Shape-Preserving C1 and C2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation
Abstract
:1. Introduction and Notations
- QI stands for quasi-interpolation;
- WENO stands for weighted essentially non-oscillatory;
- We denote by a linear QI operator on the space of a spline of degree d, with coefficient functionals defined by using m points;
- We denote by the nonlinear QI operator obtained by applying WENO techniques to ;
- We denote by the monotone QI operator associated with .
2. WENO Techniques
2.1. WENO with Positive Weights
2.2. WENO with Negative Weights
3. Quadratic QI Approximants
3.1. Linear QI Based on 3 Points:
3.2. Nonlinear QI:
3.3. Monotone QI:
4. Cubic QI Approximants
4.1. Linear QI Based on 3 Points:
4.2. Nonlinear QI:
4.3. Monotone QI:
5. Numerical Experiments
5.1. Approximation Properties
5.2. Graphical Properties
5.2.1. Example 1: Data Reconstruction
5.2.2. Example 2: Geophysical Application
6. Conclusions and Future Directions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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4.478 × 10−2 | 8.104 × 10−2 | 6.198 × 10−2 | ||||
4.381 × 10−2 | 0.03 | 8.055 × 10−2 | 0.01 | 6.990 × 10−2 | −0.17 | |
4.337 × 10−2 | 0.01 | 8.006 × 10−2 | 0.01 | 7.444 × 10−2 | −0.09 | |
4.316 × 10−2 | 0.01 | 7.975 × 10−2 | 0.01 | 7.687 × 10−2 | −0.05 | |
4.306 × 10−2 | 0.00 | 7.959 × 10−2 | 0.00 | 7.813 × 10−2 | −0.02 | |
1.034 × 10−2 | 4.833 × 10−4 | 4.109 × 10−3 | ||||
1.012 × 10−2 | 0.03 | 6.583 × 10−5 | 2.88 | 2.250 × 10−3 | 0.87 | |
1.002 × 10−2 | 0.01 | 1.511 × 10−5 | 2.12 | 1.185 × 10−3 | 0.93 | |
9.972 × 10−3 | 0.01 | 3.699 × 10−6 | 2.03 | 6.090 × 10−4 | 0.96 | |
9.949 × 10−3 | 0.00 | 9.200 × 10−7 | 2.01 | 3.089 × 10−4 | 0.98 | |
9.754 × 10−5 | 4.833 × 10−4 | 6.097 × 10−5 | ||||
1.031 × 10−5 | 3.24 | 3.566 × 10−5 | 3.76 | 7.884 × 10−6 | 2.95 | |
1.164 × 10−6 | 3.15 | 2.848 × 10−6 | 3.65 | 1.009 × 10−6 | 2.97 | |
1.373 × 10−7 | 3.08 | 2.456 × 10−7 | 3.54 | 1.275 × 10−7 | 2.98 | |
1.665 × 10−8 | 3.04 | 2.351 × 10−8 | 3.39 | 1.604 × 10−8 | 2.99 | |
5.425 × 10−2 | 1.082 × 10−1 | 8.798 × 10−2 | ||||
5.358 × 10−2 | 0.02 | 1.074 × 10−1 | 0.01 | 9.628 × 10−2 | −0.13 | |
5.326 × 10−2 | 0.01 | 1.067 × 10−1 | 0.01 | 1.009 × 10−1 | −0.07 | |
5.310 × 10−2 | 0.00 | 1.063 × 10−1 | 0.01 | 1.034 × 10−1 | −0.03 | |
5.302 × 10−2 | 0.00 | 1.061 × 10−1 | 0.00 | 1.046 × 10−1 | −0.02 | |
1.838 × 10−2 | 5.859 × 10−4 | 7.296 × 10−3 | ||||
1.799 × 10−2 | 0.03 | 1.141 × 10−4 | 2.36 | 3.999 × 10−3 | 0.87 | |
1.781 × 10−2 | 0.01 | 2.670 × 10−5 | 2.10 | 2.106 × 10−3 | 0.93 | |
1.773 × 10−2 | 0.01 | 6.566 × 10−6 | 2.02 | 1.083 × 10−3 | 0.96 | |
1.769 × 10−2 | 0.00 | 1.635 × 10−6 | 2.01 | 5.492 × 10−4 | 0.98 | |
5.357 × 10−5 | 5.319 × 10−4 | 7.219 × 10−6 | ||||
3.427 × 10−6 | 3.97 | 4.108 × 10−5 | 3.69 | 4.543 × 10−7 | 3.99 | |
2.198 × 10−7 | 3.96 | 2.592 × 10−6 | 3.99 | 3.049 × 10−8 | 3.90 | |
1.393 × 10−8 | 3.98 | 1.624 × 10−7 | 4.00 | 1.979 × 10−9 | 3.95 | |
8.775 × 10−10 | 3.99 | 1.016 × 10−8 | 4.00 | 1.261 × 10−10 | 3.97 |
