ИСТИНА

It is well-known that the multivariate wavelets on R d can be constructed with an arbitrary dilation integer expansive d × d matrix and arbitrary set of “digits” from the corresponding quotient sets. The unit segment in that numerical systems becomes a tile, which is a compact set with certain self-similarity properties. As in the univariate case, the tile B-splines are defined as convolutions of tiles. In spite of a complicated structure those functions possess very good approximation properties. Moreover, some of them have higher smoothness than the usual multivariate B-splines. We analyse the corresponding wavelets and present subdivision schemes based on the tile B-splines. Surprisingly, they have a faster convergence and higher regularity than many of known subdivision schemes of the same complexity. Computing the regularity is realized by the recent formula for the H¨older exponent of multivariate refinable function constructed with an arbitrary dilation matrix: M.Charina, V.Yu.Protasov, Regularity of anisotropic refinable functions, ACHA (2019), 795–821. To classify all tile splines of a given number of variables we apply the notion of boundary dimension of a compact set. We show that in R d there are between d2 and 2d B-splines of a given order, not equivalent to each other. We classify all of them in low dimensions. In the case of anysotropic dilation matrix the classification is much simple and is realized in all dimensions. Finally, we present numerical results and formulate several open problems.