多元tile B样条
Multivariate tile B-splines
摘要：
$\mathbb{R}^d$中的瓦片B样条被定义为瓦片的指示符的自卷积，瓦片是特殊的自相似紧集，其整数平移瓦片空间$\mathbb{R}^d$。这些函数不是分段多项式，但是，作为经典B样条的直接推广，它们具有许多经典B样条的性质，并具有一些优点。特别是，在这项工作中计算的瓷砖B样条的Hölder指数的精确值。它们有时超过了经典B样条的正则性。构造了基于B样条的小波规范正交系，并给出了其指数衰减的估计。由瓦片B样条构造的细分格式在实际应用中证明了其有效性。它是通过高的正则性，快速收敛，并在相应的细分方程的系数少的手段。
英文摘要：
Tile B-splines in $\mathbb{R}^d$ are defined as autoconvolutions of the indicators of tiles, which are special self-similar compact sets whose integer translates tile the space $\mathbb{R}^d$. 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.
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