
Approximation by multiple refinable functions
We consider the shiftinvariant space,
$\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$
of compactly supported distributions on $\RR$ when the vector of
distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system
of refinement equations expressed in matrix form as
$$
\phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot  \,\alpha)
$$
where $a$ is a finitely supported sequence of $r\times r$ matrices
of complex numbers. Such {\it multiple refinable functions} occur
naturally in the study of multiple wavelets.
The purpose of the present paper is to characterize the {\it accuracy}
of $\Phi$, the order of the polynomial space contained in
$\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The
accuracy determines the $L_p$approximation order of $\bbbs(\Phi)$ when
the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]).
The characterization is achieved in terms of the eigenvalues and
eigenvectors of the subdivision operator associated with the mask $a$.
In particular, they extend and improve the results of Heil, Strang
and Strela~[7], and of Plonka~[16]. In addition, a
counterexample is given to the statement of Strang and Strela~[20]
that the eigenvalues of the subdivision operator determine the
accuracy. The results do not require the linear independence of
the shifts of $\phi$.
Keywords:Refinement equations, refinable functions, approximation, order, accuracy, shiftinvariant spaces, subdivision Categories:39B12, 41A25, 65F15 