Poisson Wavelets on the Sphere

In this article we summarize the basic formulas of wavelet analysis with the help of Poisson wavelets on the sphere. These wavelets have the nice property that all basic formulas of wavelet analysis as reproducing kernels, etc. may be expressed simply with the help of higher degree Poisson wavelets. This makes them numerically attractive for applications in geophysical modeling.     

Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

This article presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory.     

Spherical Singular Integrals, Monogenic Kernels and Wavelets on the Three-Dimensional Sphere

The construction of wavelets relies on translations and dilations which are perfectly given in . On the sphere translations can be considered as rotations but it is difficult to say what are dilations. For the 2-dimensional sphere there exist two different approaches to obtain wavelets which are worth to be considered. The first concept goes back to W. Freeden and collaborators who define wavelets by means of kernels of spherical singular integrals. The other concept developed by J.P. Antoine and P. Vandergheynst is a purely group theoretical approach and defines dilations as dilations in the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define singular integrals and kernels of singular integrals on the three dimensional sphere which are also approximate identities. In particular the Cauchy kernel in Clifford analysis is a kernel of a singular integral, the singular Cauchy integral, and an approximate identity. Furthermore, we will define wavelets on the 3-dimensional sphere by means of kernels of singular integrals. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš Received: October, 2007. Accepted: February, 2008.     

Orthogonal Zonal, Tesseral and Sectorial Wavelets on the Sphere for the Analysis of Satellite Data

The spherical harmonics Y _n,k } _{n=0,1,...;k=?n,...,n} represent a standard complete orthonormal system in ?²(Ω), where Ω is the unit sphere. In view of present and future satellite missions (e.g., for the determination of the Earth's gravity field) it is of particular importance to treat the different accuracies and sizes of data in dependence of the index pairs (n,k). It is, e.g., known that the GOCE mission yields essentially less accurate data in the zonal (k=0) case. Therefore, this paper presents new ways of constructing multiresolutions for a Sobolev space of functions on Ω allowing the separate treatment of certain classes of pairs (n,k) and, in particular, the separate treatment of different orders k. Orthogonal bandlimited as well as non-bandlimited detail and scale spaces adapted to certain (geo)scientific problems and to the character of the given data can now be used. Finally, an explicit representation of a non-bandlimited wavelet on Ω yielding an orthogonal decomposition of the function space is calculated for the first time.     

A Nonlinear Galerkin Scheme Involving Vectorial and Tensorial Spherical Wavelets for Solving the Incompressible Navier-Stokes Equation on the Sphere

The spherical Navier-Stokes equation plays a fundamental role in meteorology by modelling meso-scale (stratified) atmospherical flows. This article introduces a wavelet based nonlinear Galerkin method applied to the Navier-Stokes equation on the rotating sphere. In detail, this scheme is implemented by using divergence free vectorial spherical wavelets, and its convergence is proven. To improve numerical efficiency an extension of the spherical panel clustering algorithm to vectorial and tensorial kernels is constructed. This method enables the rapid computation of the wavelet coefficients of the nonlinear advection term. Thereby, we also indicate error estimates. Finally, extensive numerical simulations for the nonlinear interaction of three vortices are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spline Wavelets on the Unit Circle

An orthonormal wavelet basis on the circle γ is constructed. By estabishing some new theorems on complex spline functions, the $\mathop L\limits^ \circ _2 (I)$ space can be decomposed into different orthogonal subspaces. Formulas of decomposition and reconstruction imply only two terms.     

Wavelets on irregular meshes

We construct wavelets associated with a mesh involving local refinements. The first stage consists in the definition of a multiresolution analysis adapted to our non-translation invariant situation. Then, we show that, modulo an optimal condition on the geometry of the mesh, the orthogonal complement of the approximation spaces can be constructed by hand and presents the homogeneous behaviour that we were looking for. Finally, we propose a fast algorithm on irregular meshes, using a second wavelet basis constructed for this purpose. The stability of the synthesis algorithm is proved whereas the stability of the analysis algorithm is still an open problem.     

Rational Points on the Sphere

Using only basic tools from the theory of modular forms, the rational points of bounded height on the sphere are counted and shown to be uniformly distributed. The more difficult case of points with a given height is also treated.     

Riemann Localisation on the Sphere

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), we mean that for a suitable subset X of \(\mathbb {L}_{p}(\mathbb {S}^{d})\), \(1\le p\le \infty \), the \(\mathbb {L}_{p}\)-norm of the Fourier local convolution of \(f\in X\) converges to zero as the degree goes to infinity. The Fourier local convolution of f at \(\mathbf {x}\in \mathbb {S}^{d}\) is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of \(\mathbf {x}\). The failure of Riemann localisation for \(d>2\) can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.     

Polyharmonic Approximation on the Sphere

The purpose of this article is to provide new error estimates for a popular type of spherical basis function (SBF) approximation on the sphere: approximating by linear combinations of Green’s functions of polyharmonic differential operators. We show that the L _p approximation order for this kind of approximation is σ for functions having L _p smoothness σ (for σ up to the order of the underlying differential operator, just as in univariate spline theory). This improves previous error estimates, which penalized the approximation order when measuring error in L _p , p>2 and held only in a restrictive setting when measuring error in L _p , p<2.     

Discrepancy Estimates on the Sphere

 For measures on the unit sphere in ℝ ^d , d≥3, we derive discrepancy estimates in terms of the quality of corresponding quadrature formulas and in terms of bounds for potential differences. (Received 1 August 1998; in revised form 30 December 1998)     

Discrepancy Estimates on the Sphere

 For measures on the unit sphere in ℝ ^d , d≥3, we derive discrepancy estimates in terms of the quality of corresponding quadrature formulas and in terms of bounds for potential differences.     

Wavelets Based on Orthogonal Polynomials

We present a unified approach for the construction of polynomial wavelets. Our main tool is orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.

     

Polynomial Wavelet-Type Expansions on the Sphere

We present a polynomial wavelet-type system on S ^d such that any continuous function can be expanded with respect to these wavelets. The order of the growth of the degrees of the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a dual wavelet system. The dual wavelets system is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulas for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.     

Wavelets on the Two-Sphere and Other Conic Sections

We survey the construction of the continuous wavelet transform (CWT) on the twosphere. Then we discuss the discretization of the spherical CWT, obtaining various types of discrete frames. Finally, we give some indications on the construction of a CWT on other conic sections.     

Minkowski Circle Packings on the Sphere

We consider n caps on the sphere such that none of them contains in its interior the center of another. We give an upper bound for the total area of the caps, which is sharp for n = 3 , 4, 6, and 12 and is asymptotically sharp for great values of n . Received September 11, 1997, and in revised form March 2, 1998.