M-band subdivision of multivariate B-spline

For any integer M⩾2 and integer k⩾0, the present paper gives the M-band refinement equations of bivariate B-splines in the space S3k+12k(Δmn(1)), S3k2k−1(Δmn(1)), S4k+23k+1(Δmn(2)), S4k+13k(Δmn(2)) and S4k+43k+2(Δmn(2)) (where Δmn(1) and Δmn(2) represent uniform type-1 and type-2 triangulation, respectively) by using Fourier transform and bivariate convolution. Thus, we obtain the subdivision generation of the spline functions defined on type-1 triangulation and type-2 triangulation.