Related papers: Approximation on the complex sphere
In this article, we present a space-frequency theory for spherical harmonics based on the spectral decomposition of a particular space-frequency operator. The presented theory is closely linked to the theory of ultraspherical polynomials on…
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm…
A method is described by which a function defined on a cubic grid (as from a finite difference solution of a partial differential equation) can be resolved into spherical harmonic components at some fixed radius. This has applications to…
We consider spectral minimal partitions. Continuing work of the the present authors about problems for planar domains, [23], we focus on the sphere and obtain a sharp result for 3-partitions which is related to questions from harmonic…
We will discuss the following results C_n complexification of R(n) spaces, C_n structure and the invariant surfaces C_n holomorphicity and harmonicity. We also consider the link between C_n holomorphicity and the origin of spin 1/n. In our…
The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the…
In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials…
We characterize homogeneous hypersurfaces in complex space forms which arise as critical points of a higher order energy functional. As a consequence, we obtain existence and non-existence results for $\mathbb{CP}^n$ and $\mathbb{CH}^n$,…
In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
This is Chapter 1 of the book {\it Approximation Theory and Harmonic Analysis on Spheres and Balls} by the authors. It provides a self-contained introduction to spherical harmonics. The book will be published as a title in {\it Springer…
We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest…
In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space R^{m+1} was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite…
In the framework of quaternionic Clifford analysis in Euclidean space $\mathbb{R}^{4p}$, which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is…
In this paper we introduced a new characteristics of the elements of a Hilbert space - generalized moduli of continuity $\omega_\varphi(x;L_{p,V}([0,\delta]))$ and obtain new exact inequalities of Jackson - Stechkin type with these moduli…
The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the H\"older spaces $H_p^{r,\alpha}$ for all $0<p\le\infty$ and $0<\alpha\le r$. By using modifications of the classical moduli of…
We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of…