18 research outputs found
A simple Proof of Stolarsky's Invariance Principle
Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful
relation that balances the sums of distances of points on the unit sphere and
their spherical cap -discrepancy to give the distance integral of
the uniform measure on the sphere a potential-theoretical quantity (Bj{\"o}rck
[Ark. Mat. 3 (1956), 255--269]). Read differently it expresses the worst-case
numerical integration error for functions from the unit ball in a certain
Hilbert space setting in terms of the -discrepancy and vice versa
(first author and Womersley [Preprint]). In this note we give a simple proof of
the invariance principle using reproducing kernel Hilbert spaces
An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge
For a positively charged insulated d-dimensional sphere we investigate how
the distribution of this charge is affected by proximity to a nearby positive
or negative point charge when the system is governed by a Riesz s-potential
1/r^s, s>0, where r denotes Euclidean distance between point charges. Of
particular interest are those distances from the point charge to the sphere for
which the equilibrium charge distribution is no longer supported on the whole
of the sphere (i.e. spherical caps of negative charge appear). Arising from
this problem attributed to A. A. Gonchar are sequences of polynomials of a
complex variable that have some fascinating properties regarding their zeros.Comment: 44 pages, 9 figure
Riesz external field problems on the hypersphere and optimal point separation
We consider the minimal energy problem on the unit sphere in
the Euclidean space in the presence of an external field
, where the energy arises from the Riesz potential (where is the
Euclidean distance and is the Riesz parameter) or the logarithmic potential
. Characterization theorems of Frostman-type for the associated
extremal measure, previously obtained by the last two authors, are extended to
the range The proof uses a maximum principle for measures
supported on . When is the Riesz -potential of a signed
measure and , our results lead to explicit point-separation
estimates for -Fekete points, which are -point configurations
minimizing the Riesz -energy on with external field . In
the hyper-singular case , the short-range pair-interaction enforces
well-separation even in the presence of more general external fields. As a
further application, we determine the extremal and signed equilibria when the
external field is due to a negative point charge outside a positively charged
isolated sphere. Moreover, we provide a rigorous analysis of the three point
external field problem and numerical results for the four point problem.Comment: 35 pages, 4 figure
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
Numerical integration over spheres of arbitrary dimension
In this paper, we study the worst-case error (of numerical integration) on the unit sphere , , for all functions in the unit ball of the Sobolev space , where s>d/2. More precisely, we consider infinite sequences of -point numerical integration rules , where (i) is exact for all spherical polynomials of degree , and (ii) has positive weights or, alternatively to (ii), the sequence satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) in has the upper bound , where the constant depends on and (and possibly the sequence ). This extends the recent results for the sphere by K.Hesse and I.H.Sloan to spheres of arbitrary dimension by using an alternative representation of the worst-case error. If the sequence of numerical integration rules satisfies an order-optimal rate of convergence is achieved