# An Optimal Property of B-Bases for the Modified Richardson Method

## Abstract

**:**

## 1. Introduction

## 2. B-Bases and Optimal Properties

**Theorem 1.**

**Theorem 2.**

- 1.
- Normalized B-bases present optimal shape-preserving properties in the sense that their control polygon is closer in shape than the control polygon with respect to another NTP basis. In the case of the space of polynomials of degree at most n on $[0,1]$, this property was first conjectured by Goodman and Said for the Bernstein basis in [24] and later proved by Carnicer and Peña in [25]. In [23], this optimal property was proved for all normalized B-bases.
- 2.
- 3.
- B-bases are the least supported bases, as shown in [40].
- 4.
- Nonsingular collocation matrices of nomalized B-bases have a minimal condition number for ${\parallel \xb7\parallel}_{\infty}$ among the corresponding matrices of all NTP bases of the space, as shown in [41].

- 1.
- In the space ${\mathcal{P}}_{n}\left([a,+\infty )\right)$ of polynomials of degree less than or equal to n on $[a,+\infty )$, the monomial basis $(1,(t-a),{(t-a)}^{2},\dots ,{(t-a)}^{n})$ is a B-basis (cf. Section 6 of [25]).
- 2.
- In the space ${\mathcal{P}}_{n}\left([a,b]\right)$ of polynomials of degree less than or equal to n on a compact interval $[a,b]$, the normalized B-basis is the corresponding Bernstein basis of polynomials $({b}_{0}^{n}(t;a,b),\dots ,{b}_{n}^{n}(t;a,b))$, with$${b}_{i}^{n}(t;a,b)\left(t\right):=\left(\genfrac{}{}{0pt}{}{n}{i}\right)\frac{{(b-t)}^{n-i}{(t-a)}^{i}}{{(b-a)}^{n}},\phantom{\rule{1.em}{0ex}}i=0,1,\dots ,n$$
- 3.
- Given a sequence ${\left({w}_{i}\right)}_{0\le i\le n}$ of positive weights, we can define the following system of rational functions $({r}_{0}^{n},\dots ,{r}_{n}^{n})$ on the compact interval $[a,b]$ by$${r}_{i}^{n}\left(t\right)=\frac{{w}_{i}{b}_{i}^{n}(t;a,b)}{{\sum}_{j=0}^{n}{w}_{j}{b}_{j}^{n}(t;a,b)},\phantom{\rule{1.em}{0ex}}i=0,1,\dots ,n,$$
- 4.
- In the space of exponential functions$$\mathcal{E}:=\{\sum _{i=0}^{n}{c}_{i}exp\left({\lambda}_{i}t\right)|\phantom{\rule{0.166667em}{0ex}}{c}_{i}\in ,\phantom{\rule{0.166667em}{0ex}}i=0,\dots ,n\},\phantom{\rule{1.em}{0ex}}{\lambda}_{0}<\cdots <{\lambda}_{n},\phantom{\rule{1.em}{0ex}}t\in ,$$
- 5.
- In the space of Müntz polynomials on $[0,+\infty )$, where$$\mathcal{M}:=\{\sum _{i=0}^{n}{c}_{i}{t}^{{\lambda}_{i}}|\phantom{\rule{0.166667em}{0ex}}{c}_{i}\in ,\phantom{\rule{0.166667em}{0ex}}i=0,\dots ,n\},\phantom{\rule{1.em}{0ex}}{\lambda}_{0}<\cdots <{\lambda}_{n},$$$${p}_{i}\left(t\right)={(-1)}^{n-i}({\lambda}_{i+1}-{\lambda}_{0})({\lambda}_{i+2}-{\lambda}_{0})\cdots ({\lambda}_{n}-{\lambda}_{0}){t}^{\lambda}[{\lambda}_{i},\dots ,{\lambda}_{n}],$$$${t}^{\lambda}[{\lambda}_{i},\dots ,{\lambda}_{i+j}]:=\frac{{t}^{\lambda}[{\lambda}_{i+1},\dots ,{\lambda}_{i+j}]-{t}^{\lambda}[{\lambda}_{i},\dots ,{\lambda}_{i+j-1}]}{{\lambda}_{i+j}-{\lambda}_{i}}.$$
- 6.
- The space of trigonometric polynomials$${\mathcal{T}}_{n}=\{1,cost,sint,cos2t,sin2t,\dots ,cosnt,sinnt\}$$$${v}_{i}\left(t\right)={d}_{i}{\left(\frac{sin\left(\frac{A+t}{2}\right)}{sinA}\right)}^{i}{\left(\frac{sin\left(\frac{A-t}{2}\right)}{sinA}\right)}^{m-i},\phantom{\rule{1.em}{0ex}}i=0,1,\dots ,m$$$${d}_{i}=\sum _{k=0}^{[i/2]}\left(\genfrac{}{}{0pt}{}{m/2}{i-k}\right)\left(\genfrac{}{}{0pt}{}{i-k}{k}\right){(2cosA)}^{i-2k},\phantom{\rule{1.em}{0ex}}i=0,1,\dots ,m.$$
- 7.
