  In the paper, we present a family of multivariate compactly supported scaling
functions, which we call as elliptic scaling functions. The elliptic scaling
functions are the convolution of elliptic splines, which correspond to
homogeneous elliptic differential operators, with distributions. The elliptic
scaling functions satisfy refinement relations with real isotropic dilation
matrices. The elliptic scaling functions satisfy most of the properties of the
univariate cardinal B-splines: compact support, refinement relation, partition
of unity, total positivity, order of approximation, convolution relation, Riesz
basis formation (under a restriction on the mask), etc. The algebraic
polynomials contained in the span of integer shifts of any elliptic scaling
function belong to the null-space of a homogeneous elliptic differential
operator. Similarly to the properties of the B-splines under differentiation,
it is possible to define elliptic (not necessarily differential) operators such
that the elliptic scaling functions satisfy relations with these operators. In
particular, the elliptic scaling functions can be considered as a composition
of segments, where the function inside a segment, like a polynomial in the case
of the B-splines, vanishes under the action of the introduced operator.
