A duality approach to spline approximation

This dissertation discusses a new approach to spline approximation. A periodic spline approximation 𝑓_M,m,N(x) = Σ_k=1^Nα_kΦ_M,k(x) to a periodic function 𝑓(x) is determined by requiring < Φ_m,j, 𝑓 - 𝑓_M,m,N > = 0 for j = 1,...,N, where the Φ_L,k's are the unique periodic spline basis functions of order 𝐿. Error estimates, examples and some relationships to wavelets are given for the case M - m = 2μ. The case M - m = 2µ + 1 is briefly discussed but not completely explored.