**MSC:**- 41A25 Rate of convergence, degree of approximation
- 41A30 Approximation by other special function classes

of $L^2(\Bbb{R})$ providing controlled approximatin order $m$ are

considered. Assuming that the vector of generating functins $\phi$ of

$S(\phi)$ is refinable, necessary and sufficient conditions for the

refinement mask are derived. In particular, if algebraic polynomials

can be exactly reproduced in $S(\phi)$, then a factorization of the

refinement mask can be given. This result is a natural generalization

of the result of the principal shift-invariant subspaces of $L^2(\Bbb{R})$

where the refinement mask contains a power of $\frac{1+e^{-iu}}{2}$

if the scaling function is regular.