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Optimal approximants and orthogonal polynomials in several variables

Part of: Linear function spaces and their duals Approximations and expansions

Published online by Cambridge University Press:  26 November 2020

Meredith Sargent  and
Alan A. Sola
Meredith Sargent
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, AR, USA e-mail: [email protected]
Alan A. Sola*
Affiliation:
Department of Mathematics, Stockholm University, Stockholm, Sweden
*
e-mail: [email protected]
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Abstract

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We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

Keywords

Optimal approximantsorthogonal polynomialsweakly inner functionszero sets

MSC classification

Primary: 41A10: Approximation by polynomials
Secondary: 46E22: Hilbert spaces with reproducing kernels (=
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 428 - 456
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

A.A.S acknowledges support from Ivar Bendixons stipendiefond för docenter.

References

Agler, J. and McCarthy, J. E., Pick interpolation and Hilbert function spaces . American Mathematical Society, Providence, RI, 2000.Google Scholar
Arveson, W., Subalgebras of ${C}^{\ast }$ -algebras III: multivariable operator theory . Acta Math. 181(1998), 159–228.10.1007/BF02392585CrossRefGoogle Scholar
Bénéteau, C., Condori, A. A., Liaw, C., Seco, D., and Sola, A. A., Cyclicity in Dirichlet-type spaces and extremal polynomials . J. Anal. Math. 126(2015), 259–286.10.1007/s11854-015-0017-1CrossRefGoogle Scholar
Bénéteau, C., Condori, A. A., Liaw, C., Seco, D., and Sola, A. A., Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk . Pac. J. Math. 276(2015), no. 1, 35–58.10.2140/pjm.2015.276.35CrossRefGoogle Scholar
Bénéteau, C., Fleeman, M., Khavinson, D., Seco, D., and Sola, A. A., Remarks on inner functions and optimal approximants . Can. Math. Bull. 61(2018), no. 4, 704–716.10.4153/CMB-2017-058-4CrossRefGoogle Scholar
Bénéteau, C., Ivrii, O., Manolaki, M., and Seco, D., Simultaneous zero-free approximation and universal optimal polynomial approximants . J. Approx. Theory 256(2020), 105389.10.1016/j.jat.2020.105389CrossRefGoogle Scholar
Bénéteau, C., Khavinson, D., Liaw, C., Seco, D., and Simanek, B., Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems . Rev. Mat. Iberoam 35(2019), no. 2, 607–642.10.4171/rmi/1064CrossRefGoogle Scholar
Bénéteau, C., Khavinson, D., Liaw, C., Seco, D., and Sola, A. A., Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants . J. Lond. Math. Soc. 94(2016), no. 3, 726–746.CrossRefGoogle Scholar
Bénéteau, C., Knese, G., Kosiński, L., Liaw, C., Seco, D., and Sola, A. A., Cyclic polynomials in two variables . Trans. Am. Math. Soc. 368(2016), no. 12, 87378–87754.CrossRefGoogle Scholar
Bénéteau, C., Manolaki, M., and Seco, D., Boundary behavior of optimal polynomial approximants. Constr. Approx., to appear.Google Scholar
Bickel, K., Pascoe, J. E., and Sola, A. A., Level curve portraits of rational inner functions. Ann. Scuola Norm. Sup. di Pisa, Cl. Scienze, to appear.Google Scholar
Brown, L. and Shields, A. L., Cyclic vectors in the Dirichlet space . Trans. Am. Math. Soc. 285(1984), 269–304.10.1090/S0002-9947-1984-0748841-0CrossRefGoogle Scholar
Cheng, R., Mashreghi, J., and Ross, W. T., Inner functions in reproducing kernel spaces . In: Aleman, A., Hedenmalm, H., Khavinson, D., and Putinar, M. (eds.), Analysis of operators on function spaces, Birkhäuser/Springer, Cham, 2019, pp. 167–211.10.1007/978-3-030-14640-5_6CrossRefGoogle Scholar
Chui, C. K., Approximation by double least-squares inverses . J. Math. Anal. Appl. 75(1980), 149–163.10.1016/0022-247X(80)90312-1CrossRefGoogle Scholar
Delsarte, P., Genin, Y. V., and Kamp, Y. G., Planar least squares inverse polynomials: Part I – algebraic properties . IEEE Trans. Circuits Syst. CAS-26(1979), no. 1, 59–66.CrossRefGoogle Scholar
Delsarte, P., Genin, Y. V., and Kamp, Y. G., Planar least squares inverse polynomials: Part II – asymptotic behavior . SIAM J. Alg. Disc. Meth. 1(1980), no. 3, 336–344.CrossRefGoogle Scholar
