrium measure on a ball in Rm. In Approximation Theory VI, Vol. I (College Station, TX, 1989). Academic Press, Boston, MA, pp. 97100.Google Scholar
Bos, L. P. 1994. Asymptotics for the Christoffel function for Jacobi like weights on a ball in Rm. New Zealand J. Math., 23(2), 99109.Google Scholar
Bos, L. P., Della Vecchia, B. and Mastroianni, G. 1998. On the asymptotics of Christoffel functions for centrally symmetric weight functions on the ball in Rd. In Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. I (Acquafredda di Maratea, 1996). I, no. 52. Sede della Società Palermo, pp. 277–290.Google Scholar
Bos, L. P., Brudnyi, A. and Levenberg, N. 2010. On polynomial inequalities on exponential curves in Cn. Constr. Approx., 31(1), 139147.Google Scholar
Breiding, P., Kališnik, S., Sturmfels, B. and Weinstein, M. 2018. Learning algebraic varieties from samples. Rev. Mat. Complut., 31(3), 545593.CrossRefGoogle Scholar
Brennan, Dzh. È. 2005. Thomson’s theorem on mean-square polynomial approximation. Algebra i Analiz, 17(2), 132.Google Scholar
Brudnyi, A. 2008. On local behavior of holomorphic functions along complex submanifolds of CN . Invent. Math., 173(2), 315363.Google Scholar
Bubenik, P. 2015. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res., 16(1), 77102.Google Scholar
Bueno, M. I. and Marcellán, F. 2004. Darboux transformation and perturbation of linear functionals. Linear Algebra Appl., 384, 215242.CrossRefGoogle Scholar
Burns, D., Levenberg, N., Ma’u, S. and Révész, Sz. 2010. Monge–Ampère measures for convex bodies and Bernstein–Markov type inequalities. Trans. Amer. Math. Soc., 362(12), 63256340.CrossRefGoogle Scholar
Butzer, P. L. and Fehér, F. (eds.). 1981. E. B. Christoffel: The Influence of his Work on Mathematics and the Physical Sciences. Birkhäuser, Basel–Boston, MA. Including expanded versions of lectures given at the International Christoffel Symposium held in Aachen and Monschau, November 8–11, 1979.Google Scholar
Cantero, M. J., Marcellán, F., Moral, L. and Velázquez, L. 2016. Darboux transformations for CMV matrices. Adv. Math., 298, 122206.Google Scholar
Carleman, T. 1932. Application de la théorie des équations intégrales linéaires aux systèmes d’équations différentielles non linéaires. Acta Math., 59(1), 6387.CrossRefGoogle Scholar
Carlsson, G. 2009. Topology and data. Bull. Amer. Math. Soc., 46(2), 255308.CrossRefGoogle Scholar
Chatterjee, S. and Hadi, A.S. 1986. Influential observations, high leverage points, and outliers in linear regression. Statis. Sci., 1(3), 379393.Google Scholar
Chazal, F., Cohen-Steiner, D. and Mérigot, Q. 2011. Geometric inference for probability measures. Found. Comput. Math., 11(6), 733751.Google Scholar
Chazal, F., Glisse, M., Labruère, C. and Michel, B. 2014. Convergence rates for persistence diagram estimation in topological data analysis. In International Conference on Machine Learning. PMLR, pp. 163171.Google Scholar
Chevalier, J. 1976. Estimation du support et du contour du support d’une loi de probabilité. Ann. Inst. Henri Poincaré Probab. Stat., 12(4), 339364.Google Scholar
Clarkson, K. L. and Woodruff, D. P. 2013. Low rank approximation and regression in input sparsity time. In ACM Symposium on Theory of Computing. ACM, pp. 8190.Google Scholar
Coman, D. and Poletsky, E. A. 2010. Polynomial estimates, exponential curves and Diophantine approximation. Math. Res. Lett., 17(6), 11251136.Google Scholar
Conway, J. B. 1991. The Theory of Subnormal Operators. Mathematical Surveys and Monographs, vol. 36. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Cox, D., Little, J. and O’Shea, D. 2007. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer Science and Business Media, Berlin.CrossRefGoogle Scholar
Cucker, F. and Smale, S. 2002. On the mathematical foundations of learning. Bull. Amer. Math. Soc., 39(1), 149.CrossRefGoogle Scholar
