Interpolating distributed approximating functionals

In this paper, we present a class of distributed approximating functionals (DAF's) for solving various problems in the sciences and engineering. Previous DAF's were specifically constructed to avoid interpolation in order to achieve the ``well-tempered'' limit, in which the same order error is made both on and off the grid points. These DAF's are constructed by combining the DAF concept with various interpolation schemes. The approach then becomes the same as the ``moving least squares'' method, but the specific ``interpolating DAF's'' obtained are new, to our knowledge. These interpolating DAF's are illustrated using Lagrange interpolation (the ``LDAF'') and a Gaussian weight function. Four numerical tests are used to illustrate the LDAF's: differentiation on and off a grid, fitting a function off a grid, time-dependent quantum dynamical evolution, and solving nonlinear Burgers' equation.