Delsarte, P., Genin, Y. V., and Kamp, Y. G., Comments on “proof of a modified form of Shanks’ conjecture on the stability of 2-D planar least square inverse polynomials and its implications.” IEEE Trans. Circuits Syst. CAS-32(1985), no. 9, 966–968.CrossRefGoogle Scholar
Duren, P. L., Theory of Hp spaces . Dover Publications Inc., Mineola, NY, 2000.Google Scholar
El-Fallah, O., Kellay, K., Mashreghi, J., and Ransford, T., A primer on the Dirichlet space . Cambridge University Press, Cambridge, MA, 2015.Google Scholar
Fricain, E., Mashreghi, J., and Seco, D., Cyclicity in reproducing kernel Hilbert spaces of analytic functions . Comput. Methods Funct. Theory 14(2014), no. 4, 665–680.CrossRefGoogle Scholar
Genin, Y. V. and Kamp, Y. G., Counter example in the least-squares inverse stabilisation of 2D recursive filters . Electron. Lett. 11(1975), 330–331.CrossRefGoogle Scholar
Genin, Y. V. and Kamp, Y. G., Two-dimensional stability and orthogonal polynomials on the hypercircle . Proc. IEEE 65(1977), 873–881.CrossRefGoogle Scholar
Geronimo, J. S. and Woerdeman, H. J., Two variable orthogonal polynomials on the bicircle and structured matrices . SIAM J. Matrix Anal. Appl. 29(2007), 796–825.CrossRefGoogle Scholar
Hedenmalm, H., Korenblum, B., and Zhu, K., Theory of Bergman spaces . Springer-Verlag, New York, NY, 2000.CrossRefGoogle Scholar
Izumino, S., Generalized inverses of Toeplitz operators and inverse approximation in H2 . Tohoku J. Math. 37(1985), no. (2), 95–99.CrossRefGoogle Scholar
Justice, J. and Shanks, J., Stability criterion for n-dimensional digital filters . IEEE Trans. Automat. Contr. AC-18(1973), 284–286.CrossRefGoogle Scholar
Knese, G., Kosiński, L., Ransford, T., and Sola, A. A., Cyclic polynomials in anisotropic Dirichlet spaces . J. Anal. Math. 130(2019), 23–47.CrossRefGoogle Scholar
Le, T., Inner functions in weighted Hardy spaces . Anal. Math. Phys. 10(2020), 25.CrossRefGoogle Scholar
Neuwirth, J. H., Ginsberg, J., and Newman, D. J., Approximation by $\left\{f(kx)\right\}$ . J. Funct. Anal. 5(1970), 194–203.CrossRefGoogle Scholar
Reddy, P., Reddy, D. R. R., and Swamy, M., Proof of a modified form of shanks’ conjecture on the stability of $2$ -d planar least square inverse polynomials and its implications . IEEE Trans. Circuits Syst. 31(1984), 1009.CrossRefGoogle Scholar
Richter, S., and Sundberg, C., Cyclic vectors in the Drury-Arveson space. Slides from ESI talk, 2012.Google Scholar
Richter, S. and Sunkes, J., Hankel operators, invariant subspaces, and cyclic vectors in the Drury-Arveson space . Proc. Am. Math. Soc. 144(2016), 2575–2586.CrossRefGoogle Scholar
Robinson, E. A., Structural properties of stationary stochastic processes with applications . In: Rosenblatt, M. (ed.), Time series analysis , Wiley, New York, NY, 1963.Google Scholar
Rudin, W., Function theory in polydisks . W. A. Benjamin, Inc., New York, NY, 1969.Google Scholar
Sargent, M. and Sola, A. A., Optimal approximants and orthogonal polynomials in several variables II: families of polynomials in the unit ball. Preprint, 2020.CrossRefGoogle Scholar
Seco, D., Some problems on optimal approximants . In: Bénéteau, C., Condori, A. A., Liaw, C., Ross, W. T., and Sola, A. A. (eds.), Recent progress on operator theory and approximation in spaces of analytic functions , Amer. Math. Soc., Providence, RI, 2016, pp. 193–205.CrossRefGoogle Scholar
Shalit, O. M., Operator theory and function theory in Drury-Arveson space and its quotients . In: Alpay, D. (ed.), Handbook of operator theory, Springer-Verlag, New York, NY, 2015, pp. 1125–1180.CrossRefGoogle Scholar
Shanks, J. L., Treitel, S., and Justice, J. H., Stability and synthesis of two-dimensional recursive filters . IEEE Trans. Audio Electroacoustics AU-20(1972), no. 2, 115–128.CrossRefGoogle Scholar
Shapiro, H. S. and Shields, A. L., On the zeros of functions with finite Dirichlet integral and some related function spaces . Math. Z. 80(1962), 217–229.CrossRefGoogle Scholar
Simon, B., Orthogonal polynomials on the unit circle, Part 1: classical theory . American Mathematical Society, Providence, RI, 2005.Google Scholar
Sola, A., A note on Dirichlet-type spaces and cyclic vectors in the unit ball of ${\mathbb{C}}^2$ . Arch. Math. 104(2015), 247–257.CrossRefGoogle Scholar