Spherical scattered data quasi-interpolation by Gaussian radial basis function

Abstract

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions are usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasiinterpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.

Spatial Distribution and Influence Analysis of Soil Heavy Metals in a Hilly Region of Sichuan Basin

“…This suggests that atmospheric particles settling into the soil was one of the external sources of heavy metals in the topsoil. By comparing the spatial distribution of heavy metals between the topsoil elements and the atmospheric dustfall, the spatial distribution of elements had a better consistency [37][38].…”

Optimal order Jackson type inequality for scaled Shepard approximation

“…However, for most other choices of base functions, the interpolation feature is lost, and the resulted approximants are called "quasi-interpolants" in the literature; see [4], [17], [18]. Except for the cases in which the base function is compactly supported (see [16], [17], [18]), deriving optimal order error estimates for a Shepard approximation scheme has been uncommon.…”

“…Here φ : x → |x| −λ , λ ≥ 1, and Φ n (x) = n j=1 φ(x − x j ), in which |x| denotes the Euclidean norm of x ∈ R 2 . This procedure has since become known as "Shepard approximation", and variations of it have been studied in [2], [4], [7], [14], [15], [16], [17], [18], and the references therein.…”

“…In a nutshell, a Shepard approximation scheme employs rational formations of shifts of an appropriately-selected base function to approximate a target function, and its efficiency epitomizes in the reproduction of constants. Because of the singularity of the base function at zero, the original Shepard approximant interpolates values of the target function at x j , 1 ≤ j ≤ n. That is, S Φ,n (x j ) = f (x j ), 1 ≤ j ≤ n. However, for most other choices of base functions, the interpolation feature is lost, and the resulted approximants are called "quasi-interpolants" in the literature; see [4], [17], [18]. Except for the cases in which the base function is compactly supported (see [16], [17], [18]), deriving optimal order error estimates for a Shepard approximation scheme has been uncommon.…”