# Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Auxiliary Tools

#### 2.1. Radial Basis Functions

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

#### 2.2. Multiply Monotonicity

**Definition**

**3.**

**Theorem**

**2.**

#### 2.3. Native Spaces

**Definition**

**4.**

## 3. Truncated Exponential Function

**Theorem**

**3.**

**Proof.**

## 4. Errors in Native Spaces

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 5. Numerical Experiments

#### 5.1. Single-Level Approximation

#### 5.2. Multilevel Approximation

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**TERBF approximation of the Beethoven data. From top left to bottom right: 163 (

**a**), 663 (

**b**), 1163 (

**c**), and 2663 (

**d**) points.

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 3.633326 × ${10}^{-1}$ | - | 1.000028 × ${10}^{+0}$ |

25 | 3.138226 × ${10}^{-1}$ | 0.211341 | 1.006645 × ${10}^{+0}$ |

81 | 2.003929 × ${10}^{-1}$ | 0.647118 | 3.170400 × ${10}^{+0}$ |

289 | 6.616318 × ${10}^{-2}$ | 1.598731 | 3.761572 × ${10}^{+1}$ |

1089 | 1.205109 × ${10}^{-2}$ | 2.456865 | 1.925205 × ${10}^{+5}$ |

4225 | 2.908614 × ${10}^{-4}$ | 5.372688 | 2.687885 × ${10}^{+16}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 3.256546 × ${10}^{-1}$ | - | 1.129919 × ${10}^{+0}$ |

25 | 1.722746 × ${10}^{-1}$ | 0.918633 | 1.667637 × ${10}^{+0}$ |

81 | 5.465624 × ${10}^{-2}$ | 1.656252 | 2.601726 × ${10}^{+1}$ |

289 | 1.391350 × ${10}^{-2}$ | 1.973901 | 7.316820 × ${10}^{+4}$ |

1089 | 3.273510 × ${10}^{-4}$ | 5.409503 | 1.179104 × ${10}^{+16}$ |

4225 | 1.135157 × ${10}^{-6}$ | 8.171803 | 1.906108 × ${10}^{+20}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 1.224583 × ${10}^{-1}$ | - | 5.366051 × ${10}^{+1}$ |

25 | 5.646454 × ${10}^{-2}$ | 1.116874 | 3.124063 × ${10}^{+2}$ |

81 | 6.998841 × ${10}^{-3}$ | 3.012157 | 5.534539 × ${10}^{+3}$ |

289 | 1.418117 × ${10}^{-3}$ | 2.303139 | 2.324743 × ${10}^{+5}$ |

1089 | 3.627073 × ${10}^{-4}$ | 1.967099 | 8.803829 × ${10}^{+7}$ |

4225 | 4.969932 × ${10}^{-5}$ | 2.867508 | 5.331981 × ${10}^{+11}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 1.146184 × ${10}^{-1}$ | - | 8.464360 × ${10}^{+1}$ |

25 | 5.193997 × ${10}^{-2}$ | 1.141921 | 6.680998 × ${10}^{+2}$ |

81 | 4.534144 × ${10}^{-3}$ | 3.517943 | 2.158362 × ${10}^{+4}$ |

289 | 9.608696 × ${10}^{-4}$ | 2.238418 | 5.033541 × ${10}^{+6}$ |

1089 | 1.506154 × ${10}^{-4}$ | 2.673471 | 3.025049 × ${10}^{+10}$ |

4225 | 4.603113 × ${10}^{-6}$ | 5.032116 | 5.613893 × ${10}^{+16}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 2.491443 × ${10}^{-1}$ | - | 2.733942 × ${10}^{+0}$ |

25 | 9.914856 × ${10}^{-2}$ | 1.329318 | 6.933813 × ${10}^{+0}$ |

81 | 3.257319 × ${10}^{-2}$ | 1.605907 | 5.444834 × ${10}^{+1}$ |

289 | 1.159691 × ${10}^{-2}$ | 1.489945 | 1.022341 × ${10}^{+3}$ |

1089 | 3.420734 × ${10}^{-3}$ | 1.761362 | 1.850967 × ${10}^{+5}$ |

4225 | 6.703871 × ${10}^{-4}$ | 2.351240 | 5.607685 × ${10}^{+8}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 2.065836 × ${10}^{-1}$ | - | 5.995564 × ${10}^{+0}$ |

25 | 5.366442 × ${10}^{-2}$ | 1.944688 | 2.312141 × ${10}^{+1}$ |

81 | 1.517723 × ${10}^{-2}$ | 1.822057 | 4.053520 × ${10}^{+2}$ |

289 | 5.181480 × ${10}^{-3}$ | 1.550472 | 3.889766 × ${10}^{+4}$ |

1089 | 9.630601 × ${10}^{-4}$ | 2.427667 | 1.155244 × ${10}^{+8}$ |

4225 | 4.615820 × ${10}^{-5}$ | 4.382967 | 1.158439 × ${10}^{+14}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 1.951235 × ${10}^{-1}$ | - | 6.639719 × ${10}^{+0}$ |

25 | 5.018953 × ${10}^{-2}$ | 1.958929 | 2.405994 × ${10}^{+1}$ |

81 | 1.628459 × ${10}^{-2}$ | 1.623879 | 1.669026 × ${10}^{+2}$ |

289 | 6.727682 × ${10}^{-3}$ | 1.275326 | 1.250365 × ${10}^{+3}$ |

1089 | 2.402630 × ${10}^{-3}$ | 1.485495 | 1.058555 × ${10}^{+4}$ |

4225 | 9.728457 × ${10}^{-4}$ | 1.304332 | 9.410946 × ${10}^{+4}$ |

N | RMS-Error | Rate | cond(A) |
---|---|---|---|

9 | 1.728785 × ${10}^{-1}$ | - | 1.275042 × ${10}^{+1}$ |

25 | 4.535991 × ${10}^{-2}$ | 1.930269 | 5.066809 × ${10}^{+1}$ |

81 | 1.335521 × ${10}^{-2}$ | 1.764015 | 3.608813 × ${10}^{+2}$ |

289 | 5.013012 × ${10}^{-3}$ | 1.413653 | 2.719227 × ${10}^{+3}$ |

1089 | 1.773595 × ${10}^{-3}$ | 1.499001 | 2.305630 × ${10}^{+4}$ |

4225 | 7.107796 × ${10}^{-4}$ | 1.319203 | 2.050036 × ${10}^{+5}$ |

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**MDPI and ACS Style**

Xu, Q.; Liu, Z.
Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function. *Mathematics* **2019**, *7*, 1101.
https://doi.org/10.3390/math7111101

**AMA Style**

Xu Q, Liu Z.
Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function. *Mathematics*. 2019; 7(11):1101.
https://doi.org/10.3390/math7111101

**Chicago/Turabian Style**

Xu, Qiuyan, and Zhiyong Liu.
2019. "Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function" *Mathematics* 7, no. 11: 1101.
https://doi.org/10.3390/math7111101