ON THE REPRESENTATION OF OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS
ON THE REPRESENTATION OF OPERATORS IN BASES OF COMPACTLY SUPPORTED WAVELETS
复制标题
DOI：
10.1137/0729097
复制
发表时间：
1992-12-01
影响因子：
2.9
通讯作者：
BEYLKIN, G
中科院分区：
文献类型：
--
作者：
BEYLKIN, G
本文描述了紧支持小波正交基中微分算子 d(n)/dx(n), n = 1,2,... 的精确和显式表示，以及希尔伯特变换和分数阶导数的表示。计算这些表示的方法直接适用于多维卷积算子。此外，还讨论了紧支持小波正交基上移位算子的稀疏表示，并构造了一种需要 O(N log N) 运算的快速算法，用于计算长度为 N = 2n 的向量的所有 N 个循环移位的小波系数。作为该算法的应用示例，表明将伪微分算子的标准形式应用于向量的快速算法的存储要求（参见 [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141-183]）可以从 O(N) 个有效条目减少到 O(log2 N) 个有效条目。
This paper describes exact and explicit representations of the differential operators, d(n)/dx(n), n = 1,2,..., in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring O(N log N) operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length N = 2n is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141-183]) may be reduced from O(N) to O(log2 N) significant entries.