Shape-Preserving C^1 and C^2 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 based on B-splines. Our methods maintain computational efficiency while better handling discontinuities, achieving C^1 and C^2 reconstructions and preserving essential shape properties. We demonstrate theoretical frameworks showing optimal approximation order O(h^(d+1)), d=2,3, 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.