  The Strang-Fix conditions are necessary and sufficient to reproduce spaces of
algebraic polynomials up to some degree by integer shifts of compactly
supported functions. W. Dahmen and Ch. Micchelli (Linear Algebra Appl.
52/3:217-234, 1983) introduced a generalization of the Strang-Fix conditions to
affinely invariant subspaces of higher degree polynomials. C. de Boor (Constr.
Approx. 3:199-208, 1987) raised a question on the necessity of scale-invariance
of polynomial space for an arbitrary function; and he omitted the
scale-invariance restriction on the space. In the paper, we present a matrix
approach to determine (not necessarily scale-invariant) polynomial space
contained in the span of integer shifts of a compactly supported function.
Also, in the paper, we consider scaling functions that we call the elliptic
scaling functions (Int. J. Wavelets Multiresolut. Inf. Process. To appear);
and, using the matrix approach, we prove that the elliptic scaling functions
satisfy the generalized Strang-Fix conditions and reproduce only affinely
invariant polynomial spaces. Namely we prove that any algebraic polynomial
contained in the span of integer shifts of a compactly supported elliptic
scaling function belongs to the null-space of a homogeneous elliptic
differential operator. However, in the paper, we present nonstationary elliptic
scaling functions such that the scaling functions reproduce not scale-invariant
(only shift-invariant) polynomial spaces.
