Symmetric multi-box splines for higher degree

We consider the space S _n =S _n (v ₀,…,v _n+r ) of compactly supported C ^n?1 piecewise polynomials on a mesh M of lines through ?² in directions v ₀,…,v _n+r . A sequence ψ=(ψ ₁,…,ψ _r ) of elements of S _n is called a multi-box spline if every element of S _n is a finite linear combination of shifts of (the components of) ψ. For the case n=2, 3 we give some examples for multi-box splines and show that they are not always stable. It is further shown that any C ^n?1 piecewise polynomial of degree n≥2 on M, is possibly a symmetric multi-box spline.