Edward B. Saff

Johann S. Brauchart, Edward B. Saff, Ian H. Sloan et al.

We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space $H^s(S^d)$ with smoothness parameter $s>d/2$ defined over the unit sphere $S^d$ in $R^{d+1}$. Focusing on $N$-point sets that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of $N$-point node sets $X_N$ on $S^d$ such that the worst-case error of the corresponding QMC rules satisfy a bound of order $O(N^{-s/d})$ as $N\to\infty$ with an implied constant that depends on the $H^s(S^d)$-norm. We provide methods for generation and numerical testing of QMC designs. As a consequence of a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the $N$-point energy for the reproducing kernel for $H^s(S^d)$, $s>d/2$, form a sequence of QMC designs for $H^s(S^d)$. Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for $H^s(S^d)$ with $s\in(d/2,d/2+1)$. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area, spiral, minimal [Coulomb or log.] energy, and Fekete points) are QMC designs for appropriate values of $s$. For comparison purposes we show that sets of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any $s>d/2$. If $(X_N)$ is a sequence of QMC designs for $H^s(S^d)$, we prove that it is also a sequence of QMC designs for $\mathbb{H}^{s'}(S^d)$ for all $s'\in(d/2,s)$. This leads to the question of determining the supremum of such $s$, for which we provide estimates based on computations for the aforementioned sequences.

Edward B. Saff, Nikos Stylianopoulos

Let G be a bounded Jordan domain in the complex plane and consider the infinite upper Hessenberg matrix M associated with the Bergman orthogonal polynomials of G. This matrix represents the Bergman shift operator of G. The main purpose of the paper is to describe and analyze a close relation between M and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of the closure of G. Our results are based on the strong asymptotics of the Bergman polynomials. As an application, we describe and analyze an algorithm for recovering the shape of G from its area moments.

Johann S. Brauchart, Josef Dick, Edward B. Saff et al.

We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $S^d \subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\frac{1}{N}\sum_{\mathbf{x} \in X_N}f(\mathbf{x}) \approx \int_{S^d} f \, \mathrm{d} σ_d$ for functions in the Sobolev space $\mathbb{W}_p^s(S^d)$, where $σ_d$ denotes normalized area measure on $S^d.$ These bounds are close to optimal when $s$ is close to $d/p$. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for $\mathbb{W}_p^s(S^d)$, which have previously been introduced only in the Hilbert space setting $p=2$. We say that a sequence $(X_N)$ of $N$-point configurations is a QMC-design sequence for $\mathbb{W}_p^s(S^d)$ with $s > d/p$ provided the worst-case equal weight cubature error for $X_N$ has order $N^{-s/d}$ as $N \to \infty$, a property that holds, in particular, for a sequence of spherical $t$-designs in which each design has order $t^d$ points. For the case $p = 1$, we deduce that any QMC-design sequence $(X_N)$ for $\mathbb{W}_1^s(S^d)$ with $s > d$ has the optimal covering property; i.e., the covering radius of $X_N$ has order $N^{-1/d}$ as $N \to \infty$. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of $X_N$. As a consequence we prove that any QMC-design sequence for $\mathbb{W}_p^s(S^d)$ is also a QMC-design sequence for $\mathbb{W}_{p^\prime}^s(S^d)$ for all $1 \leq p < p^\prime \leq \infty$ and, furthermore, if $(X_N)$ is a quasi-uniform QMC-design sequence for $\mathbb{W}_p^s(S^d)$, then it is also a QMC-design sequence for $\mathbb{W}_p^{s^\prime}(S^d)$ for all $s > s^\prime > d/p$.