9.370 × 10−2 | 1.148 × 10−1 | 1.065 × 10−1 | ||||
9.385 × 10−2 | −0.00 | 1.145 × 10−1 | 0.00 | 1.124 × 10−1 | −0.08 | |
9.321 × 10−2 | 0.01 | 1.144 × 10−1 | 0.00 | 1.139 × 10−1 | −0.02 | |
9.361 × 10−2 | −0.01 | 1.144 × 10−1 | 0.00 | 1.142 × 10−1 | −0.00 | |
9.373 × 10−2 | −0.00 | 1.144 × 10−1 | 0.00 | 1.143 × 10−1 | −0.00 | |
3.452 × 10−2 | 2.124 × 10−3 | 3.673 × 10−3 | ||||
3.440 × 10−2 | 0.00 | 1.503 × 10−4 | 3.82 | 9.309 × 10−4 | 1.98 | |
3.437 × 10−2 | 0.00 | 9.964 × 10−6 | 3.91 | 2.334 × 10−4 | 2.00 | |
3.436 × 10−2 | 0.00 | 8.014 × 10−7 | 3.64 | 5.839 × 10−5 | 2.00 | |
3.436 × 10−2 | 0.00 | 7.092 × 10−8 | 3.50 | 1.460 × 10−5 | 2.00 | |
2.332 × 10−4 | 2.124 × 10−3 | 2.746 × 10−4 | ||||
2.523 × 10−5 | 3.21 | 1.503 × 10−4 | 3.82 | 2.786 × 10−5 | 3.30 | |
2.851 × 10−6 | 3.15 | 9.964 × 10−6 | 3.91 | 3.014 × 10−6 | 3.21 | |
3.346 × 10−7 | 3.09 | 8.014 × 10−7 | 3.64 | 3.446 × 10−7 | 3.13 | |
4.033 × 10−8 | 3.05 | 7.092 × 10−8 | 3.50 | 4.095 × 10−8 | 3.07 | |
9.512 × 10−2 | 1.435 × 10−1 | 1.348 × 10−1 | ||||
9.481 × 10−2 | 0.00 | 1.431 × 10−1 | 0.00 | 1.410 × 10−1 | −0.06 | |
9.493 × 10−2 | −0.00 | 1.430 × 10−1 | 0.00 | 1.425 × 10−1 | −0.01 | |
9.496 × 10−2 | −0.00 | 1.430 × 10−1 | 0.00 | 1.428 × 10−1 | −0.00 | |
9.493 × 10−2 | 0.00 | 1.429 × 10−1 | 0.00 | 1.429 × 10−1 | −0.00 | |
5.983 × 10−2 | 1.751 × 10−3 | 6.370 × 10−3 | ||||
5.963 × 10−2 | 0.00 | 1.894 × 10−4 | 3.21 | 1.614 × 10−3 | 1.98 | |
5.958 × 10−2 | 0.00 | 1.203 × 10−5 | 3.98 | 4.046 × 10−4 | 2.00 | |
5.956 × 10−2 | 0.00 | 7.517 × 10−7 | 4.00 | 1.012 × 10−4 | 2.00 | |
5.956 × 10−2 | 0.00 | 4.689 × 10−8 | 4.00 | 2.531 × 10−5 | 2.00 | |
2.172 × 10−4 | 1.751 × 10−3 | 2.639 × 10−4 | ||||
1.239 × 10−5 | 4.13 | 1.894 × 10−4 | 3.21 | 1.546 × 10−5 | 4.09 | |
7.500 × 10−7 | 4.05 | 1.203 × 10−5 | 3.98 | 9.531 × 10−7 | 4.02 | |
4.700 × 10−8 | 4.00 | 7.517 × 10−7 | 4.00 | 6.049 × 10−8 | 3.98 | |
3.021 × 10−9 | 3.96 | 4.689 × 10−8 | 4.00 | 3.851 × 10−9 | 3.97 |
depth | 0 | 0.3 | 0.5 | 0.8 | 1.2 | 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 2.4 | 2.8 | 3.5 | 4.0 | 4.2 | 4.5 | 5.0 |
cond. | 0.15 | 0.18 | 0.22 | 0.25 | 0.28 | 0.30 | 0.32 | 0.35 | 0.85 | 0.88 | 0.90 | 0.92 | 0.95 | 0.97 | 0.98 | 0.99 | 1.0 |
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Aràndiga, F.; Remogna, S. Shape-Preserving C1 and C2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation. Mathematics 2025, 13, 1237. https://doi.org/10.3390/math13081237
Aràndiga F, Remogna S. Shape-Preserving C1 and C2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation. Mathematics. 2025; 13(8):1237. https://doi.org/10.3390/math13081237
Chicago/Turabian StyleAràndiga, Francesc, and Sara Remogna. 2025. "Shape-Preserving C1 and C2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation" Mathematics 13, no. 8: 1237. https://doi.org/10.3390/math13081237
APA StyleAràndiga, F., & Remogna, S. (2025). Shape-Preserving C1 and C2 Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation. Mathematics, 13(8), 1237. https://doi.org/10.3390/math13081237