Related papers: A Survey on Spherical Designs: Existence, Numerica…

If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such…

In this paper, we study spherical $T$-designs and their harmonic strength $\text{Hst}(X)$ on the unit circle $S^1$. For any finite set $T\subset\mathbb{N}$, we constructively demonstrate the existence of a finite design $X$ such that…

This article provides an overview of the techniques related to classification of spherical and more general objects within triangulated categories, and its relationship with algebraic geometry, representation theory and symplectic geometry.…

In this paper optimal designs for regression problems with spherical predictors of arbitrary dimension are considered. Our work is motivated by applications in material sciences, where crystallographic textures such as the missorientation…

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We…

Approximating a function with a finite series, e.g., involving polynomials or trigonometric functions, is a critical tool in computing and data analysis. The construction of such approximations via now-standard approaches like least squares…

A finite subset $Y$ on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\sum_{\mathbf{x}\in Y}f(\mathbf{x})=0$ for all real homogeneous harmonic…

We develop a systematic framework for constructing spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and analytic properties…

This short paper is concerned with the use of spherical t-designs as optimal designs for the spherical harmonic regression model in three dimensions over a range of specified criteria. The nature of the designs is explored and their…

Fourier analysis and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry etc. in dealing with 2-dimensional…

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.

Optimizing over the cone of nonnegative polynomials, and its dual counterpart, optimizing over the space of moments that admit a representing measure, are fundamental problems that appear in many different applications from engineering and…

Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast,…

How to distribute a set of points uniformly on a spherical surface is a very old problem that still lacks a definite answer. In this work, we introduce a physical measure of uniformity based on the distribution of distances between points,…

This paper proposes a sketching strategy based on spherical designs, which is applied to the classical spherical basis function approach for massive spherical data fitting. We conduct theoretical analysis and numerical verifications to…

In analogy to classical spherical t-design points, we introduce the concept of t-design curves on the sphere. This means that the line integral along a t-design curve integrates polynomials of degree t exactly. For low degrees we construct…

Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4,…

A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal…

The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to…

In this paper, we investigate in detail the structures of the variational characterization $A_{N,t}$ of the spherical $t$-design, its gradient $\nabla A_{N,t}$, and its Hessian $\mathcal{H}(A_{N,t})$ in terms of fast spherical harmonic…