Abstract
In this paper we derive several new results involving matrix-valued radial basis functions (RBFs). We begin by introducing a class of matrix-valued RBFs which can be used to construct interpolants that are curl-free. Next, we offer a characterization of the native space for divergence-free and curl-free kernels based on the Fourier transform. Finally, we investigate the stability of the interpolation matrix for both the divergence-free and curl-free cases, and when the kernel has finite smoothness we obtain sharp estimates.
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Devore, R.A., Sharpley, R.C.: Besov spaces on domains in \(\mathbb{R}^{d}\). Trans. Amer. Math. Soc. 335, 843–864 (1993)
Fasshauer, G.: Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv. Comput. Math. 11, 139–159 (1999)
Fuselier, E.: Refined error estimates for matrix-valued radial basis functions. Ph.D. thesis, Texas A&M University (2006)
Kansa, E.J.: Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – I. Surface approximations and partial derivative estimates. Comput. Math. Appl. 19, 127–145 (1990)
Lowitzsch, S.: Approximation and interpolation employing divergence-free radial basis functions with applications. Ph.D. thesis, Texas A&M University (2002)
Lowitzsch, S.: Matrix-valued radial basis functions: stability estimates and applications. Adv. Comput. Math. 23, 299–315 (2005)
Lowitzsch, S.: A density theorem for matrix-valued radial basis functions. Numer. Algorithms 39, 253–256 (2005)
Lowitzsch, S.: Error estimates for matrix-valued radial basis function interpolation. J. Approx. Theory 137, 238–249 (2005)
Narcowich, F.J., Ward, J.D.: Norms of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64, 69–94 (1991)
Narcowich, F.J., Ward, J.D.: Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices. J. Approx. Theory 69, 84–109 (1992)
Narcowich, F.J., Ward, J.D.: Generalized Hermite interpolation via matrix-valued conditionally positive definite functions. Math. Comp. 63, 661–687 (1994)
Schaback, R.: Error estimates and condition number for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ (1971)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, London (1966)
Watson, G.N., Whittaker, E.T.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1965)
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2004)
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Communicated by J.M. Pena.
The results are part of the author’s dissertation written at Texas A&M University, College Station, TX 77843, USA.
An erratum to this article can be found at http://dx.doi.org/10.1007/s10444-008-9091-6
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Fuselier, E.J. Improved stability estimates and a characterization of the native space for matrix-valued RBFs. Adv Comput Math 29, 269–290 (2008). https://doi.org/10.1007/s10444-007-9046-3
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DOI: https://doi.org/10.1007/s10444-007-9046-3