Cuevas, A. and Fraiman, R. 1997. A plug-in approach to support estimation. Ann. Statist., 25(6), 23002312.Google Scholar
Cuevas, A., González-Manteiga, W. and Rodríguez-Casal, A. 2006a. Plug-in estimation of general level sets. Aust. N. Z. J. Stat., 48(1), 719.CrossRefGoogle Scholar
Cuevas, A., González-Manteiga, W. and Rodríguez-Casal, A. 2006b. Plug-in estimation of general level sets. Aust. N. Z. J. Stat., 48(1), 719.CrossRefGoogle Scholar
Curto, R. and Fialkow, L.A. 2005. Truncated K-moment problems in several variables. J. Operator Theory, 54, 189226.Google Scholar
Daras, N. J. 2014. Markov-type inequalities with applications in multivariate approximation theory. In Topics in Mathematical Analysis and Applications. Springer Optimization and its Applications, vol. 94. Springer, Cham, pp. 277314.Google Scholar
Das, S. and Giannakis, D. 2019. Delay-coordinate maps and the spectra of Koopman operators. J. Stat. Phys., 175(6), 11071145.Google Scholar
Davis, J. and Goadrich, M. 2006. The relationship between Precision-Recall and ROC curves. In Proceedings of the 23rd International Conference on Machine Learning. Association for Computing Machnery, pp. 233240.Google Scholar
de Branges, L. 1968. Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
de Castro, Y., Gamboa, F., Henrion, D., Hess, R. and Lasserre, J. B. 2019. Approximate optimal designs for multivariate polynomial regression. Ann. Statist., 47, 127155.Google Scholar
Demailly, J.-P. 1985. Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines. Mém. Soc. Math. Fr. (N.S.), 19, 124.Google Scholar
Demailly, J.-P. 2013. Applications of pluripotential theory to algebraic geometry. In Pluripotential Theory. Lecture Notes in Mathematics, vol. 2075. Springer, Heidelberg, pp. 143263.Google Scholar
DeVore, R. A. 1972. The Approximation of Continuous Functions by Positive Linear Operators. Lecture Notes in Mathematics, vol. 293. Springer, Berlin.Google Scholar
Devroye, L. and Wise, G. L. 1980. Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math., 38(3), 480488.Google Scholar
Dinew, S. 2019. Lectures on pluripotential theory on compact Hermitian manifolds. In Complex Non-Kähler Geometry. Lecture Notes in Mathemtaics, vol. 2246. Springer, Cham, pp. 156.CrossRefGoogle Scholar
Donner, K. 1982. Extension of Positive Operators and Korovkin Theorems. Lecture Notes in Mathematics, vol. 904. Springer, berlin.CrossRefGoogle Scholar
Drineas, P., Magdon-Ismail, M., Mahoney, M. W. and Woodruff, D. P. 2012. Fast approximation of matrix coherence and statistical leverage. J. Mach. Learn. Res., 13, 34753506.Google Scholar
Dubois, D. W. and Efroymson, G. 1970. Algebraic theory of real varieties. I. In Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, April 1, 1970). Mathematical research Center, National Taiwan University, Taipei, pp. 107135.Google Scholar
Dunford, N. and Schwartz, J. 1958. Linear Operators Part I: General Theory. Interscience, New York, NY.Google Scholar
Dunkl, C. F. and Xu, Y. 2014. Orthogonal Polynomials of Several Variables, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 155. Cambridge University Press, Cambridge.Google Scholar
Durrett, R. 2019. Probability: Theory and Examples, vol. 49. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Eckhoff, K. S. 1993. Accurate and efficient reconstruction of discontinuous functions from truncated series expansions. Math. Comp., 61(204), 745763.CrossRefGoogle Scholar
Edelsbrunner, H. and Harer, J. 2008. Persistent homology-a survey. Contemp. Math., 453, 257282.Google Scholar
Fefferman, C., Mitter, S. K. and Narayanan, H. 2016. Testing the manifold hypothesis. J. Amer. Math. Soc., 29(4), 9831049.CrossRefGoogle Scholar
Foias, C. and Frazho, A. E. 1990. The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, vol. 44. Birkhäuser, Basel.Google Scholar
