On bounding spline interpolation

as a function of s at the k+ 1 points ti, . . . , ti+k. The elements of $k,t are called polynomial splines of order k with knot sequence t. Let τ := (τi) n 1 be a strictly increasing real sequence. As is shown in [12], there exists, for given f , exactly one s ∈ $k,t, such that s(τi) = f(τi), i = 1, . . . , n, if and only if Ni,k(τi) 6= 0, i = 1, . . . , n, i.e., if and only if ti < τi < ti+k, i = 1, . . . , n. (1) Hence, assuming τ to satisfy (1), the conditions