Fourier interpolation in 1D. Let m ?{1, 2, }, h=(2 m+1) ^?1, x _?= ?h, ?=0, 1, 2, Suppose that ?(x) is a 1-periodic function, ?(x) ??( x+1), defined at all gridpoints x=x _? . We want to interpolate ?(x) by a Fourier polynomial

at all grid points:

Theorem A.2.1. The interpolation problem (A.2.2) has a unique solution (A.2.1).

Proof. If we introduce the inner product

then the gridfunctions

form an orthonormal system, i.e.,

The above result is obvious for k= l and follows from the geometric-sum formula otherwise. Consequently, if w(x) is an interpolant of the form (A.2.1), then

Thus the coefficients a _l are uniquely determined by the data function ?(x). Also, we can consider the interpolation-condition

as a linear system of 2 m+1 equations for the 2 m+1 unknown a _k. Since the solution vector is unique (if it exists), the system-matrix is nonsingular, and existence follows.

Fourier interpolation is very useful since the interpolant w(x) inherits smoothness from the data ?(x), with estimates independent of h. More precisely, if we introduce norms by

then the following result holds:

Theorem A.2.2. If w(x) denotes the Fourier interpolant of the data ?(x) then

and

It is important that the factor ( ?/2) ^{2 p} does not depend on h.

Proof.