10 research outputs found
Menke points on the real line and their connection to classical orthogonal polynomials
AbstractWe investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1]
Minimal Riesz energy on the sphere for axis-supported external fields
We investigate the minimal Riesz s-energy problem for positive measures on
the d-dimensional unit sphere S^d in the presence of an external field induced
by a point charge, and more generally by a line charge. The model interaction
is that of Riesz potentials |x-y|^(-s) with d-2 <= s < d. For a given
axis-supported external field, the support and the density of the corresponding
extremal measure on S^d is determined. The special case s = d-2 yields
interesting phenomena, which we investigate in detail. A weak* asymptotic
analysis is provided as s goes to (d-2)^+.Comment: 42 pages, 2 figure
Point sets on the sphere with small spherical cap discrepancy
In this paper we study the geometric discrepancy of explicit constructions of
uniformly distributed points on the two-dimensional unit sphere. We show that
the spherical cap discrepancy of random point sets, of spherical digital nets
and of spherical Fibonacci lattices converges with order . Such point
sets are therefore useful for numerical integration and other computational
simulations. The proof uses an area-preserving Lambert map. A detailed analysis
of the level curves and sets of the pre-images of spherical caps under this map
is given
Quasi-Monte Carlo rules for numerical integration over the unit sphere
We study numerical integration on the unit sphere using equal weight quadrature rules, where the weights are such
that constant functions are integrated exactly.
The quadrature points are constructed by lifting a -net given in the
unit square to the sphere by means of an area
preserving map. A similar approach has previously been suggested by Cui and
Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2].
We prove three results. The first one is that the construction is (almost)
optimal with respect to discrepancies based on spherical rectangles. Further we
prove that the point set is asymptotically uniformly distributed on
. And finally, we prove an upper bound on the spherical cap
-discrepancy of order (where denotes the
number of points). This slightly improves upon the bound on the spherical cap
-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm.
Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the
-nets lifted to the sphere have spherical cap
-discrepancy converging with the optimal order of
Analysis of framelet transforms on a simplex
In this paper, we construct framelets associated with a sequence of quadrature rules on the simplex T2 in ℝ2. We give the framelet transforms- decomposition and reconstruction of the coefficients for framelets of a function on T2. We prove that the reconstruction is exact when the framelets are tight. We give an example of construction of framelets and show that the framelet transforms can be computed as fast as FFT
Analysis of framelet transforms on a simplex
In this paper, we construct framelets associated with a sequence of quadrature rules on the simplex T2 in ℝ2. We give the framelet transforms- decomposition and reconstruction of the coefficients for framelets of a function on T2. We prove that the reconstruction is exact when the framelets are tight. We give an example of construction of framelets and show that the framelet transforms can be computed as fast as FFT