Generalized orthogonal series for natural tensor product interpolation

This 326 page doctoral thesis studies the problem of interpolation by a natural tensor product (NTP), and does so at two contrasting levels of generality: over concrete function spaces to express the NTP interpolant in closed form as a vector-matrix-vector product, the thesis uses a Newton paradigm to express the interpolant in series form. This series is generated by a simple iterative algorithm; the next term is obtained by splitting the previous remainder into a rank-one tensor product The series form is much better suited than the traditional closed form for applications of interpolation to the approximation of a given function or vector by a NTP. (See, for example, Bateman's method for solving Fredholm linear integral equations.); In the tradition of Bruno Buchberger, the author has named the above series expansions Geddes series to honor his thesis supervisor, Professor Keith O. Geddes. Geddes series are orthogonal series expansions in the sense of generalized inner product (GIP) spaces. The theory of GIP spaces used here is purely algebraic and is original to the author. Note that all classical inner product spaces are also GIP spaces. In addition, the Geddes series includes all classical orthogonal eigenfunction expansions of Hilbert-Schmidt kernels of Fredholm linear integral operators as special cases.; The thesis studies two concrete interpolation problems in depth: (1) ordinary interpolation of scalar-valued functions of two general variables on the lines of a two-dimensional grid; (2) Taylor interpolation of real-valued functions of two real variables on the two rectilinear coordinate lines through a specified point. For problem (2) the thesis develops remainder formulas in both integral and mean-value forms; the thesis also gives explicit, rigorous, pointwise and uniform error estimates, as well as a sufficient condition for the uniform convergence of infinite Geddes series solutions over compact rectangles. The thesis uses these new infinite series techniques to provide an original derivation and proof of the well-known Neumann addition formula for the order-zero Bessel function of the first kind.; The thesis develops the Geddes series solution of problem (2) as a special case of the dual asymptotic expansion (DAE). (Abstract shortened by UMI.)...