Freud, G. 1969. Orthogonale Polynome. Birkhäuser, Basel–Stuttgart. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 33.Google Scholar
Gaier, D. 1987. Lectures on Complex Approximation, trans. Renate McLaughlin. Birkhäuser Boston, Boston, MA.Google Scholar
Garza, L. and Marcellán, F. 2009. Szegö transformations and rational spectral transformations for associated polynomials. J. Comput. Appl. Math., 233(3), 730– 738.Google Scholar
Gautschi, W. 1986. On the sensitivity of orthogonal polynomials to perturbations in the moments. Numer. Math., 48(4), 369382.Google Scholar
Geffroy, J. 1964. Sur un probleme d’estimation géométrique. Publ. Inst. Statist. Univ. Paris, 13, 191210.Google Scholar
Genovese, C. R., Perone-Pacifico, M., Verdinelli, I. and Wasserman, L. 2012a. Manifold estimation and singular deconvolution under Hausdorff loss. Ann. Statist., 40(2), 941963.Google Scholar
Genovese, C. R., Perone-Pacifico, M., Verdinelli, I. and Wasserman, L. 2012b. Minimax manifold estimation. J. Mach. Learn. Res., 13(1), 12631291.Google Scholar
Geronimo, J. S. and Woerdeman, H. 2007. Two variable orthogonal polynomials on the bicircle and structured matrices. SIAM J. Matrix Anal. Appl., 29(3), 796825.CrossRefGoogle Scholar
Geronimo, J. S. and Woerdeman, H. J. 2004. Positive extensions, Fejér–Riesz factorization and autoregressive filters in two variables. Ann. of Math. (2), 160(3), 839906.CrossRefGoogle Scholar
Ghrist, R. 2008. Barcodes: the persistent topology of data. Bull. Amer. Math. Soc., 45(1), 6175.Google Scholar
Gómez-Ullate, D., Kamran, N. and Milson, R. 2010. Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A, 43(43), 434016, 16.Google Scholar
Gorbachuk, M. L. and Gorbachuk, V. I. 1997. M. G. Krein’s lectures on entire operators. Operator Theory: Advances and Applications, vol. 97. Birkhäuser, Basel.CrossRefGoogle Scholar
Grant, M. and Boyd, S. 2014. CVX: Matlab Software for Disciplined Convex Programming, version 2.1. CVX Research. Available at: http://cvxr.com/cvx/Google Scholar
Gröbner, W. 1948. Über die Konstruktion von Systemen orthogonaler Polynome in einune zwei-dimensionalen Bereichen. Monatsh. Math., 52, 3854.Google Scholar
Gustafsson, B., Putinar, M., Saff, E. B. and Stylianopoulos, N. 2009. Bergman polynomials on an archipelago: estimates, zeros and shape reconstruction. Adv. Math., 222(4), 14051460.CrossRefGoogle Scholar
Gustafsson, B. and Putinar, M. 2017. Hyponormal Quantization of Planar Domains: Exponential Transform in Dimension Two. Lecture Notes in Mathematics, vol. 2199. Springer, Cham.Google Scholar
Hein, M. and Audibert, J.-Y. 2005. Intrinsic dimensionality estimation of submanifolds in Rd. In Proceedings of the 22nd International Conference on Machine Learning. Association for Computing Machinery, pp. 289296.Google Scholar
Hein, M., Audibert, J.-Y. and Von Luxburg, U. 2005. From graphs to manifolds–weak and strong pointwise consistency of graph Laplacians. In International Conference on Computational Learning Theory. COLT 2005. Springer, pp. 470485.Google Scholar
Hellinger, E. 1922. Zur Stieltjesschen Kettenbruchtheorie. Math. Ann., 86(1–2), 1829.Google Scholar
Hellinger, E. and Toeplitz, O. 1953. Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten. Chelsea, New York, NY.Google Scholar
Henrion, D., Korda, M. and Lasserre, J. B. 2020. The Moment-SOS Hierarchy: Lectures in Probability, Statistics, Computational Geometry and Nonlinear PDEs. World Scientific, Singapore.CrossRefGoogle Scholar
Herglotz, G., Schur, I., Pick, G., Nevanlinna, R. and Weyl, H. 1991. Ausgewählte Arbeiten zu den Ursprüngen der Schur-Analysis. Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], vol. 16. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart. Gewidmet dem großen Mathematiker Issai Schur (1875–1941). [Dedicated to the great mathematician Issai Schur (1875–1941)], Edited and with a foreword and afterword by Bernd Fritzsche and Bernd Kirstein, With contributions by W. Ledermann and W. K. Hayman, With English, French and Russian summaries.Google Scholar
