Interpolation of Odd Periodic Functions on Uniform Meshes

TL;DRAbstract

Let g ϵ C2π have an absolutely convergent Fourier series. For n ϵ ℕ we define the uniform mesh tk = 2πk/n, k ϵ ℤ, and the translates gk = g(·−tk), 0 ≦ k < n, of g. Locher [4] presented a method of interpolation of periodic functions f at the uniform mesh tk, k ϵ ℤ, by functions h from the linear space Vn (g) = lin { g0,g1,...gn−1} of translates of g. Locher’s method is only applicable if Bk (0) ≠ 0 for k = 0,1,...,n−1 where the functions Bk, k = 0,1,...,n−1, are defined by $${B_k}(t)\;\, = \;\,\sum\limits_{j = 0}^{n - 1} {\;\,g(t - {t_j})\;\exp (ik{t_j})}$$ . In [1] we derived a modified method of interpolation by translation which is applicable under the hypothesis Bk(0) ≠ 0 for k=1,...,n−1. It is the objective this paper to develop a related method of interpolation of odd periodic functions which works under the assumption Bk (0) ≠ 0, 0 < k < m = n/2.

Chat with Paper

AI Agents for this Paper