Uniform Lebesgue Constants of Local Spline Approximation

Abstract

Let a function ϕ ∈ C¹[−h, h] be such that ϕ(0) = ϕ'(0) = 0, ϕ(−x) = ϕ(x) for x ∈ [0; h], and ϕ(x) is nondecreasing on [0; h]. For any function f: ℝ → ℝ, we consider local splines of the form

$$S(x) = {S_\varphi }(f,x) = \sum\limits_{j \in mathbb{Z}} {{y_i}{B_\varphi }(x + \frac{{3h}}{2} - jh)(x \in \mathbb{R}),} $$

where yj = f(jh), m(h) > 0, and

$$B_{\varphi}(x)=m(h) \begin{cases}\varphi(x), & x \in [0;h],\\ 2\varphi (h) - \varphi(x-h) - \varphi(2h-x),& x \in [h;2h],\\ \varphi(3h -x), & x\in[2h;3h],\\ 0, &x\notin [0; 3h].\end{cases}$$

These splines become parabolic, exponential, trigonometric, etc., under the corresponding choice of the function ϕ. We study the uniform Lebesgue constants L_ϕ = ||S|| ^C_C (the norms of linear operators from C to C) of these splines as functions depending on ϕ and h. In some cases, the constants are calculated exactly on the axis ℝ and on a closed interval of the real line (under a certain choice of boundary conditions from the spline S_ϕ(f, x)).