Some properties of multi-box splines

We consider the space Sn=Sn(v0,…,vn+r)Sn=Sn(v0,…,vn+r) of compactly supported Cn−1Cn−1 piecewise polynomials on a mesh M   of lines through Z2Z2 in directions v0,…,vn+rv0,…,vn+r, possibly satisfying some restrictions on the jumps of the nnth-order derivatives. A sequence ϕ=(ϕ1,…,ϕr)ϕ=(ϕ1,…,ϕr) of elements of SnSn is called a multi-box spline if every element of SnSn is a finite linear combination of shifts of (the components of) ϕϕ. Here we generally assume v0=(1,0),v1=(0,1)v0=(1,0),v1=(0,1). For the case n=0n=0 we give explicit formulas for multi-box splines and show that their shifts are linearly independent. It is then shown that, for any n≥0n≥0, if the shifts of a multi-box spline form a Riesz basis, then they are linearly independent. It is further shown that any Cn−1Cn−1 piecewise polynomial of degree nn on MM, satisfying the jump conditions, is a (possibly infinite) linear combination of shifts of a multi-box spline.