Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation
This paper addresses fundamental challenges in numerical approximation methods, focusing on balancing accuracy with shape-preserving properties. We present novel approaches that combine traditional spline methods with modern numerical techniques, extending existing quasi-interpolation techniques bas...
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2025-04-01
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| author | Francesc Aràndiga Sara Remogna |
| author_facet | Francesc Aràndiga Sara Remogna |
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| collection | DOAJ |
| description | This paper addresses fundamental challenges in numerical approximation methods, focusing on balancing accuracy with shape-preserving properties. We present novel approaches that combine traditional spline methods with modern numerical techniques, extending existing quasi-interpolation techniques based on B-splines. Our methods maintain computational efficiency while better handling discontinuities, achieving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula> reconstructions and preserving essential shape properties. We demonstrate theoretical frameworks showing optimal approximation order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula>, with local reconstruction. Numerical experiments confirm significant improvements in accuracy and smoothness near discontinuities compared to existing methods, particularly in image processing and shock-capturing applications. |
| format | Article |
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| publishDate | 2025-04-01 |
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| spelling | doaj-art-5d59656bc79842ca809c535c989f9b1f2025-08-20T02:18:14ZengMDPI AGMathematics2227-73902025-04-01138123710.3390/math13081237Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-InterpolationFrancesc Aràndiga0Sara Remogna1Departament de Matemàtiques, Universitat de València, Av. Vicent Andrés Estellés, E-46100 Burjassot, SpainDepartment of Mathematics “G. Peano”, University of Torino, Via Carlo Alberto 10, 10123 Torino, ItalyThis paper addresses fundamental challenges in numerical approximation methods, focusing on balancing accuracy with shape-preserving properties. We present novel approaches that combine traditional spline methods with modern numerical techniques, extending existing quasi-interpolation techniques based on B-splines. Our methods maintain computational efficiency while better handling discontinuities, achieving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula> reconstructions and preserving essential shape properties. We demonstrate theoretical frameworks showing optimal approximation order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula>, with local reconstruction. Numerical experiments confirm significant improvements in accuracy and smoothness near discontinuities compared to existing methods, particularly in image processing and shock-capturing applications.https://www.mdpi.com/2227-7390/13/8/1237spline quasi-interpolationWENOpiecewise smooth functions approximationshape-preserving |
| spellingShingle | Francesc Aràndiga Sara Remogna Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation Mathematics spline quasi-interpolation WENO piecewise smooth functions approximation shape-preserving |
| title | Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation |
| title_full | Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation |
| title_fullStr | Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation |
| title_full_unstemmed | Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation |
| title_short | Shape-Preserving <i>C</i><sup>1</sup> and <i>C</i><sup>2</sup> Reconstructions of Discontinuous Functions Using Spline Quasi-Interpolation |
| title_sort | shape preserving i c i sup 1 sup and i c i sup 2 sup reconstructions of discontinuous functions using spline quasi interpolation |
| topic | spline quasi-interpolation WENO piecewise smooth functions approximation shape-preserving |
| url | https://www.mdpi.com/2227-7390/13/8/1237 |
| work_keys_str_mv | AT francescarandiga shapepreservingicisup1supandicisup2supreconstructionsofdiscontinuousfunctionsusingsplinequasiinterpolation AT sararemogna shapepreservingicisup1supandicisup2supreconstructionsofdiscontinuousfunctionsusingsplinequasiinterpolation |