Approximation order of bivariate spline interpolation for arbitrary smoothness
Consider the partition \(\Delta^1\) of the rectangle \(R= [a_0,b_0]\times [c_0,d_0]\) into triangles. This partition is formed by the subrectangles \(R_{i,j}= (x_{i-1}, x_i)\times (y_{i- 1},y_i)\), \(i= 1,\dots, n_1\), \(j= 1,\dots, n_2\), and the diagonals from the vertex \(z_{i- 1,j-1}= (x_{i-1}, y_{j-1})\) to \(z_{i,j}\). Let integers \(r\) and \(q\) with \(0\leq r< q\) be given and let \(\widetilde\Pi_q\) be the space of bivariate polynomials of total degree \(q\). The space \(S^r_q(\Delta^1)\) of bivariate splines of degree \(q\) and smoothness \(r\) is that of all functions \(s\in C^r(R)\) such that the restriction to each subtriangle of the partition \(\Delta^1\) is in \(\widetilde\Pi_q\). The paper under review deals with a Hermite interpolation problem where the space \(S^r_q(\Delta^1)\) of bivariate splines is involved. Theorem 1. Let \(T\) be an arbitrary subtriangle of \(\Delta^1\), and \(\widetilde T\) denote the adjacent subtriangle left of \(T\) in the same row if it exists and up of \(T\) otherwise. For each sufficiently differentiable function \(f\in C(R)\), there exists a unique solution \(s_f\in S^r_q(\Delta^1)\), \(q\geq 3.5r+ 1\), which satisfies the Hermite interpolation conditions described below. Condition A (Starting condition): \(D^\omega p(z)= D^\omega f(z)\), \(\omega= 0,\dots, q\), where \(z\) is a vertex of \(T\). Condition B: \(D^\omega p(z)= D^\omega f(z)\), \(\omega= 0,\dots, q-r-1\), where \(z\) is a vertex of \(T\) not belonging to \(\widetilde T\). Condition C: \(p_{x^\alpha y^\beta}(z)= f_{x^\alpha y^\beta}(z)\), \(\alpha\geq 0\), \(\beta\geq 0\), \(\alpha+ \beta\leq q-r-1\), \(\alpha+ 2\beta\leq 2q- 3r-2\), where \(z\) is the vertex of \(T\) not belonging to \(\widetilde T\). Condition D: \(D^\omega p(z)= D^\omega f(z)\), \(\omega= 0,\dots, q- 2r- 2\), where \(z\) is the midpoint of the diagonal of \(T\). Theorem 2. Let \(q,r\in\mathbb{N}\cup \{0\}\) be given such that \(q\geq 3.5r+1\). For each function \(f\in C^{q+ 1}(R)\), there exists a constant \(K>0\), such that for the unique interpolating spline \(s_f\in S^r_q(\Delta^1)\) in theorem 1 and for all \(\omega\in \{0,\dots, q\}\), the following estimate holds \(\|D^\omega(f- s_f)\|\leq Kh^{q+ 1-\omega}\), where \(h\) is the norm of \(\Delta^1\) and \(K\) does not depend on \(h\).
scientific article; zbMATH DE number 1326310
scientific article