Hilbert, D. 1920. Gaston Darboux. 1842–1917 Acta Math., 42(1), 269273.Google Scholar
Hilbert, D. 1953. Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Chelsea, New York, NY.Google Scholar
Hoaglin, D. C. and Welsch, R. E. 1978. The hat matrix in regression and ANOVA. Amer. Statist., 32(1), 1722.Google Scholar
Hoel, P. G. 1961/62. Some properties of optimal spacing in polynomial estimation. Ann. Inst. Statist. Math., 13, 18.Google Scholar
Hoel, P. G. and Levine, A. 1964. Optimal spacing and weighting in polynomial prediction. Ann. Math. Statist., 35, 15531560.Google Scholar
Kiefer, J. 1961. Optimum designs in regression problems. II. Ann. Math. Statist., 32, 298325.Google Scholar
Kiefer, J. and Wolfowitz, J. 1959. Optimum designs in regression problems. Ann. Math. Statist., 30, 271294.CrossRefGoogle Scholar
Kiefer, J. andWolfowitz, J. 1960. The equivalence of two extremum problems. Canadian J. Math., 12, 363366.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. 1965. On a theorem of Hoel and Levine on extrapolation designs. Ann. Math. Statist., 36, 16271655.CrossRefGoogle Scholar
Kim, A. K H. and Zhou, H. H. 2015. Tight minimax rates for manifold estimation under Hausdorff loss. Electron. J. Stat., 9(1), 15621582.CrossRefGoogle Scholar
Klimek, M. 1991. Pluripotential Theory. London Mathematical Society Monographs, New Series, vol. 6. Clarendon Press, Oxford.Google Scholar
Knill, O. 1998. A remark on quantum dynamics. Helv. Phys. Acta, 71(3), 233241.Google Scholar
Kolmogorov, A. N. 1941. Stationary sequences in Hilbert’s space. Vestnik Moskovskogo Gosudarstvennogo Universiteta. Matematika [Moscow Univ. Math. Bull.], 2, 40pp.Google Scholar
Koopman, B. O. 1931. Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. USA, 17(5), 315318.CrossRefGoogle ScholarPubMed
Koopman, B. O. and Von Neumann, J. 1932. Dynamical systems of continuous spectra. Proc Natl Acad. Sci. USA, 18(3), 255263.Google Scholar
Koornwinder, T. 1975. Two-variable analogues of the classical orthogonal polynomials. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975). Academic Press, New York, pp. 435495.Google Scholar
Korda, M., Putinar, M. and Mezić, I. 2020. Data-driven spectral analysis of the Koopman operator. Appl. Comput. Harmon. Anal., 48(2), 599629.Google Scholar
Krein, M. G. 1957. On a continual analogue of a Christoffel formula from the theory of orthogonal polynomials. Dokl. Akad. Nauk SSSR (N.S.), 113, 970973.Google Scholar
Krein, M. G. 1951. The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development. Uspekhi Mat. Nauk (N.S.), 6(4 (44)), 3120.Google Scholar
Kroó, A. 2015. Christoffel functions on convex and starlike domains in Rd. J. Math. Anal. Appl., 421(1), 718729.Google Scholar
Kroó, A. and Lubinsky, D. S. 2013a. Christoffel functions and universality in the bulk for multivariate orthogonal polynomials. Canad. J. Math., 65(3), 600620.CrossRefGoogle Scholar
Kroó, A. and Lubinsky, D. S. 2013b. Christoffel functions and universality on the boundary of the ball. Acta Math. Hungar., 140(1–2), 117133.Google Scholar
Lacey, M. and Terwilleger, E. 2008. A Wiener–Wintner theorem for the Hilbert transform. Ark. Mat., 46(2), 315336.Google Scholar
Lam, T. Y. 1984. An introduction to real algebra. Rocky Mountain J. Math., 14(4), 767814.CrossRefGoogle Scholar
Lasserre, J. B. 2015. A generalization of Löwner–John’s Ellipsoid theorem. Math. Program. Ser. A, 152(1), 559591.Google Scholar
Lasserre, J. B. 2019. The Moment–SOS Hierarchy. In Sirakov, B., Ney de Sousa, P. and Viana, M. (eds.), Proceedings of the International Congress of Mathematicians (ICM 2018). World Scientific, Singapore, pp. 37733794.Google Scholar
