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) in L/sub 2/(R) is obtained by projecting g(t) on S/sup n/(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 S/sup n/(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. >