# Rational Approximation on Exponential Meshes

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## Abstract

**:**

## 1. Introduction

## 2. Rational Approximation on Exponential-Type Meshes

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Remark**

**2.**

**Theorem**

**3.**

**Remark**

**3.**

**Theorem**

**4.**

**Remark**

**4.**

## 3. Shepard-Type Curves and Discrete Fourier Transform

**Theorem**

**5.**

**Remark**

**5.**

**Theorem**

**6.**

**Remark**

**6.**

#### 3.1. Shepard-Type Curves via DFT and PIA

## 4. Proofs

**Proof of Theorem 1.**

**Proof of Theorem 2.**

**Proof of Theorem 3.**

**Proof of Theorem 4.**

**Proof of Theorem 6.**

## 5. Numerical Experiments

#### 5.1. Example 1

#### 5.2. Example 2

- (a)
- $n=9,\phantom{\rule{4pt}{0ex}}j=7,\phantom{\rule{4pt}{0ex}}\lambda ={10}^{-5}$ (Figure 1);
- (b)
- $n=11,\phantom{\rule{4pt}{0ex}}j=3,\phantom{\rule{4pt}{0ex}}\lambda ={10}^{-5}$ (Figure 2, left);
- (c)
- $n=11,\phantom{\rule{4pt}{0ex}}j=10,\phantom{\rule{4pt}{0ex}}\lambda ={10}^{-5}$ (Figure 2, right);
- (d)
- $n=13,\phantom{\rule{4pt}{0ex}}j=3,\phantom{\rule{4pt}{0ex}}\lambda ={10}^{-6}$ (Figure 3, left);
- (e)
- $n=13,\phantom{\rule{4pt}{0ex}}j=5,\phantom{\rule{4pt}{0ex}}\lambda ={10}^{-6}$ (Figure 3, right).

#### 5.3. Example 3

#### 5.4. Example 4

#### 5.5. Example 5

#### 5.6. Example 6

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | Discrete Fourier Transform |

CAGD | Computer Aided Geometric Design |

FT | Fourier Transform |

PIA | Progressive Iterative Approximation |

## References

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**Figure 4.**Basic Shepard curves for $n+1=7,$ $\lambda =5\phantom{\rule{4pt}{0ex}}\times {10}^{-5}$ and $j=1,\dots ,6$.

**Figure 5.**Archimedes spiral, $n=10$, $\lambda ={10}^{-5}$, $k=9$, 10, 11 and $r=1$ (

**left**), $r=2$ (

**right**).

**Figure 6.**Archimedes spiral, $n=18$, $\lambda =3\times {10}^{-6}$, $k=$17, 18, 19 and $r=1$ (

**left**), $r=3$ (

**right**).

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Amato, U.; Della Vecchia, B.
Rational Approximation on Exponential Meshes. *Symmetry* **2020**, *12*, 1999.
https://doi.org/10.3390/sym12121999

**AMA Style**

Amato U, Della Vecchia B.
Rational Approximation on Exponential Meshes. *Symmetry*. 2020; 12(12):1999.
https://doi.org/10.3390/sym12121999

**Chicago/Turabian Style**

Amato, Umberto, and Biancamaria Della Vecchia.
2020. "Rational Approximation on Exponential Meshes" *Symmetry* 12, no. 12: 1999.
https://doi.org/10.3390/sym12121999