# A Characterization of Polynomial Density on Curves via Matrix Algebra

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. New Indices of an HPD Matrix and Connections with the Polynomial Approximation

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Lemma**

**1.**

- 1.
- $\lambda \left({\mathbf{M}}_{1}\right)\le \lambda \left({\mathbf{M}}_{2}\right)$.
- 2.
- $\gamma \left({\mathbf{M}}_{1}\right)\le \gamma \left({\mathbf{M}}_{2}\right)$.

**Corollary**

**1.**

**Proof.**

**Remark**

**3.**

**Proposition**

**2.**

**Proposition**

**3.**

## 3. HPSD Matrices Which Are Moment Matrices

**Lemma**

**2.**

**Lemma**

**3.**

- 1.
- $\gamma \left(\mathbf{M}\right)=0$.
- 2.
- For all$k\in \mathbb{Z}$,${z}^{k}\in {\overline{[{z}^{k+1},{z}^{k+2},\dots ]}}^{{L}^{2}\left(\mu \right)}$.

**Proof.**

**Corollary**

**2.**

- 1.
- $\gamma \left(\mathbf{M}\right)=0$.
- 2.
- ${P}^{2}\left(\mu \right)=\mathbb{C}[z,{z}^{-1}]$.

**Theorem**

**1.**

- 1.
- $\gamma \left(\mathbf{M}\right)=0$.
- 2.
- ${P}^{2}\left(\mu \right)={L}^{2}\left(\mu \right)$.

**Proof.**

**Corollary**

**3.**

- 1.
- Polynomials are dense in${L}^{2}\left(\mathbb{T}\right)$.
- 2.
- ${\sum}_{k=0}^{\infty}{\left|{\phi}_{k}\left(0\right)\right|}^{2}=\infty $.

**Proof.**

## 4. Bounded Point Evaluations from the Matrix Algebra Point of View: Thomson’s Theorem Revisited

**Remark**

**4.**

**Lemma**

**4.**

**Proof.**

**Definition**

**3.**

**Remark**

**5.**

**Definition**

**4.**

**Remark**

**6.**

**Lemma**

**5.**

- 1.
- ${z}_{0}$ is a bounded point evaluation for ${P}^{2}\left(\mathbf{M}\right)$.
- 2.
- ${\gamma}_{{z}_{0}}\left(\mathbf{M}\right)>0$.

**Proof.**

**Remark**

**7.**

**Lemma**

**6.**

**Proof.**

**Corollary**

**4.**

- 1.
- ${z}_{0}$ is a bounded point evaluation of ${P}^{2}\left(\mathbf{M}\right)$.
- 2.
- ${K}_{\mathbf{M}}({z}_{0},{z}_{0})={\sum}_{n=0}^{\infty}{\left|{\phi}_{n}\left({z}_{0}\right)\right|}^{2}<\infty $
- 3.
- ${\gamma}_{{z}_{0}}\left(\mathbf{M}\right)>0$.
- 4.
- ${\displaystyle \frac{1}{{\gamma}_{{z}_{0}}\left(\mathbf{M}\right)}}=\underset{n\to \infty}{lim}(1,{z}_{0},{z}_{0}^{2},\dots ,{z}_{0}^{n}){\mathbf{M}}_{n}^{-1}\left(\right)open="("\; close=")">\begin{array}{c}1\\ {z}_{0}\\ \vdots \\ {z}_{0}^{n}\end{array}0$.

**Proof.**

**Theorem**

**2.**

- 1.
- ${\gamma}_{{z}_{0}}\left(\mathbf{M}\right)>0$.
- 2.
- ${P}^{2}\left(\mu \right)\ne {L}^{2}\left(\mu \right)$.
- 3.
- ${z}_{0}$ is a bounded point evaluation of ${P}^{2}\left(\mathbf{M}\right)$.

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gaier, D. Lectures on Complex Approximation; Birkhauser: Boston, MA, USA, 1987. [Google Scholar]
- Szegö, G. Orthogonal Polynomials; American Mathematical Society, Colloquium Publications: Providence, RI, USA, 1975; Volume 23. [Google Scholar]
- Simon, B. Orthogonal Polynomials on the Unit Circle, Part 1. Classical Theory; 54 Part 1; American Mathematical Society, Colloquium Publications: Providence, RI, USA, 2003. [Google Scholar]
- Escribano, C.; Gonzalo, R.; Torrano, E. Small Eigenvalues of Large Hermitian moment matrices. J. Math. Anal. Appl.
**2011**, 374, 470–480. [Google Scholar] [CrossRef] - Berg, C.; Duran, A.J. Orthogonal Polynomials and analytic functions associated with positive definite matrices. J. Math. Anal. Appl.
**2006**, 315, 54–67. [Google Scholar] [CrossRef] [Green Version] - Daruis, L.; Hernández, J.; Marcellán, F. Spectral transformations for Hermitian Toeplitz matrices. J. Comput. Appl. Math.
**2007**, 202, 155–176. [Google Scholar] [CrossRef] [Green Version] - Brennan, J.E. Thomson’s Theorem on mean square polynomial approximation. St. Petesburg Math. J.
**2006**, 17, 217–238. [Google Scholar] [CrossRef] - Thomson, J.E. Appoximation in the mean by polynomials. Ann. Mater.
**1991**, 133, 477–507. [Google Scholar] [CrossRef] - Grenander, U.; Szegö, G. Toeplitz Forms and Their Applications; Chelsea Publishing Company: New York, NY, USA, 1958. [Google Scholar]
- Nikishin, E.M.; Sorokin, V.N. Rational Approximations and Orthogonality; Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 1991; Volume 92. [Google Scholar]
- Conway, J.B. The Theory of Subnormal Operators; Mathematical Surveys and Monographs; American Mathematical Society: Providence, RI, USA, 1985; Volume 36. [Google Scholar]
- Escribano, C.; Sastre, M.A.; Torrano, E. A fixed point theorem for moment matrices self-similar measures. J. Comput. Appl. Math.
**2007**, 207, 352–359. [Google Scholar] [CrossRef] [Green Version]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Escribano, C.; Gonzalo, R.; Torrano, E.
A Characterization of Polynomial Density on Curves via Matrix Algebra. *Mathematics* **2019**, *7*, 1231.
https://doi.org/10.3390/math7121231

**AMA Style**

Escribano C, Gonzalo R, Torrano E.
A Characterization of Polynomial Density on Curves via Matrix Algebra. *Mathematics*. 2019; 7(12):1231.
https://doi.org/10.3390/math7121231

**Chicago/Turabian Style**

Escribano, Carmen, Raquel Gonzalo, and Emilio Torrano.
2019. "A Characterization of Polynomial Density on Curves via Matrix Algebra" *Mathematics* 7, no. 12: 1231.
https://doi.org/10.3390/math7121231