AbstractLet E = ∥ eij ∥j = 0,…,n−1i = 1,…,k be a given n-incidence matrix and suppose knots x1 < x2 < ··· < xk are given. This paper studies the following problem related to the matrix E: if p is an integer, 1 ⩽ p ⩽ n − 1, and g(x) ϵ C(n−p) [x1, xk], does there exist a function ƒ(x) satisfying 1.(i) ƒ(j)(xi) = 0 when eij = 1 and2.(ii) ƒ(p)(x) ≡ g(x)?Certain functions W1(t),…, Wn−1(t) which do not depend on g are constructed with the result that for almost all choices of the knots xi a solution exists if ∝x1xkg(q)(t) Wq(n−q)(t)dt = 0 for q = p,…, n − 1.This result is applied to the nonhomogenous problem where data yij is prescribed and (i) is replaced with ƒ(j)(xi) = yij. Also, the concept of a simple matrix is introduced, and some results on the relation between poised and simple matrices are given.