The Construction of Wavelet Sets
The Construction of Wavelet Sets
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小波集的构造
DOI：
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发表时间：
2011
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通讯作者：
R. Benedetto
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文献类型：
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作者：
J. Benedetto;R. Benedetto
d维欧几里得空间中的集合Ω是用集合Ω的特征函数1 Ω的傅里叶反变换是单个并矢正交小波的性质构造的。迭代构造的特点是它的通用性，它的计算实现和它的简单性。这种构造被转移到局部紧致阿贝尔群G具有紧致开子群H的情况下。这种群最著名的例子是（G = {mathbb{Q}}_{p}）， p进有理数的域（作为加法中的群），它具有紧致开子群（H = {mathbb{Z}}_{p}）， p进整数环。令人着迷的错综复杂出现了。经典的小波理论要求平移有一个非平凡的离散子群，但不适用于G，因为G可能没有这样的子群。然而，我们的小波理论是用新的群论算子在G上表述的，这可以被认为是欧几里得平移的类似物。因此，我们关于G的理论在结构上具有凝聚力和显著的普遍性。从角度来看，哈尔和香农小波在欧几里得环境中自然是对映的，而它们对G的类似物是等效的。
Sets Ω in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1 Ω of the set Ω is a single dyadic orthonormal wavelet. The iterative construction is characterized by its generality, its computational implementation, and its simplicity. The construction is transported to the case of locally compact abelian groups G with compact open subgroups H. The best known example of such a group is (G = {mathbb{Q}}_{p}), the field of p-adic rational numbers (as a group under addition), which has the compact open subgroup (H = {mathbb{Z}}_{p}), the ring of p-adic integers. Fascinating intricacies arise. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. However, our wavelet theory is formulated on G with new group theoretic operators, which can be thought of as analogues of Euclidean translations. As such, our theory for G is structurally cohesive and of significant generality. For perspective, the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, whereas their analogues for G are equivalent.