${\text{PDL}}_g $ Splines Defined by Partial Differential Operators with Initial and Boundary Value Conditions

As a natural generalization of $L_g $ splines and thin-plate splines, ${\text{PDL}}_g $ splines are introduced in this paper. A ${\text{PDL}}_g $ spline is defined as a solution of the optimal scattered-data interpolation problem described by a general higher-order separable linear partial differential operator (which may or may not incorporate time), with (initial-terminal and) boundary value conditions over (the time domain and) an arbitrary bounded spatial domain with a continuous and piecewise smooth boundary. A closed-form expression for the ${\text{PDL}}_g $ spline is obtained by means of the reproducing kernel of a related Hilbert space. A feasible constructive method for finding the reproducing kernel via a fundamental solution (or Green’s function) of an induced linear and self-adjoint partial differential operator is also included. This method provides a way to take care of the initial-terminal and boundary value conditions. Moreover, an explicit formulation for finding the best approximation, in the sense of A. Sard [“Linear Approximation,” American Mathematical Society, Providence, RI, 1963], to a given linear functional over the related Hilbert space is described. This formulation is a multivariate analogue of the standard result of Sard and a precise realization of a general result of P. J. Laurent [“Approximation et optimisation,” Hermann, Paris, 1972]. Finally, the corresponding smoothing ${\text{PDL}}_g $ splines are briefly discussed.