Variational Theory of Splines by Anatoly Yu. Bezhaev , Vladimir A. Vasilenko
English | PDF | 2001 | 291 Pages | ISBN : 0306466422 | 20.5 MB
Th e variational spline theory which originates f rom th e well-known paper by J.e. Holliday (1957) is today a well-developed field in approxi- mat ion theory. The gene ra l definition of splines in the Hilbert space, ex- istence, uniqueness, and characterization theorems were obt ained about 35 years ago by M. Atteia, P.J. L aur ent, and P.M. Anselone, but in rec ent years impor tant new results have been obtained in th e abst ract variational spline theory. Th ese concern the convergence in th e Hilb ert spaces, general techniques for error es tim ation of abs trac t splines includ- ing multi-dimensional splines on scatt ered meshes, new kinds of charac- teriz ation th eorems based on reproducing kernels and m appin gs, and the theory and algorithms for the splines on subspaces (finite element me th od for complic at ed non-polynomial splines). New variatio na l for- mulations arose for vec tor splines, rational splines, te nsor and blending splines, trac es of splines on the smooth manifold, discontinuous splines, etc. Optim al approxi mat ions of linear functionals and ope rat ors are also developed from th ese concepts.
