Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon´s sampling theorem

The least-squares polynomial spline approximation of a signal g(t) ∈ L₂(R) is obtained by projecting g(t) on Sⁿ( R) (the space of polynomial splines of order n). It is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g(t) with a B-spline of order n. More specifically, the coefficients of the B-spline interpolation of order 2n+1 of the sampled filtered sequence are identical to the coefficients of the least-squares approximation of g(t) of order n. It is shown that this approximation can be obtained from a succession of three basic operations: prefiltering, sampling, and postfiltering, which confirms the parallel with the classical sampling/reconstruction procedure for bandlimited signals. The frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift-invariant basis functions of Sⁿ(R ): the standard expansion in terms of B-spline coefficients, a representation in terms of sampled signal values, and a representation using orthogonal basis functions