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Remarks on Inner Functions and Optimal Approximants

Published online by Cambridge University Press:  20 November 2018

Catherine Bénéteau ,
Matthew C. Fleeman ,
Dmitry S. Khavinson ,
Daniel Seco  and
Alan A. Sola
Catherine Bénéteau
Affiliation:
Department of Mathematics, University of South Florida, Tampa, FL 33620, USA, e-mail : cbenetea@usf.edu
Matthew C. Fleeman
Affiliation:
Baylor University, Waco, TX 76710, USA, e-mail : matthew_fleeman@baylor.edu
Dmitry S. Khavinson
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA, e-mail : dkhavins@usf.edu
Daniel Seco
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Facultat de Matemàtiques, Universitat de Barcelona, 08007 Barcelona, Spain, e-mail : dseco@mat.uab.cat
Alan A. Sola
Affiliation:
Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden, e-mail : sola@math.su.se
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Abstract

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We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to ${1}/{f}\;$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions.

Keywords

46E2230J05inner functionreproducing kernel hilbert spaceoperator-theoretic function theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 4 , 01 December 2018 , pp. 704 - 716
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Aleman, A., Richter, S., and Sundberg, C., Beurling's theorem for the Bergman space. Acta Math. 177 (1996), 275–310. http://dx.doi.org/10.1007/BF02392623Google Scholar
[2] Bénéteau, C., Condori, A. A., Liaw, C., Seco, D., and Sola, A. A., Cydicity in Dirichlet-type spaces and extremal polynomials. J. Anal. Math. 126 (2015), 259–286. http://dx.doi.Org/10.1007/s11854-015-0017-1Google Scholar
[3] Bénéteau, C., Khavinson, D., Liaw, C., Seco, D., and Sola, A. A., Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants. J. London Math. Soc. 94 (2016), 726–746. http://dx.doi.Org/10.1112/jlms/jdwO57Google Scholar
[4] Bénéteau, C., Khavinson, D., Liaw, C., Seco, D., and Simanek, B., Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems. arxiv:1606.08615Google Scholar
[5] Beurling, A., On two problems concerning linear transformations in Hilbert space. Acta Math. 81 (1948), 239–255. http://dx.doi.org/10.1007/BF02395019Google Scholar
[6] Carswell, B., Duren, P., and Stessin, M., Multiplication invariant subspaces of the Bergman space. Indiana Univ. Math. J. 51 (2002), no. 4, 931-961. http://dx.doi.Org/10.1512/iumj.2002.51.2184Google Scholar
[7] Duren, P., Theory of Hp spaces. Second Edition, Dover Publications, Mineola, NY, 2000.Google Scholar
[8] Duren, P., Khavinson, D., Shapiro, H., and Sundberg, C., Contractive zero-divisors in Bergman spaces. Pacific J. Math. 157 (1993), 37–56. http://dx.doi.Org/10.2140/pjm.1993.157.37Google Scholar
[9] Duren, P., Khavinson, D., Shapiro, H., and Sundberg, C., Invariant subspaces in Bergman spaces and the biharmonic equation, Michigan Math. J. 41 (1994), 247–259. http://dx.doi.org/10.1307/mmjV1029004992Google Scholar
[10] Duren, P. and Schuster, A., Bergman spaces. Mathematical Surveys and Monographs, 100. American Mathematical Society, Providence, RI, 2004. http://dx.doi.Org/10.1090/surv/100Google Scholar
[11] El-Fallah, O., Kellay, K., Mashreghi, J., and Ransford, T., A primer on the Dirichlet space. Cambridge Tracts in Mathematics, 203. Cambridge University Press, Cambridge, 2014.Google Scholar
[12] Fricain, E., Mashreghi, J., and Seco, D., Cydicity in reproducing kernel Hilbert spaces of analytic functions. Comput. Methods Funct. Theory 14 (2014), 665–680. http://dx.doi.org/!0.1007/s40315-014-0073-zGoogle Scholar
[13] Halmos, P. R., Shifts on Hilbert spaces. J. Reine Angew. Math. 208 (1961), 102–112. http://dx.doi.Org/10.1515/crll.1961.208.102Google Scholar
[14] Hansbo, J., Reproducing kernels and contractive divisors in Bergman spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). 232 (1996), no. Issled. po Linein. Oper. i Teor. Funktsii. 24,174-198.Google Scholar
[15] Hedenmalm, H., A factorization theorem for square area-integrable analytic functions. J. Reine Angew. Math. 422 (1991), 45–68. http://dx.doi.Org/10.1515/crll.1991.422.45Google Scholar
[16] Hedenmalm, H., Korenblum, B., and Zhu, K., Theory of Bergman spaces. Graduate Texts in Mathematics, 199. Springer-Verlag, New York, 2000. http://dx.doi.Org/10.1007/978-1-4612-0497-8Google Scholar
[17] Hedenmalm, H. and Zhu, K., On the failure of optimal factorization for certain weighted Bergman spaces. Complex Variables Theory Appl. 19 (1992), 165–176.Google Scholar
[18] Khavinson, D., Lance, T., and Stessin, M., Wandering property in the Hardy space. Michigan Math. J. 44 (1997), no. 3, 597–606. http://dx.doi.org/10.1307/mmj71029005790Google Scholar
[19] Korenblum, B., Outer functions and cyclic elements in Bergman spaces. J. Funct. Anal. 115 (1993), 104–118. http://dx.doi.org/10.1006/jfan.1993.1082Google Scholar
[20] Lance, T., and Stessin, M., Multiplication invariant subspaces of Hardy spaces. Canad. J. Math. 49 (1997), no. 1, 100–118. http://dx.doi.org/10.4153/CJM-1997-005-9Google Scholar
[21] Paulsen, V. and Raghupathi, M., An introduction to the theory of reproducing kernel Hilbert spaces. Cambridge Studies in Advanced Mathematics, 152. Cambridge University Press, Cambridge, 2016. http://dx.doi.Org/10.1017/CBO9781316219232Google Scholar
[22] Richter, S., Invariant subspaces of the Dirichlet shift. J. Reine Angew. Math. 386 (1988), 205–220. http://dx.doi.Org/1 0.1 51 5/crll.1 988.386.205Google Scholar
[23] Seco, D., Some problems on optimal approximants. In: Recent progress on operator theory and approximation in spaces of analytic functions. Contemp. Math., 679. American Mathematical Society, Providence, RI, 2016, pp. 193-205.Google Scholar
[24] Shapiro, H. and Shields, A. L., On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80 (1962), 217–229. http://dx.doi.Org/10.1007/BF01162379Google Scholar
[25] Shields, A. I., Weighted shift operators and analytic function theory. In: Topics in operator theory. Math. Surveys, 13. American Mathematical Society, Providence, RI, 1974, pp. 49-128.Google Scholar