POLYNOMIAL SPLINE SIGNAL APPROXIMATIONS - FILTER DESIGN AND ASYMPTOTIC EQUIVALENCE WITH SHANNONS SAMPLING THEOREM

The least-squares polynomial spline approximation of a signal g(t) is-an-element-of L2(R) is obtained by projecting g(t) on S(n)(R) (the space of polynomial splines of order n). We show that this process can be linked to the classical problem of cardinal spline interpolation [1] 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. We then show 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. We determine the frequency responses of these filters for three equivalent spline representations using alternative sets of shift-invariant basis functions of S(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. For the two latter cases, we prove that the frequency response of these filters converge to the ideal lowpass filter pointwise and in all L(p)-norms with 1 less-than-or-equal-to p less-than-or-equal-to infinity as the order of the spline tends to infinity, which establishes the asymptotic equivalence with Shannon's sampling theorem.