Let \(M \geq 2\) be an integer, and let \(\varphi\) be a compactly supported function on \(R\) with total mass 1 and which satisfies the refinement equation \[ \varphi(x) = \sum_{j=0}^N c_j \varphi(Mx - j) \] for some coefficients \(c_j\). For each \(0 \leq l < M\), assume that the matrices with entries \(c_{Mi-j+l}\) for \(0 \leq i,j < N/(M-1)\) are invertible. Define \(\zeta_0\) to be the number of times the polynomial \((1-z^M)/(1-z)\) divides \(\sum_{j=0}^N c_j z^j\). Assume that the set of integer translates of \(\varphi\) are linearly independent. Let \(V_{(0,1)}(\varphi)\) denote the span of all translates \(\varphi(\cdot+j)\) of \(\varphi\), where \(0 \leq (M-1)j \leq N\). This is a space of functions supported in \([0,1]\) which contains \(\Pi_{\zeta_0}\), the restriction to \((0,1)\) of all polynomials of degree less than \(\zeta_0\). The main result of the paper is the following. Let \(D\) be any subspace of \(V_{(0,1)}(\varphi)\) which contains \(\Pi_{\zeta_0}\) and is closed under the scaling operators \(f \mapsto f((\cdot + l)/M)\) for all \(0 \leq l < M\). Then \(D\) is either equal to \(\Pi_{\zeta_0}\) or \(V_{(0,1)}(\varphi)\). The authors give various technical applications, which have the following general form: if the matrix invertibility condition mentioned earlier holds, and \(\varphi\) satisfies some regularity property on a small interval, then \(\varphi\) must satisfy the same property globally.