Uniform Lebesgue Constants of Local Spline Approximation

Let a function phi C-1[-h, h] be such that phi(0) = phi'(0) = 0, phi(-x) = phi(x) for x [0; h], and phi(x) is nondecreasing on [0; h]. For any function f: , we consider local splines of the form S(x)=S-phi(f, x)= Sigma(j is an element of Z) y(j)B(phi)(x+3h/2 - jh) (x is an element of R), where y(j) = f(jh), m(h) > 0, and B-phi(x) = m(h) { phi(x), x is an element of[0; h], 2 phi(h) - phi(x - h)-phi(2h - x), x is an element of[h; 2h], phi(3h - x), x is an element of[2h; 3h], 0, x is an element of[0; 3h]. These splines become parabolic, exponential, trigonometric, etc., under the corresponding choice of the function phi. We study the uniform Lebesgue constants L phi = parallel to S parallel to(C)(C) (the norms of linear operators from C to C) of these splines as functions depending on phi and h. In some cases, the constants are calculated exactly on the axis and on a closed interval of the real line (under a certain choice of boundary conditions from the spline S-phi(f, x)).