Mathematical interpolation theory considers a function *f*, defined on
a regular grid *N*. The problem is to find *f* in a continuum, which
includes *N*. I am not defining the dimensionality of *N* and *f* here
because it is not essential for the derivations. Most of the examples
in this paper use one-dimensional functions, but the general theory
applies equally well to a higher number of dimensions. Furthermore, I
am not specifying the exact meaning of "regular grid", since it will
become clear from the further analysis. The function *f* is assumed to
belong to a Hilbert space with a defined dot product.

If we restrict our consideration to a linear case, the desired solution will take the following general form

(1) |

**Property 1189**

(2) |

Equality (2) is necessary to assure that the interpolation
of a single spike at some point *n* does not change the value *f* (*n*)
at the spike.

**Property 1193**

(3) |

This property is the normalization condition. Formula (3)
assures that interpolation of a constant function *f*(*n*) remains
constant.

One classic example of the interpolation weight *W* (*x*, *n*) is the
Lagrange polynomial, which has the form

(4) |

(5) |

(6) |

11/11/1997