Lasserre, J. B. and Pauwels, E. 2016. Sorting out typicality with the inverse moment matrix SOS polynomial. In Lee, D. D., Sugiyama, M., Luxburg, U. V., Guyon, I. and Garnett, R. (eds.), Advances in Neural Information Processing Systems. Curran Associates, New York, NY, pp. 190198.Google Scholar
Lasserre, J. B. and Pauwels, E. 2019. The empirical Christoffel function with applications in data analysis. Adv. Comput. Math., 45(3), 14391468.Google Scholar
Levin, E. and Lubinsky, D. S. 2015. Christoffel functions, Lp universality, and Paley-Wiener spaces. J. Anal. Math., 125, 243283.CrossRefGoogle Scholar
Levina, E. and Bickel, P. 2004. Maximum likelihood estimation of intrinsic dimension. Advances in neural information processing systems, 17, 777784.Google Scholar
Lichman, J. 2013. UCI Machine Learning Repository. School of Information and Computer Sciences, University of California, Irvine, CA. Available at: http://archive.ics.uci.edu/ml.Google Scholar
Liesen, J. and Strakoš, Z. 2013. Krylov Subspace Methods: Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford.Google Scholar
Lippmann, G., Poincaré, H., Appell, P., Lavisse, E., Volterra, V., Belugou, L., Picard, R E., Lèvy, L., Guichard, C. and Darboux, G. 1912. Le jubilé de M Gaston Darboux. Revue internationale de l’enseignement, 6, 97125.Google Scholar
Lovell, S. C., Davis, I. W., Arendall III, W. B., De Bakker, P. IW., Word, J.M., Prisant, M. G., Richardson, J. S. and Richardson, D. C. 2003. Structure validation by Cα geometry: φ, ψ and Cβ deviation. Proteins, 50(3), 437450.Google Scholar
Ma, P., Mahoney, M. W. and Yu, B. 2015. A statistical perspective on algorithmic leveraging. J. Mach. Learn. Res., 16(1), 861911.Google Scholar
Magnani, A., Lall, S. and Boyd, S. 2005. Tractable fitting with convex polynomials via sum-of-squares. In Proceedings of the 44th IEEE Conference on Decision and Control. Curran Associates, New York, NY, pp. 16721677.Google Scholar
Mahoney, M. W. 2011. Randomized algorithms for matrices and data. Found. Trends Mach. Learn., 3(2), 123224.Google Scholar
Mahoney, M. W. and Drineas, P. 2009. CUR matrix decompositions for improved data analysis. Proc. Natl Acad. Sci USA, 106(3), 697702.Google Scholar
Mammen, E. and Tsybakov, A. B. 1995. Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist, 502524.Google Scholar
Marx, S., Pauwels, E., Weisser, T., Henrion, D. and Lasserre, J. B. 2021. Tractable semi-algebraic approximation using Christoffel–Darboux kernel. Constr. Approx., 54, 391429.CrossRefGoogle Scholar
Máté, A., Nevai, P. and Totik, V. 1987. Strong and weak convergence of orthogonal polynomials. Amer. J. Math., 109(2), 239281.Google Scholar
Máté, A., Nevai, P. and Totik, V. 1991. Szegö’s extremum problem on the unit circle. Ann. of Math. (2), 134(2), 433453.Google Scholar
Matveev, V. B. and Salle, M. A. 1991. Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin.Google Scholar
Mauroy, A., Mezić, I. and Susuki, Yoshihiko. 2020. The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol. 484. Springer, Cham.Google Scholar
Molchanov, I. S. 1998. A limit theorem for solutions of inequalities. Scand. J. Stat., 25(1), 235242.Google Scholar
Nevai, P. 1986. Géza Freud, orthogonal polynomials and Christoffel functions: A case study. J. Approx. Theory, 48(1), 3167.Google Scholar
Nie, J. 2013. Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. Ser. A, 142(1–2), 485510.CrossRefGoogle Scholar
Nikolski, N. 2019. Hardy Spaces, French edn. Cambridge Studies in Advanced Mathematics, vol. 179. Cambridge University Press, Cambridge.Google Scholar
Niyogi, P., Smale, S. and Weinberger, S. 2008. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom., 39(1–3), 419441.Google Scholar
Niyogi, P., Smale, S. and Weinberger, S. 2011. A topological view of unsupervised learning from noisy data. SIAM J. Comput., 40(3), 646663.Google Scholar