- The space of even trigonometric functions given by$${\mathcal{C}}_{n}=span\{1,cos\phantom{\rule{0.166667em}{0ex}}t,cos\phantom{\rule{0.166667em}{0ex}}2t,\dots ,cos\phantom{\rule{0.166667em}{0ex}}nt\}$$$${u}_{i}^{n}\left(t\right)=\left(\genfrac{}{}{0pt}{}{n}{i}\right){cos}^{2(n-i)}(t/2){sin}^{2i}(t/2),\phantom{\rule{1.em}{0ex}}i=0,1,\dots ,n.$$
- 8.
- Given a sequence of positive weights ${\left({w}_{i}\right)}_{0\le i\le n}$ and a knots vector $({t}_{0},\dots ,{t}_{n+d})$ with ${t}_{i}\le {t}_{i+1}$ for all $i=0,1,\dots ,n+d-1$, we can define the B-spline basis $({N}_{0,d},{N}_{1,d},\dots ,{N}_{n,d})$ over the previous knots vector by$$\begin{array}{cc}\hfill {N}_{i,0}\left(t\right)& =\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{t}_{i}\le t<{t}_{i+1},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \\ \hfill {N}_{i,k}\left(t\right)& =\frac{t-{t}_{i}}{{t}_{i+k}-{t}_{i}}{N}_{i,k-1}\left(t\right)+\frac{{t}_{i+k+1}-t}{{t}_{i+k+1}-{t}_{i+1}}{N}_{i+1,k-1}\left(t\right),\phantom{\rule{1.em}{0ex}}k=1,\dots ,d,\hfill \end{array}$$$${r}_{i}\left(t\right)=\frac{{w}_{i}{N}_{i,d}\left(t\right)}{{\sum}_{j=0}^{n}{w}_{j}{N}_{j,d}\left(t\right)},\phantom{\rule{1.em}{0ex}}i=0,1\dots ,n,$$
- 9.
- The space given by$${\mathcal{P}}_{1}=\mathrm{span}\{1,t,cost,sint\}$$$$\mathrm{sin}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right):=\mathrm{t}-sin\mathrm{t},\phantom{\rule{1.em}{0ex}}\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right):=1-cos\mathrm{t},\phantom{\rule{1.em}{0ex}}\mathrm{tan}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right):=\frac{\mathrm{sin}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right)}{\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right)}.$$Using these cycloidal functions, we now define the functions of the normalized B-basis:$$\begin{array}{ccc}{Z}_{3}\left(t\right)\hfill & =& \frac{\mathrm{sin}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right)}{S},\phantom{\rule{1.em}{0ex}}{Z}_{0}\left(t\right)={Z}_{3}(p-t),\hfill \\ {Z}_{2}\left(t\right)\hfill & =& M\left(\frac{\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{t}\right)}{C}-{Z}_{3}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}{Z}_{1}\left(t\right)={Z}_{2}(p-t),\hfill \end{array}$$$$S=\mathrm{sin}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{p}\right),\phantom{\rule{1.em}{0ex}}\mathrm{C}=\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{p}\right),\phantom{\rule{1.em}{0ex}}\mathrm{T}=\mathrm{tan}\phantom{\rule{0.166667em}{0ex}}\mathrm{C}\left(\mathrm{p}\right),$$$$M=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4pt}{0ex}}p=\pi ,\hfill \\ \frac{sinp}{p-2T}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

## 3. A New Optimal Property of Normalized B-Bases

**Theorem 3.**

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Remark 1.**

## 4. Tensor Product Case

**Theorem 6.**

**Proof.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PIA | progressive iterative approximation |

TP | totally positive |

NTP | normalized totally positive |

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Peña, J.M.
An Optimal Property of B-Bases for the Modified Richardson Method. *Axioms* **2024**, *13*, 223.
https://doi.org/10.3390/axioms13040223

**AMA Style**

Peña JM.
An Optimal Property of B-Bases for the Modified Richardson Method. *Axioms*. 2024; 13(4):223.
https://doi.org/10.3390/axioms13040223

**Chicago/Turabian Style**

Peña, Juan Manuel.
2024. "An Optimal Property of B-Bases for the Modified Richardson Method" *Axioms* 13, no. 4: 223.
https://doi.org/10.3390/axioms13040223