ИСТИНА

The regularity of re nable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case o ered a great resistance. It was done only recently by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very e cient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in Wk 2 (Rn) by means of the Perron eigenvalues of linear operators on special cones. Applied those results to recently introduces tile B-splines, we analyse the corresponding multivariate Haar and wavelets systems and the tile subdivision schemes. In particular, we construct the tile B-splines of a higher smoothness than the classical ones of the same order. The optimality of two-digit tile B-splines among all re nable functions is proved. This implies that the tile subdivision schemes have the lowest algorithmic complexity. Examples and numerical results are provided.