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BY 4.0 license Open Access Published by De Gruyter Open Access July 7, 2022

Optimal Polynomial Approximants in Lp

  • Raymond Centner EMAIL logo
From the journal Concrete Operators

Abstract

Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space Lp, 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L2. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in Lp. This allows us to study OPAs in Lp with the additional tools from the L2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, fHp, and f(0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in Lp. To inspire further research in the general theory, we pose several open questions throughout our discussions.

References

[1] G. Ammar, Classical foundations of algorithms for solving positive definite Toeplitz equations, Calcolo 33 (1996), no. 1-2, 99-113.Search in Google Scholar

[2] C. Bénéteau and R. Centner, A survey of optimal polynomial approximants, applications to digital filter design, and related open problems, Complex Anal. Synerg. 7 (2021), no. 2, Paper No. 16.Search in Google Scholar

[3] C. Bénéteau, A. Condori, C. Liaw, D. Seco, and A. Sola, Cyclicity in Dirichlet-type spaces and extremal polynomials, J. Anal. Math. 126 (2015), 259-286.10.1007/s11854-015-0017-1Search in Google Scholar

[4] C. Bénéteau, O. Ivrii, M. Manolaki, and D. Seco, Simultaneous zero-free approximation and universal optimal polynomial approximants, J. Approx. Theory 256 (2020).10.1016/j.jat.2020.105389Search in Google Scholar

[5] C. Bénéteau, D. Khavinson, C. Liaw, D. Seco, and B. Simanek, Optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems, Rev. Mat. Iberoam. 35 (2019), 607-642.10.4171/rmi/1064Search in Google Scholar

[6] C. Bénéteau, D. Khavinson, C. Liaw, D. Seco, and A. Sola, Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 726-746.Search in Google Scholar

[7] C. Bénéteau, M. Manolaki, and D. Seco, Boundary behavior of optimal polynomial approximants, Constr. Approx. 54 (2021), no. 1, 157-183.Search in Google Scholar

[8] R. Cheng, W. Ross, and D. Seco, Zeros of optimal polynomial approximants in pA, arXiv:2104.08014.Search in Google Scholar

[9] C. Chui, Approximation by double least-squares inverses, J. Math. Anal. Appl. 75 (1980), no. 1, 149-163.Search in Google Scholar

[10] C. Chui and A. Chan, Application of approximation theory methods to recursive digital filter design, IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-30 (1982), no. 1, 18-24.Search in Google Scholar

[11] P. Delsarte, Y. Genin, and Y. Kamp, Planar least-squares inverse polynomials. II. Asymptotic behavior, SIAM J. Algebraic Discrete Methods 1 (1980), no. 3, 336-344.Search in Google Scholar

[12] P. Duren, “Theory of Hp Spaces”, Academic Press, New York, 1970.Search in Google Scholar

[13] Y. Genin and Y. Kamp, Counterexample in the least-squares inverse stabilisation of 2D recursive filters, Elec. Letters 11 (1975), 330-331.10.1049/el:19750252Search in Google Scholar

[14] Y. Genin and Y. Kamp, Two-dimensional stability and orthogonal polynomials of the hypercircle, Proceedings of the IEEE 65 (1977), no. 6, 873-881.Search in Google Scholar

[15] I. Gonos, N. Mastorakis, M. Swamy, and L. Virirakis, Evolutionary design of 2-dimensional recursive filters via the computer language GENETICA, IEEE Transactions on Circuits and Systems II: Express Briefs 53 (2006), no. 4, 254-258.Search in Google Scholar

[16] S. Izumino, Generalized inverses of Toeplitz operators and inverse approximation in H2, Tohoku Math. J. (2) 37 (1985), no. 1, 95-99.Search in Google Scholar

[17] J. Justice, J. Shanks, and S. Treitel, Stability and synthesis of two-dimensional recursive filters, IEEE Transactions on Audio and Electroacoustics AU-20 (1972), no. 2, 115-128.Search in Google Scholar

[18] E. Kreyszig, “Introductory Functional Analysis with Applications”, Wiley, New York, 1978.Search in Google Scholar

[19] E. Robinson, Structural properties of stationary stochastic processes with applications, in “Proceedings of the Symposium on Time Series Analysis”, Wiley, New York, 1963.Search in Google Scholar

[20] D. Seco and R. Téllez, Polynomial approach to cyclicity for weighted pA, Banach J. Math. Anal. 15 (2021), no.1, Paper No. 1.10.1007/s43037-020-00085-8Search in Google Scholar

[21] H. Shapiro, “Topics in approximation theory”, Springer, New York, 1971.10.1007/BFb0058976Search in Google Scholar

Received: 2021-12-27
Accepted: 2022-05-10
Published Online: 2022-07-07

© 2022 Raymond Centner, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 2.12.2023 from https://www.degruyter.com/document/doi/10.1515/conop-2022-0131/html
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