2022/12/25 by Zaitseva, Tatyana
#41A15 #42C40 #52C22 #68U05 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2212.12945
Tile B-splines in ℝd are defined as autoconvolutions of the indicators of tiles, which are special self-similar compact sets whose integer translates tile the space ℝd. These functions are not piecewise-polynomial, however, being direct generalizations of classical B-splines, they enjoy many of their properties and have some advantages. In particular, the precise values of the Hölder exponents of the tile B-splines are computed in this work. They sometimes exceed the regularity of the classical B-splines. The orthonormal systems of wavelets based on the tile B-splines are constructed and the estimates of their exponentional decay are obtained. Subdivision schemes constructed by the tile B-splines demonstrate their efficiency in applications. It is achieved by means of the high regularity, the fast convergence, and small number of the coefficients in the corresponding refinement equation.