© 2004 by Institute of Mathematics and its Applications
The gradient iteration for approximation in reproducing kernel Hilbert spaces
Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield S1 3JD, UK
Consider the bounded linear operator, L: F 
Z, where Z 
RN and F are Hilbert spaces defined on a common field X. L is made up of a series of N bounded linear evaluation functionals, Li: F R. By the Riesz representation theorem, there exist functions k(xi, ·) 
F : Lif = 
f, k(xi, ·)
F. The functions, k(xi, ·), are known as reproducing kernels and F is a reproducing kernel Hilbert space (RKHS). This is a natural framework for approximating functions given a discrete set of observations. In this paper the computational aspects of characterizing such approximations are described and a gradient method presented for iterative solution. Such iterative solutions are desirable when N is large and the matrix computations involved in the basic solution become infeasible. This is also exactly the case where the problem becomes ill-conditioned. An iterative approach to Tikhonov regularization is therefore also introduced. Unlike iterative solutions for the more general Hilbert space setting, the proofs presented make use of the spectral representation of the kernel.
Keywords: reproducing kernel Hilbert space; gradient iteration; function approximation.
Received 18 July 2003.
* Email: t.j.dodd{at}shef.ac.uk
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