Continuous wavelets on compact manifolds
Continuous wavelets on compact manifolds
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紧流形上的连续小波
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
A. Mayeli
中科院分区：
文献类型：
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作者：
D. Geller;A. Mayeli
令 M 为光滑紧致黎曼流形，令 ΔM 为 M 上的拉普拉斯-贝尔特拉米算子。假设 $${0 \neq f \in \mathcal{S}(\mathbb {R}^+)}$$ ，且 f (0) = 0。对于 t > 0，令 Kt(x, y) 表示 f (t2 ΔM) 的核。我们证明 Kt 在对角线附近很好地定位，从某种意义上说，它满足类似于 $${\mathbb {R}^n}$$ 上卷积算子 f (t2Δ) 的内核所满足的估计。我们定义 M 上的连续 $${\mathcal {S}}$$ 小波，使得 Kt(x, y) 满足这个定义，因为它位于对角线附近。 M 上的连续 $${\mathcal {S}}$$ 小波类似于 $${\mathcal {S}}$$ ($${\mathbb {R}^n}$$) 中 $${\mathbb {R}^n}$$ 上的连续小波。特别是，我们能够通过连续 $${\mathcal {S}}$$ 小波变换的大小来表征 M 上的 Hölder 连续函数，Hölder 指数严格介于 0 和 1 之间。如果 M 是圆环 $${\mathbb T^2}$$ 或球体 S2，并且 f (s) = se−s （“墨西哥帽”情况），我们得到 Kt 的两个显式近似公式，当 t 大时使用一种，当 t 小时使用一种。
Let M be a smooth compact oriented Riemannian manifold, and let ΔM be the Laplace–Beltrami operator on M. Say $${0 \neq f \in \mathcal{S}(\mathbb {R}^+)}$$ , and that f (0) = 0. For t > 0, let Kt(x, y) denote the kernel of f (t2 ΔM). We show that Kt is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t2Δ) on $${\mathbb {R}^n}$$ . We define continuous $${\mathcal {S}}$$-wavelets on M, in such a manner that Kt(x, y) satisfies this definition, because of its localization near the diagonal. Continuous $${\mathcal {S}}$$-wavelets on M are analogous to continuous wavelets on $${\mathbb {R}^n}$$ in $${\mathcal {S}}$$ ($${\mathbb {R}^n}$$). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous $${\mathcal {S}}$$-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus $${\mathbb T^2}$$ or the sphere S2, and f (s) = se−s (the “Mexican hat” situation), we obtain two explicit approximate formulas for Kt, one to be used when t is large, and one to be used when t is small.