A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

• Main Authors: Rabia Hameed, Ghulam Mustafa, Dumitru Baleanu, Yu-Ming Chu
• Format: Article
• Language: English
• Published: SpringerOpen 2021-03-01
• Series: Advances in Difference Equations
• Subjects: Refinement scheme; Divided difference; Smoothness; Continuity; Subdivision scheme; Inequalities
• Online Access: https://doi.org/10.1186/s13662-021-03336-6

Abstract In this article, we present the continuity analysis of the 3D models produced by the tensor product scheme of ( m + 1 ) $(m+1)$ -point binary refinement scheme. We use differences and divided differences of the bivariate refinement scheme to analyze its smoothness. The C 0 $C^{0}$ , C 1 $C^{1}$ and C 2 $C^{2}$ continuity of the general bivariate scheme is analyzed in our approach. This gives us some simple conditions in the form of arithmetic expressions and inequalities. These conditions require the mask and the complexity of the given refinement scheme to analyze its smoothness. Moreover, we perform several experiments by using these conditions on established schemes to verify the correctness of our approach. These experiments show that our results are easy to implement and are applicable for both interpolatory and approximating types of the schemes.