Multiscale radial kernels with high-order generalized Strang-Fix conditions
- PDF / 1,154,537 Bytes
- 22 Pages / 439.642 x 666.49 pts Page_size
- 83 Downloads / 177 Views
Multiscale radial kernels with high-order generalized Strang-Fix conditions Wenwu Gao1,2,3 · Xuan Zhou4 Received: 8 June 2018 / Accepted: 25 September 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasiinterpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasiinterpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders. Keywords Radial function · Multiscale radial kernel · Generalized Strang-Fix condition · Generator · Fourier transform Mathematics Subject Classification (2010) 41A05 · 41065 · 65D05 · 65D10 · 65D15
Xuan Zhou
[email protected]
Extended author information available on the last page of the article.
Numerical Algorithms
1 Introduction Let φ be a univariate function defined on [0, +∞). A d-variate function d : R d → R is called a radial function if it can be expressed as d (·) = φ(|| · ||), where || · || denotes the Euclidean norm in R d and φ is called the generator of d . Moreover, in many cases, we write d (x) = φ(||x||) := f (r 2 /2), r = ||x||, and call f the f -form of d [26]. Radial functions have been extensively studied and widely used both in approximation theory and its applications (see [2–8, 11–13, 28–32] and the references therein). Among them, construction of new radial kernels with fair properties from some given univariate generator is always a hot topic. These include positive definiteness [3, 21, 30], compact support [5, 8, 28, 30], Strang-Fix conditions or polynomial reproduction [1, 10, 27], generalized Strang-Fix conditions [16, 34], moment conditions [22], and polynomial reproduction in the convolution sense [
Data Loading...
