Error estimate for minimal norm SBF interpolation
© Wang et al.; licensee Springer. 2013
Received: 25 February 2013
Accepted: 27 August 2013
Published: 8 November 2013
By the method of spherical splitting, the interpolation capability of the spherical basis function (SBF) is investigated. As the main result, we deduce the error estimate for the minimal norm SBF interpolation in the metric of the p th Lebesgue integral function space on the sphere. The result shows that the interpolation capability of SBF depends not only on the smoothness of the target function, but also on the geometric distributions of the interpolation knots.
1 Introductions and main results
Over the past decades, research in the field of function approximation of scattered data points gradually drifted from the polynomials to the radial basis functions (RBFs), and later to the spherical basis functions (SBFs) in a spherical coordinate. Recently, people, such as Wang and Li , Freeden et al. , and Muller , have moved their interests further to the topics of spherical approximation [1–7]. Based on the developments, spherical harmonic analysis was established and had some considerable progress. Meanwhile, error estimations for the SBF approximation were studied hereafter and stepped forward in the studies of Le Gia et al. , Hubbert and Morton , and Sloan and Wendland  after 2004. In 2007, Chen  established error bound for the minimal norm interpolation on a sphere. With this plentiful foundation, Lin and Cao  embedded firstly the smooth SBFs in a native space and specified the error bound between the best approximation and the target function via metric. As a consecutive study, the paper thus aims to derive a minimal norm interpolation with the measure.
where represents a function space constructed by a complex function, , fulfilling . An identical function in is characterized as a function identical to the functions which have the same output values everywhere with the same inputs.
if satisfies (see the reference ).
where is the n th order Legendre polynomial. Here is obviously a spherical radial basis function.
where is the spherical distance between ξ and η, i.e., , .
for an arbitrary , if is interpolated via the minimal norm.
for a given arbitrary . It implies that is an orthogonal projection of f to the span in the space .
for an arbitrary (refer to ). Coming with these contributions, this study is sought to extend the significant results to a more general case of ().
where C denotes a positive constant independent of f and h.
To completely prove Theorems 1.1 and 1.2, five lemmas are given as follows.
where . Furthermore, there exists a constant independent of h such that .
Lemma 2.3 Assume that is an evaluation functional corresponding to in , in can alternatively be expressed as .
It should be noted that Lemmas 2.3 and 2.4 can be directly obtained from .
3 Proof of theorems
where is a constant dependent only on q and satisfying .
With the fact that is the orthogonal projection of f in , the proof of Theorem 1.1 is thus completed. □
Together with the results of Theorem 1.1, Theorem 1.2 is thus established. □
The first author and the third author would like to thank the Natural Science Foundation of China (Nos. 61273020, 11001227), the Fundamental Research Funds for the Central Universities (No. XDJK2010B005).
- Wang KY, Li LQ: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing; 2000.Google Scholar
- Freeden W, Gervens T, Schreiner M: Constructive Approximation on the Sphere. Oxford University Press, New York; 1998.Google Scholar
- Müller C Lecture Notes in Mathematics 17. In Spherical Harmonics. Springer, New York; 1966.Google Scholar
- Le Gia QL, Narcowich F, Ward J, Wendland H: Continuous and discrete least-squares approximation by basis functions on spheres. J. Approx. Theory 2006, 143: 124–133. 10.1016/j.jat.2006.03.007MathSciNetView ArticleGoogle Scholar
- Hubbert S, Morton TM:-Error estimates for radial basis function interpolation on the sphere. J. Approx. Theory 2004, 129: 58–77. 10.1016/j.jat.2004.04.006MathSciNetView ArticleGoogle Scholar
- Hubbert S, Morton TM: A Duchon framework for the sphere. J. Approx. Theory 2004, 129: 28–57. 10.1016/j.jat.2004.04.005MathSciNetView ArticleGoogle Scholar
- Sloan LH, Wendland H: Inf-sup condition for spherical polynomials and radial basis functions on spheres. Math. Comput. 2009, 267: 1319–1331.MathSciNetView ArticleGoogle Scholar
- Chen ZX: Error estimates of minimal norm interpolant approximation by the spherical radial basis functions. Acta Math. Appl. Sin. 2007, 30(1):159–167. (in Chinese)MathSciNetGoogle Scholar
- Lin SB, Cao FL: Approximation by shifts of smooth kernels on spheres. Chin. Ann. Math., Ser. A 2010, 31(5):615–624. (in Chinese)MathSciNetGoogle Scholar
- Wendland H: Scattered Data Approximation. Cambridge University Press, Cambridge; 2005.Google Scholar
- Narcowich F, Schaback R, Ward J: Approximation in Sobolev spaces by kernel expansions. J. Approx. Theory 2002, 114(1):70–83. 10.1006/jath.2001.3637MathSciNetView ArticleGoogle Scholar
- Brenner SC, Scott RL: The Mathematical Theory of Finite Element Methods. Springer, New York; 1994.View ArticleGoogle Scholar
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