Increasing the polynomial reproduction of a quasi-interpolation operator
Let \(\Pi_m(\mathbb R^d)\) stand for the space of \(d\)-variate polynomials of degree less or equal than \(m\), and let \(L\) be a quasi-interpolant from the space of sufficiently smooth functions onto \(\Pi_m(\mathbb R^d)\) such that \(LP=P\) for any polynomial \(P\) there, given by \[ Lf(x)=\sum_{i=1}^n\lambda_i(f)\varphi_i(x), \qquad \lambda_i(f)=\int q_i(x,D^{k_i}f(x))\,d\mu_i(x), \] where the \(q_i\) are continuous functions of \(x\) and the derivatives \(D^{k_i}f(x)\), and \(\mu_i\) are finite Borel measures on \(\mathbb R^d\) with compact support. The main result of this paper is the following: for such a quasi-interpolant \(L\) and \(r\geq 0\), it is constructed another quasi-interpolant \(R\) of the form \(R_m^{m+r}L=L+A\), such that \(R\) is the identity map in \(\Pi_{m+r}(\mathbb R^d)\), which increases the order of approximation of \(L\) by \(r\), and comes with an integral error formula which involves only the \(m+r+1\)-th derivative of the function approximated. Several interesting examples are presented, and to finish the paper the authors show that the interpolant \(R\) is well-behaved in the following sense: for given \(r_1,r_2\geq 0\), and a quasi-interpolant \(L\), then \[ R_{m+r_1}^{m+r_1+r_2}R_m^{m+r_1}L=R_m^{m+r_1+r_2}L. \]
scientific article; zbMATH DE number 5639035