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Exponentials of de Branges–Rovnyak kernels
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- Canadian Mathematical Bulletin / Volume 65 / Issue 2 / June 2022
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- 03 June 2021, pp. 447-455
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- June 2022
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LOCAL RADIAL BASIS FUNCTION APPROXIMATION ON THE SPHERE
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- Bulletin of the Australian Mathematical Society / Volume 77 / Issue 2 / April 2008
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- 01 April 2008, pp. 197-224
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- April 2008
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In this paper we derive local error estimates for radial basis function interpolation on the unit sphere
. More precisely, we consider radial basis function interpolation based on data on a (global or local) point set 
for functions in the Sobolev space 
with norm 
, where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with . Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of 
satisfies 
, where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.
. More precisely, we consider radial basis function interpolation based on data on a (global or local) point set 
for functions in the Sobolev space 
with norm 
, where 
satisfies 
, where 