Paulsen, V. I. and Raghupathi, M. 2016. An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge Studies in Advanced Mathematics, vol. 152. Cambridge University Press, Cambridge.Google Scholar
Pauwels, E., Bach, F. and Vert, J.-P. 2018. Relating leverage scores and density using regularized christoffel functions. In Advances in Neural Information Processing Systems. MIT Press, Cambridge, MA, pp. 16631672.Google Scholar
Pauwels, E., Putinar, M. and Lasserre, J. B. 2021. Data analysis from empirical moments and the Christoffel function. Found. Comput. Math., 21(1), 243273.CrossRefGoogle Scholar
Pawłucki, W. and Pleśniak, W. 1986. Markov’s inequality and C functions on sets with polynomial cusps. Math. Ann., 275(3), 467480.CrossRefGoogle Scholar
Peherstorfer, F. 1992. Finite perturbations of orthogonal polynomials. J. Comput. Appl. Math., 44(3), 275302.Google Scholar
Perryman, M., Lindegren, L., Kovalevsky, J., Hoeg, E., Bastian, U., Bernacca, P. et al. 1997. The Hipparcos Catalogue. Astron. Astrophys., 500, 501504.Google Scholar
Piazzon, F. 2018. The extremal plurisubharmonic function of the torus. Dolomites Res. Notes Approx., 11(Special Issue Norm Levenberg), 6272.Google Scholar
Piazzon, F. 2019a. Laplace Beltrami operator in the Baran metric and pluripotential equilibrium measure: the ball, the simplex, and the sphere. Comput. Methods Funct. Theory, 19(4), 547582.Google Scholar
Piazzon, F. 2019b. Pluripotential numerics. Constr. Approx., 49(2), 227263.Google Scholar
Poincaré, H. 1975. L’avenir des mathématiques. Scientia (Milano), 110(5-8), 357– 368. Abridged text of a paper read at the Fourth International Congress of Mathematicians, Rome, 1908, with introductions in Italian and English.Google Scholar
Polonik, W. 1995. Measuring mass concentrations and estimating density contour clusters-an excess mass approach. Ann. Statist., 23(3), 855881.Google Scholar
Prestel, A. and Delzell, C. N. 2001. Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra. Springer Monographs in Mathematics. Springer, Berlin.Google Scholar
Prymak, A. and Usoltseva, O. 2019. Christoffel functions on planar domains with piecewise smooth boundary. Acta Math. Hungar., 158(1), 216234.CrossRefGoogle Scholar
Putinar, M. 1993. Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J., 42, 969984.Google Scholar
Putinar, M. 2021a. Moment estimates of the cloud of a planar measure. Acta Appl. Math., https://doi.org/10.1007/s10440–021-00443-0.CrossRefGoogle Scholar
Putinar, M. 2021b. Spectral Analysis of 2D outlier layout. J. Spectral Theory, 11, 821– 845.Google Scholar
Rényi, A. and Sulanke, R. 1963. Über die konvexe Hülle von n zufällig gewählten Punkten. Probab. Theory. Related Fields, 2(1), 7584.Google Scholar
Ricci, P. E. 1978. Čebyšev polynomials in several variables. Rend. Mat. (6), 11(2), 295327.Google Scholar
Riesz, F. and Sz.-Nagy, B. 1990 [1955]. Functional Analysis, trans. L. F. Boron. Dover Books on Advanced Mathematics. Dover, New York, NY.Google Scholar
Riesz, M. 2013 [1988]. Collected Papers, ed. Garding, L. and Hörmander, L.. Springer Collected Works in Mathematics. Springer, Heidelberg.Google Scholar
Rodríguez Casal, A. 2007. Set estimation under convexity type assumptions. Ann. Inst. Henri Poincaré Probab. Stat., 43(6), 763774.Google Scholar
Rosenblum, M. and Rovnyak, J. 1997 [1985]. Hardy Classes and Operator Theory. Dover, Mineola, NY.Google Scholar
Ross, J. and Nyström, D. W. 2019. Applications of the duality between the homogeneous complex Monge-Ampère equation and the Hele-Shaw flow. Ann. Inst. Fourier (Grenoble), 69(1), 130.Google Scholar
Rudi, A. and Rosasco, L. 2017. Generalization properties of learning with random features. In Advances in Neural Information Processing Systems. NIPS, pp. 3218– 3228.Google Scholar
Rudi, A., Camoriano, R. and Rosasco, L. 2015. Less is more: Nyström computational regularization. In Advances in Neural Information Processing Systems. NIPS, pp. 16571665.Google Scholar
