Supersmooth tile $\mathrm B$-splines

A tile is a self-affine compact subset of $\mathbb R^n$ whose integer translates tile the space. A tile $\mathrm B$-spline is a self-convolution of the characteristic function of the tile, similarly to $\mathrm B$-splines, which are self-convolutions of the characteristic functions of closed intervals. It is known that tile $\mathrm B$-splines, even ones with ‘fractal’ support, can be ‘supersmooth’, that is, their smoothness can exceed that of classical $\mathrm B$-splines of the same order. We evaluate the smoothness of tile $\mathrm B$-splines in $W_2^k(\mathbb R^n)$ by applying a method developed recently and based on Littlewood-Paley type estimates for refinement equations. We adapt this method for tile $\mathrm B$-splines, thereby obtaining 20 families with the property of supersmoothness. We put forward the conjecture, supported by numerical experiments, that this classification is complete if the number of digits is small. Bibliography: 51 titles.