On the Gibbs phenomenon for rational spline functions

In the case of functions f(x) continuous on a given Closed interval [a, b] except for jump discontinuity points, the Gibbs phenomenon is studied for rational spline functions R-N,R-1(x) = R-N,R-1(x, f, Delta, g) defined for a knot grid Delta : a = x(0) < x(1) < ... < x(N) = b and a family of poles g(i) is not an element of [x(i-1), x(i+1)] (i = 1, 2, ..., N - 1) by the equalities R-N,R-1(x) = [R-i(x) (x - x(i-1)) + Ri-1(x) (x(i) - x)]/(x(i) - x(i-1)) for x is an element of [x(i-1), x(i)] (i = 1, 2, ..., N). Here the rational functions R-i(x) = alpha(i) + beta(i)(x - x(i)) + gamma(i)/(x - g(i)) (i = 1, 2, ..., N - 1) are uniquely defined by the conditions R-i(x(j)) = f(x(j)) (j = i - 1, i, i + 1); we assume that R-0(x) R-1(x), R-N(x) RN-1(x). Conditions on the knot grid Delta are found under which the Gibbs phenomenon occurs or does not occur in a neighborhood of a discontinuity point.