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Bergman polynomials on an archipelago: estimates, zeros and shape reconstruction. (English) Zbl 1194.42030

Summary: Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago \(|z^m - 1|<r^m, 0<r<1\), which consists of \(m\) islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom [Adv. Math. 3, 127–232 (1969; Zbl 0183.07503)] of Szegő orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
32A36 Bergman spaces of functions in several complex variables
30C40 Kernel functions in one complex variable and applications
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
30E05 Moment problems and interpolation problems in the complex plane
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)

Citations:

Zbl 0183.07503
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References:

[1] Ambroladze, A., On exceptional sets of asymptotic relations for general orthogonal polynomials, J. Approx. Theory, 82, 2, 257-273 (1995) · Zbl 0833.41008
[2] Andrievskii, V. V.; Blatt, H.-P., Erdős-Turán type theorems on quasiconformal curves and arcs, J. Approx. Theory, 97, 2, 334-365 (1999) · Zbl 0918.30006
[3] Carleman, T., Über die Approximation analytisher Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat. Astr. Fys., 17, 9, 215-244 (1923) · JFM 49.0708.03
[4] Davis, P. J., The Schwarz Function and Its Applications, Carus Math. Monogr., vol. 17 (1974), The Mathematical Association of America: The Mathematical Association of America Buffalo, NY · Zbl 0293.30001
[5] Duren, P. L., Theory of \(H^p\) Spaces, Pure Appl. Math., vol. 38 (1970), Academic Press: Academic Press New York · Zbl 0215.20203
[6] Duren, P. L.; Schuster, A., Bergman Spaces, Math. Surveys Monogr., vol. 100 (2004), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1059.30001
[7] Finn, M. D.; Cox, S. M.; Byrne, H. M., Topological chaos in inviscid and viscous mixers, J. Fluid Mech., 493, 345-361 (2003) · Zbl 1064.76105
[8] Gaier, D., Lectures on Complex Approximation (1987), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA, Translated from the German by Renate McLaughlin
[9] Golub, G.; Gustafsson, B.; He, C.; Milanfar, P.; Putinar, M.; Varah, J., Shape reconstruction from moments: Theory, algorithms, and applications, (Luk, F. T., Advanced Signal Processing, Algorithms, Architecture and Implementations, X. Advanced Signal Processing, Algorithms, Architecture and Implementations, X, Proc. SPIE, vol. 4116 (2000)), 406-416
[10] Gustafsson, B.; He, C.; Milanfar, P.; Putinar, M., Reconstructing planar domains from their moments, Inverse Problems, 16, 4, 1053-1070 (2000) · Zbl 0959.44010
[11] Hayman, W. K., Subharmonic Functions, vol. 2, London Math. Soc. Monogr. Ser., vol. 20 (1989), Academic Press: Academic Press London · Zbl 0699.31001
[12] Hedenmalm, H.; Korenblum, B.; Zhu, K., Theory of Bergman Spaces, Grad. Texts in Math., vol. 199 (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0955.32003
[13] Kuijlaars, A. B.J.; Saff, E. B., Asymptotic distribution of the zeros of Faber polynomials, Math. Proc. Cambridge Philos. Soc., 118, 437-447 (1995) · Zbl 0848.30004
[14] Landkof, N. S., Foundations of Modern Potential Theory, Grundlehren Math. Wiss., Band 180 (1972), Springer-Verlag: Springer-Verlag New York, Translated from the Russian by A.P. Doohovskoy · Zbl 0253.31001
[15] Levin, A. L.; Saff, E. B.; Stylianopoulos, N. S., Zero distribution of Bergman orthogonal polynomials for certain planar domains, Constr. Approx., 19, 3, 411-435 (2003) · Zbl 1027.30009
[16] Martínez-Finkelshtein, A.; McLaughlin, K. T.-R.; Saff, E. B., Szegő orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics, Constr. Approx., 24, 3, 319-363 (2006) · Zbl 1135.42326
[17] Maymeskul, V.; Saff, E. B., Zeros of polynomials orthogonal over regular \(N\)-gons, J. Approx. Theory, 122, 1, 129-140 (2003) · Zbl 1036.30026
[18] E. Miña-Díaz, Asymptotics for Faber polynomials and polynomials orthogonal over regions in the complex plane, PhD thesis, Vanderbilt University, August 2006; E. Miña-Díaz, Asymptotics for Faber polynomials and polynomials orthogonal over regions in the complex plane, PhD thesis, Vanderbilt University, August 2006