Sadullaev, A. 1981. Plurisubharmonic measures and capacities on complex manifolds. Uspekhi Mat. Nauk, 36(4(220)), 53105, 247.Google Scholar
Sadullaev, A. 1982. Estimates of polynomials on analytic sets. Izv. Akad. Nauk SSSR Ser. Mat., 46(3), 524534, 671.Google Scholar
Sadullaev, A. 1985. Plurisubharmonic functions. In Current Problems in Mathematics: Fundamental Directions, Vol. 8. Itogi Nauki i Tekhniki. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, pp. 65113, 274.Google Scholar
Saff, E. B., Stahl, H., Stylianopoulos, N. and Totik, V. 2015. Orthogonal polynomials for area-type measures and image recovery. SIAM J. Math. Anal., 47(3), 24422463.Google Scholar
Samsonov, B. F. and Ovcharov, I. N. 1995. The Darboux transformation and nonclassical orthogonal polynomials. Izv. Vyssh. Uchebn. Zaved. Fiz., 38(4), 5865.Google Scholar
Schmüdgen, K. 2017. The Moment Problem. Graduate Texts in Mathematics, vol. 277. Springer, Cham.Google Scholar
Schölkopf, B., Smola, A. J., Bach, F. et al. 2002. Learning with Kernels: Support Vector Machines, regularization, Optimization, and Beyond. MIT Press, Cambridge, MA.Google Scholar
Serre, J.-P. 1956. Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, 6, 142.Google Scholar
Siciak, J. 1962. On some extremal functions and their applications in the theory of analytic functions of several complex variables. Trans. Amer. Math. Soc., 105, 322357.Google Scholar
Simanek, B. 2012. Weak convergence of CD kernels: a new approach on the circle and real line. J. Approx. Theory, 164(1), 204209.Google Scholar
Simon, B. 2005a. Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, vol. 54. American Mathematical Society, Providence, RI.Google Scholar
Simon, B. 2005b. Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, vol. 54. American Mathematical Society, Providence, RI.Google Scholar
Simon, B. 2008. The Christoffel–Darboux kernel. In Perspectives in Partial Differential Equations, Harmonic Analysis and Applications. Proceedings of Symposia in Pure Mathematics, vol. 79. American Mathematical Society, Providence, RI, pp. 295335.Google Scholar
Simon, B. 2009. Weak convergence of CD kernels and applications. Duke Math. J., 146(2), 305330.Google Scholar
Simon, B. 2011. Szegő’s Theorem and its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials.. M. B. Porter Lectures. Princeton University Press, Princeton, NJ.Google Scholar
Stahl, H. and Totik, V. 1992. General Orthogonal Polynomials. Encyclopedia of Mathematics and its Applications, vol. 43. Cambridge University Press, Cambridge.Google Scholar
Stone, M. H. 1990 [1932]. Linear Transformations in Hilbert Space. American Mathematical Society Colloquium Publications, vol. 15. American Mathematical Society, Providence, RI.Google Scholar
Suetin, P. K. 1999. Orthogonal Polynomials in Two Variables. Analytical Methods and Special Functions, vol. 3. Gordon and Breach Science Publishers, Amsterdam. Translated from the 1988 Russian original by E. V. Pankratiev.Google Scholar
Szegő, G. 1975. Orthogonal Polynomials, 4th edn. American Mathematical Society, Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI.Google Scholar
Thomson, J. E. 1991. Approximation in the mean by polynomials. Ann. of Math. (2), 133(3), 477507.Google Scholar
Todd, M. J. 2016. Minimum-Volume Ellipsoids: Theory and Algorithms. SIAM, Philadelphia.Google Scholar
Totik, V. 2000. Asymptotics for Christoffel functions for general measures on the real line. J. Anal. Math., 81, 283303.Google Scholar
Totik, V. 2010. Christoffel functions on curves and domains. Trans. Amer. Math. Soc., 362, 20532087.Google Scholar
Totik, V. 2012. The polynomial inverse image method. In Neamtu, M. and Schumaker, L. (eds.), Approximation Theory XIII: San Antonio 2010. Springer Proceedings in Mathematics, vol. 13. Springer, Berlin.Google Scholar
