Semi reproducing kernel hilbert spaces and mixed precision computation.
Degree GrantorUniversity of Canterbury
Degree NameMaster of Science
Positive definite and conditionally positive definite functions are widely used in interpolation and smoothing problems, particularly when the data is scattered. This thesis concerns such functions and also concerns the somewhat related topics of reproducing kernel Hilbert spaces, and semi reproducing kernel Hilbert spaces. The thesis presents various pieces of the relevant theory, sometimes with known established methods of proof, and sometimes with novel proofs.
Chapter one concerns the history of a specific class of such functions, namely the radial basis functions.
Chapter two concerns the general properties of positive definite functions, highlighting their use in interpolation problems and establishing the existence of the corresponding native spaces.
Chapter three concerns reproducing kernel Hilbert Spaces and shows their relation to positive definite functions. Afterwards, interpolation and smoothing, along with other approximation problems are discussed within the reproducing kernel Hilbert space setting.
Chapter four concerns examples of positive definite functions in the settings of Rd and on the spheres Sd₋¹.
Chapter five concerns the basic theory of conditionally positive definite functions and their application in interpolation problems.
Chapter six concerns a variant to reproducing kernel Hilbert spaces, namely their semi Hilbert space variant and discusses interpolation and smoothing in this setting.
Chapter seven concerns interpolation via Guassian functions and the use of higher precision arithmetic to counter poor conditioning. Unfortunately, due to lack of time, this chapter goes no further.
Lastly the Appendices contain the prerequisite information concerning semi inner product spaces and convex functions.