[19] Miña-Díaz, E., An asymptotic integral representation for Carleman orthogonal polynomials, Int. Math. Res. Not., 2008 (2008), article ID rnn066, 35 pages · Zbl 1171.30004
[20] Miña-Díaz, E.; Saff, E. B.; Stylianopoulos, N. S., Zero distributions for polynomials orthogonal with weights over certain planar regions, Comput. Methods Funct. Theory, 5, 1, 185-221 (2005) · Zbl 1098.30009
[21] Olver, F. W.J., Asymptotics and Special Functions, AKP Classics (1997), A.K. Peters Ltd.: A.K. Peters Ltd. Wellesley, MA, Reprint of the 1974 original, Academic Press, New York
[22] Papamichael, N.; Warby, M. K., Stability and convergence properties of Bergman kernel methods for numerical conformal mapping, Numer. Math., 48, 6, 639-669 (1986) · Zbl 0564.30009
[23] Papamichael, N.; Saff, E. B.; Gong, J., Asymptotic behaviour of zeros of Bieberbach polynomials, J. Comput. Appl. Math., 34, 3, 325-342 (1991) · Zbl 0726.30008
[24] Ransford, T., Potential Theory in the Complex Plane, London Math. Soc. Stud. Texts, vol. 28 (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0828.31001
[25] Saff, E. B., Polynomials of interpolation and approximation to meromorphic functions, Trans. Amer. Math. Soc., 143, 509-522 (1969) · Zbl 0185.31602
[26] Saff, E. B., Orthogonal polynomials from a complex perspective, (Orthogonal Polynomials. Orthogonal Polynomials, Columbus, OH, 1989 (1990), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 363-393 · Zbl 0697.42021
[27] Saff, E. B.; Stylianopoulos, N. S., Asymptotics for polynomial zeros: Beware of predictions from plots, Comput. Methods Funct. Theory, 8, 2, 185-221 (2008) · Zbl 1163.30010
[28] Saff, E. B.; Totik, V., Logarithmic Potentials with External Fields (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0881.31001
[29] Shapiro, H. S., The Schwarz Function and Its Generalization to Higher Dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9 (1992), John Wiley & Sons: John Wiley & Sons New York · Zbl 0784.30036
[30] Stahl, H.; Totik, V., \(n\) th root asymptotic behavior of orthonormal polynomials, (Orthogonal Polynomials. Orthogonal Polynomials, Columbus, OH, 1989 (1990), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 395-417
[31] Stahl, H.; Totik, V., General Orthogonal Polynomials (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0791.33009
[32] N.S. Stylianopoulos, The use of orthogonal Bergman polynomials for recovering planar domains from their moments, preprint; N.S. Stylianopoulos, The use of orthogonal Bergman polynomials for recovering planar domains from their moments, preprint
[33] Suetin, P. K., Order comparison of various norms of polynomials in a complex region, Ural. Gos. Univ. Mat. Zap., 5, 4, 91-100 (1966), (in Russian)
[34] Suetin, P. K., Polynomials Orthogonal over a Region and Bieberbach Polynomials (1974), American Mathematical Society: American Mathematical Society Providence, RI, Translated from the Russian by R.P. Boas · Zbl 0282.30034
[35] Szegő, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. XXIII (1975), American Mathematical Society: American Mathematical Society Providence, RI · JFM 65.0278.03
[36] Totik, V., Orthogonal polynomials, Surv. Approx. Theory, 1, 70-125 (2005) · Zbl 1105.42017
[37] V. Totik, Christoffel functions on curves and domains, preprint; V. Totik, Christoffel functions on curves and domains, preprint · Zbl 1189.26028
[38] L.N. Trefethen, Ten-digits algorithms, Report 05/13, Oxford University Computing Laboratory, 2005; L.N. Trefethen, Ten-digits algorithms, Report 05/13, Oxford University Computing Laboratory, 2005
[39] Walsh, J. L., A sequence of rational functions with application to approximation by bounded analytic functions, Duke Math. J., 30, 177-189 (1963) · Zbl 0116.28202
[40] Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Colloq. Publ., vol. XX (1965), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0146.29902
[41] Widom, H., Polynomials associated with measures in the complex plane, J. Math. Mech., 16, 997-1013 (1967) · Zbl 0182.09201
[42] Widom, H., Extremal polynomials associated with a system of curves in the complex plane, Adv. Math., 3, 127-232 (1969) · Zbl 0183.07503
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