Splines with maximal zero sets

Consider a spline s(x) of degree n with L knots of specified multiplicities R₁, ..., R_L, which satisfies r sign consistent mixed boundary conditions in addition to s⁽ⁿ⁾(a) = 1. Such a spline has at most n + 1 -r + ∑_{j = 1}^L R_j zeros in (a, b) which fulfill an interlacing condition with the knots if s(x) ≢ = 0 everywhere. Conversely, given a set of n -r + ∑_{j = 1}^L R_j zeros then for any choice η₁ < ··· < η_L of the knot locations which fulfills the interlacing condition with the zeros, the unique spline s(x) possessing these knots and zeros and satisfying the boundary conditions is such that s⁽ⁿ⁾(x) vanishes nowhere and changes sign at η_j if and only if R_j is odd. Moreover there exists a choice of the knot locations, not necessarily unique, which makes |s⁽ⁿ⁾(x)| ≡ 1. In particular, this establishes the existence of monosplines and perfect splines with knots of given multiplicities, satisfying the mixed boundary conditions and possessing a prescribed maximal zero set. An application is given to double-precision quadrature formulas with mixed boundary terms and a certain polynomial extremal problem connected with it.