Lagrange Interpolation by C 1 Cubic Splines on Triangulated Quadrangulations

We describe local Lagrange interpolation methods based on C ¹ cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.     

Jackson-Type Inequality for Doubling Weights on the Sphere

In the one-dimensional case, Jackson's inequality and its converse for weighted algebraic polynomial approximation, as well as many important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumption on the weights. In most cases this minimal assumption is the doubling condition. In this paper, we establish Jackson's theorem and its Stechkin-type converse for spherical polynomial approximation with respect to doubling weights on the unit sphere.     

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.     

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.     

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.     

Convergence for Summation Methods with Multipliers on the Sphere

In this article, we present convergence rates for summation methods with multipliers related to certain approximation methods on the sphere we previously introduced and discussed in Menegatto and Piantella [13 V.A. Menegatto and A.C. Piantella ( 2005 ). Approximation on the sphere by weighted Fourier expansions . J. Appl. Math. 2005 ( 4 ): 321 – 340 . [Google Scholar]]. The results are based upon spherical moduli of smoothness defined via the Laplace–Beltrami derivative.     

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)     

Spin Wavelets on the Sphere

In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In particular, we propose needlet-type spin wavelets as an extension of the needlet approach recently introduced by Narcowich et al. in SIAM J. Math. Anal. 38, 574–594 (2006) and J. Funct. Anal. 238, 530–564 (2006) and then considered for more general manifolds by Geller and Mayeli in Math. Z. 262, 895–927 (2009), Math. Z. 263, 235–264 (2009), and Indiana Univ. Math. J. (2009). We discuss localization properties in the real and harmonic domains, and investigate stochastic properties for the analysis of spin random fields. Our results are strongly motivated by cosmological applications, in particular in connection to the analysis of Cosmic Microwave Background polarization data.     

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.     

Sufficient Conditions for Sampling and Interpolation on the Sphere

We obtain sufficient conditions for arrays of points, \(\mathcal {Z}=\{\mathcal {Z}(L) \}_{L\ge 1}\) , on the unit sphere \(\mathcal {Z}(L)\subset \mathbb {S}^d\) , to be Marcinkiewicz–Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the separation radius of \(\mathcal {Z}(L)\) .     

Trace Formula for Integral Points on the Three-Dimensional Sphere

Doklady Mathematics - Average values over integral points on a three-dimensional sphere with an arbitrary smooth weight function are studied. For them, an expansion of the mean product of two...     

Shortest Inspection Curves for the Sphere

What is the form of the shortest curve C going outside the unit sphere S in ?³ such that passing along C we can see all points of S from outside? How will the form of C change if we require that C has one (or both) of its endpoints on S? A solution to the latter problem also answers the following question. You are in a half-space at a unit distance from the boundary plane P, but you do not know where P is. What is the shortest space curve C such that going along C you will certainly come to P? Geometric arguments suggest that the required curves should be looked for in certain classes depending on several parameters. A computer-aided analysis yields the best curves in the classes. Some other questions are solved in a similar way. Bibliography: 4 titles.     

Quadrangulations on 3-Colored Point Sets with Steiner Points and Their Winding Numbers

Let P be a point set on the plane, and consider whether P is quadrangulatable, that is, whether there exists a 2-connected plane graph G with each edge a straight segment such that V(G) = P, that the outer cycle of G coincides with the convex hull Conv(P) of P, and that each finite face of G is quadrilateral. It is easy to see that it is possible if and only if an even number of points of P lie on Conv(P). Hence we give a k-coloring to P, and consider the same problem, avoiding edges joining two vertices of P with the same color. In this case, we always assume that the number of points of P lying on Conv(P) is even and that any two consecutive points on Conv(P) have distinct colors. However, for every k ≥ 2, there is a k-colored non-quadrangulatable point set P. So we introduce Steiner points, which can be put in any position of the interior of Conv(P) and each of which may be colored by any of the k colors. When k = 2, Alvarez et al. proved that if a point set P on the plane consists of \({\frac{n}{2}}\) red and \({\frac{n}{2}}\) blue points in general position, then adding Steiner points Q with \({|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}\) , P ∪ Q is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the winding number for a 3-colored point set P, and prove that a 3-colored point set P in general position with a finite set Q of Steiner points added is quadrangulatable if and only if the winding number of P is zero. When P ∪ Q is quadrangulatable, we prove \({|Q| \leq \frac{7n+34m-48}{18}}\) , where |P| = n and the number of points of P in Conv(P) is 2m.     

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.     

A Tree Algorithm for Isotropic Finite Elements on the Sphere

The earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0.4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modeling of geoscientifically relevant data on the sphere plays an important role. In this paper, we deal with isotropic kernel functions showing a local support (briefly called isotropic finite elements) for reconstructing square-integrable functions on the sphere. An essential tool is the concept of multiresolution analysis by virtue of the spherical up function. Because the up function is built by an infinite convolution product, we do not know an explicit representation of it. However, the tree algorithm for the multiresolution analysis based on the up functions can be formulated by convolutions of isotropic kernels of low-order polynomial structure. For these kernels, we are able to find an explicit representation, so that the tree algorithm can be implemented efficiently.     

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.     

A General Multiresolution Method for Fitting Functions on the Sphere

In [7], Lyche and Schumaker have described a method for fitting functions of class C ¹ on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three associated with uniform knots. In this paper, we present a multiresolution method leading to C ²-functions on the sphere, using tensor products of polynomial and trigonometric splines of odd order with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the general tensor product decomposition and reconstruction algorithms in matrix form which are convenient for the compression of surfaces. We give the different steps of the computer implementation of these algorithms and, finally, we present a test example.