Constructing radial kernels with higher-order generalized Strang-Fix conditions
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Constructing radial kernels with higher-order generalized Strang-Fix conditions Wenwu Gao1,2,3 · Zongmin Wu2,4
Received: 6 February 2016 / Accepted: 15 March 2017 © Springer Science+Business Media New York 2017
Abstract The paper provides an approach for constructing multivariate radial kernels satisfying higher-order generalized Strang-Fix conditions from a given univariate generator. There are three key features of the approach. First, the kernels are explicitly expressed only by the derivatives of the f -form of the generator without computing any Fourier transforms. Second, it includes the radial kernels with the highest-order generalized Strang-Fix conditions. Finally, it requires only computing univariate derivatives of the f -form. Therefore, the approach is simple, efficient and easy to implement. As examples, the paper constructs radial kernels from some commonly used generators, including the Gaussian functions, the inverse multiquadric functions and compactly supported positive definite functions. Keywords Radial functions · Generalized Strang-Fix conditions · Quasi-interpolation · Fourier transforms · Generators
Communicated by: Tomas Sauer Zongmin Wu
[email protected] Wenwu Gao [email protected] 1
School of Economics, Anhui University, Hefei, People’s Republic of China
2
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China
3
Anhui Engineering Laboratory of Agro-Ecological Big Data, Anhui University, Hefei, People’s Republic of China
4
Shanghai Center for Mathematical Sciences, Shanghai, People’s Republic of China
W. Gao, Z. Wu
Mathematics Subject Classification (2010) 41A05 · 41063 · 41065 · 65D05 · 65D10 · 65D15
1 Introduction Let ϕ : [0, +∞) → R be a univariate function. 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 . Besides, for some reasons that will be apparent soon, we write d (x) = ϕ(||x||) := f (r 2 /2), r = ||x||. Here f is called the f -form of d in [42], a concept that is motivated from [50]. Radial functions have been studied extensively in the literature (see [4, 5, 8, 17, 28, 37, 47, 48, 50–53], and the references therein). Among them, how to construct new radial kernels with fair properties (i.e., positive definiteness, compact support, rapid decay, Strang-Fix conditions, etc) from some known kernels is always a hot topic that attract lots of attention, for example, [8, 10, 23, 42, 47, 50, 54], and plenty of others. Recently, Bozzini, Rossini and Schaback [4] constructed positive definite radial kernels from the scaled Whittle-Matern ´ kernels by taking divided differences of the f -form with respect to the scale variable. Besides, by taking scale derivatives of the f -form of radial functions, Bozzini et al. [5] constructed several new radial kernels. The resulting kernels are positive definite if the f -form satisfies certain c
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