Tsybakov, A. B. 1997. On nonparametric estimation of density level sets. Ann. Statist., 25(3), 948969.CrossRefGoogle Scholar
Velleman, P. F. and Welsch, R. E. 1981. Efficient computing of regression diagnostics. Amer. Statis., 35(4), 234242.Google Scholar
Vershynin, R. 2010. Introduction to the non-asymptotic analysis of random matrices. arXiv preprint arXiv:1011.3027.Google Scholar
von Neumann, J. 1932. Zur Operatorenmethode in der klassischen Mechanik. Ann. of Math. (2), 33(3), 587642.Google Scholar
Vu, Mai Trang, Bachoc, F. and Pauwels, E. 2019. Rate of convergence for geometric inference based on the empirical Christoffel function. arXiv preprint arXiv:1910.14458.Google Scholar
Walther, G. 1999. On a generalization of Blaschke’s rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci., 22(4), 301316.Google Scholar
Wang, S. and Zhang, Z. 2013. Improving CUR matrix decomposition and the Nyström approximation via adaptive sampling. J. Mach. Learn. Res., 14(1), 27292769.Google Scholar
Weisse, A., Wellein, G., Alvermann, A. and Fehske, H. 2006. The kernel polynomial method. Rev. Modern Phys., 78, 275306.Google Scholar
Widom, H. 1967. Polynomials associated with measures in the complex plane. J. Math. Mech., 16, 9971013.Google Scholar
Widom, H. 1969. Extremal polynomials associated with a system of curves in the complex plane. Adv. Math., 3, 127232.CrossRefGoogle Scholar
Wiener, N. and Wintner, A. 1941. Harmonic analysis and ergodic theory. Amer. J. Math., 63, 415426.CrossRefGoogle Scholar
Williams, G., Baxter, R., He, H., Hawkins, S. and Gu, L. 2002. A comparative study of RNN for outlier detection in data mining. In Proceedings of the IEEE International Conference on Data Mining. IEEE Computer Society, p. 709.Google Scholar
Xu, Y. 1995. Christoffel functions and Fourier series for multivariate orthogonal polynomials. J. Approx. Theory, 82(2), 205239.Google Scholar
Xu, Y. 1996. Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal., 3(2), 257272.Google Scholar
Xu, Y. 1999. Asymptotics of the Christoffel functions on a simplex in Rd. J. Approx. Theory, 99(1), 122133.Google Scholar
Yamanishi, K., Takeuchi, J.-I., Williams, G. and Milne, P. 2004. On-line supervised outlier detection using finite mixtures with discounting learning algorithms. Data Min. Knowl. Discov., 8(3), 275300.Google Scholar
Yoon, G. J. 2002. Darboux transforms and orthogonal polynomials. Bull. Korean Math. Soc., 39(3), 359376.Google Scholar
Zhedanov, A. 1997. Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math., 85(1), 6786.Google Scholar
Zhou, Y. 2016. Asymptotics of Lp Christoffel functions. J. Math. Anal. Appl., 433(2), 13901408.Google Scholar

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  • References
  • Jean Bernard Lasserre, Edouard Pauwels, Institut de Recherche en Informatique, Toulouse, Mihai Putinar, University of California, Santa Barbara
  • Book: The Christoffel–Darboux Kernel for Data Analysis
  • Online publication: 31 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108937078.017
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  • References
  • Jean Bernard Lasserre, Edouard Pauwels, Institut de Recherche en Informatique, Toulouse, Mihai Putinar, University of California, Santa Barbara
  • Book: The Christoffel–Darboux Kernel for Data Analysis
  • Online publication: 31 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108937078.017
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  • References
  • Jean Bernard Lasserre, Edouard Pauwels, Institut de Recherche en Informatique, Toulouse, Mihai Putinar, University of California, Santa Barbara
  • Book: The Christoffel–Darboux Kernel for Data Analysis
  • Online publication: 31 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108937078.017
Available formats
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