Essential closures and supports of multivariate copulas

We generalize the notion of essential closures and use them to formulate a geometric necessary condition for a set to be the support of a multivariate copula. Furthermore, in some special cases, we derive an explicit formula of the support in terms of essential closures and obtain a stronger necessary condition.     

Estimations of Hölder Regularities and Direction of Singularity by Hart Smith and Curvelet Transforms

Using Hart Smith’s and curvelet transforms, new necessary and new sufficient conditions for an L ²(?²) function to possess Hölder regularity, uniform and pointwise, with exponent α>0 are given. Similar to the characterization of Hölder regularity by the continuous wavelet transform, the conditions here are in terms of bounds of the transforms across fine scales. However, due to the parabolic scaling, the sufficient and necessary conditions differ in both the uniform and pointwise cases. We also investigate square-integrable functions with sufficiently smooth background. Specifically, sufficient and necessary conditions, which include the special case with 1-dimensional singularity line, are derived for pointwise Hölder exponent. Inside their “cones” of influence, these conditions are practically the same, giving near-characterization of direction of singularity.     

Tight frames and geometric properties of wavelet sets

A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝ^d is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to generate an orthonormal basis for L²(ℝ^d) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions. Dedicated to Charles A. Micchelli for his 60th birthday Mathematics subject classification (2000) 42C40. John J. Benedetto: Both authors gratefully acknowledge support from ONR Grant N000140210398. The first named author also gratefully acknowledges support from NSF DMS Grant 0139759.     

On Supersets of Wavelet Sets

Considering a single dyadic orthonormal wavelet ψ in L ²(?), it is still an open problem whether the support of $\widehat{\psi}$ always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T ₂ and D ₂ of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if $A\cup B\not\subseteq T_{2}\cap D_{2}$ then S does not contain a wavelet set; and (2) if A∪B?T ₂∩D ₂ then every wavelet subset of S must be in S?(A∪B) and if S?(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S?(A∪B). In particular, if the set S?(A∪B) is of the right size then it must be a wavelet set.