Multiquadric quasi-interpolation for integral functionals

Multiquadric (MQ) quasi-interpolation has been extensively studied in approximation theory and its applications. However, most of them only focus on the case that the sampling data are point-evaluation functionals (discrete function values). This restricts its applications since a more general form of data in real applications is integral functionals (integration of an approximand over subdomains that form a partition of the considering region). In addition, integral functionals also demonstrate characteristics of sampling devices. Therefore it is more meaningful to extend multiquadric quasi-interpolation to integral functionals. The purpose of the paper is to construct such a new multiquadric quasi-interpolation scheme. The scheme possesses the constant reproduction property and thus preserves positivity of an approximand. Moreover, if the approximand is a density function of a random variable, then the resulting quasi-interpolant is also a density function. Thus, as a byproduct, we also provide a scheme for nonparametric kernel density estimation. However, unlike classical kernel density estimation that usually has a boundary bias problem over a bounded domain, our scheme circumvents the problem. Numerical simulations including posteriori error estimates and density estimation are demonstrated at the end of the paper to show efficiency and robustness of the scheme.