# A Characterization of Polynomial Density on Curves via Matrix Algebra

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## 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.
- ${\displaystyle \frac{1}{{\gamma}_{{z}_{0}}\left(\mathbf{M}\right)}}=\underset{n\to \infty}{lim}(1,{z}_{0},{z}_{0}^{2},\dots ,{z}_{0}^{n}){\mathbf{M}}_{n}^{-1}\left(\begin{array}{c}1\\ {z}_{0}\\ \vdots \\ {z}_{0}^{n}\end{array}\right)>0$.

**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

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**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