Narrow Results

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation.

We consider, for which subgroup G′ of G, the restriction π|_G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p₁) × U(p₂) × U(q) for p₁, p₂ ≥ 1 with p₁ + p₂ = p is multiplicity-free, whereas the restriction to U(p₁) × U(p₂) × U(q₁) × U(q₂) for q₁, q₂ ≥ 1 with q₁ + q₂ = q has infinite multiplicities.

It is shown that the SO(3) isometries of the Euclidean Taub–NUT space combine a linear three-dimensional representation with one induced by an SO(2) subgroup, giving the transformation law of the fourth coordinate under rotations. This explains the special form of the angular momentum operator on this manifold which leads to a new type of spherical harmonics and spinors.

Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Y_{n m}(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u(1,1). Then by applying the lowering operator of the Lie algebra u(1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Y_{m m}(θ,φ) and Y_{m+1 m}(θ,φ).

It is shown that the space of spherical harmonics whose 2l - m = p - 1 is given, represent irreducibly a cubic deformation of su(2) algebra, the so-called su_Φp(2), with deformation function as . The irreducible representation spaces are classified in three different bunches, depending on one of values 3k - 2, 3k - 1 and 3k, with k as a positive integer, to be chosen for p. So, three different methods for generating the spectrum of spherical harmonics are presented by using the cubic deformation of su(2). Moreover, it is shown that p plays the role of deformation parameter.

The associated Legendre functions for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics .

This paper proposes an extension of the variational theory of complex rays (VTCR) to three-dimensional linear acoustics, The VTCR is a Trefftz-type approach designed for mid-frequency range problems and has been previously investigated for structural dynamics and 2D acoustics. The proposed 3D formulation is based on a discretization of the amplitude portrait using spherical harmonics expansions. This choice of discretization allows to substantially reduce the numerical integration work by taking advantage of well-known analytical properties of the spherical harmonics. It also permits (like with the previous 2D Fourier version) an effective a priori selection method for the discretization parameter in each sub-region, and allows to estimate the directivity of the pressure field by means of a natural definition of rescaled amplitude portraits. The accuracy and performance of the proposed formulation are demonstrated on a set of numerical examples that include results on an actual case study from the automotive industry.

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential for power p being in some regime when the initial data are radial. This result extends the well-posedness result in Planchon, Stalker, and Tahvildar-Zadeh.

The Beauville–Fujiki relation for a compact Hyperkähler manifold $X$ of dimension $2k$ allows to equip the symmetric power ${\mathrm{\text{Sym}}}^k{H}^2(X)$ with a symmetric bilinear form induced by the Beauville–Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere ${𝕊}^d$, or equivalently, over ${ℝ}^{d+1}$ with a Gaussian kernel.

A theory of spherical harmonics associated to the Laplace–Bessel differential operator is developed. Natural analogs of the Plancherel theory, the Laplace formula, the Funk–Hecke formula, the product formula, and the addition theorem are obtained. Symmetry properties of the Fourier–Bessel transform, decompositions of smooth functions, and convolution operators are studied.

$K$-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and $K$-functionals in $L^p$ spaces with $p\ge 1$. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the $K$-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly.

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a $K$-functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness.

The main purpose of this paper is to survey some of the work on spherical approximation done by the BNU group under the direction of Professor Sun.

The equiconvergent operators of Cesàro means, and their interesting applications are described. The Jackson inequality for spherical polynomials and some moduli of smoothness on the sphere are investigated. The equivalence between moduli of smoothness and K-functionals is also discussed. We also describe several weighted polynomial inequalities on the sphere, including the Remez-type and the Nikolskii-type inequalities, the Marcinkiewicz–Zygmund inequality, the Bernstein-type and the Schur-type inequalities. Positive cubature formulas on the sphere, and their relation to the Marcinkiewicz–Zygmund inequality are also discussed. A survey on recent results on asymptotic orders of the n-widths of Sobolev's classes on the sphere is also given.

This paper surveys some of the scientific work on positive polynomial sums, Fourier analysis and spherical approximation on the sphere that Kunyang Wang did in the past 20 years.

We consider Eigen-functions of the Laplace–Beltrami Operator on n-Spheres and characterize them in terms of their local plane wave behavior. We estimate the local spectrum of wave numbers by approximating the Spherical harmonics in the locally flat neighborhood around a point on the Spheres. These local wave numbers are shown to obey an interesting Pythagorean type relation. Based on this relation, we propose a question whether there are integer triples for 2-spheres and their generalization to n-spheres. We apply the local spectrum to define quantities such as phase velocity and group velocity on a sphere and outline the relevance of the analysis for the case fields on de Sitter space.

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

The generalized spectral-analytical method is based on the use of orthogonal decompositions of signals and analytic transformations in the space of expansion coefficients. The theory of classical orthogonal bases is a generalization of the theory of Fourier series to algebraic orthogonal polynomials of continuous and discrete arguments. The choice of the optimal set of features is carried out on the basis of integral signal estimates, and the mathematical operations on the signals studied are performed in the space of the expansion coefficients. Within the framework of the generalized spectral-analytical method, the problems of adaptive analytical description of one-dimensional and multidimensional digital information arrays are effectively solved in order to reduce the redundancy of signals’ description, as well as recognition problems on the basis of pairwise comparison of objects in Fourier coefficients features space.

This chapter describes the application of this method in the following problems:

• image recognition (by using invariants when recognizing contour objects),
• bioinformatics (recognition of repeats in sequences and complex spatial configurations of biomacromolecules),
• biomedicine (analysis of the spatial and temporal organization of electrical and magnetic activity of the brain).

The recent development of image analysis and visualization tools allowing explicit 3D depiction of both normal anatomy and pathology provides a powerful means to obtain morphological descriptions and characterizations. Shape analysis offers the possibility of improved sensitivity and specificity for detection and characterization of structural differences and is becoming an important tool for the analysis of medical images. It provides information about anatomical structures and disease that is not always available from a volumetric analysis. In this chapter we present our own shape analysis work directed to the study of anatomy and disease as seen on medical images. Section 2 presents a new comprehensive method to establish correspondences between morphologically different 3D objects. The correspondence mapping itself is performed in a geometry- and orientation-independent parameter space in order to establish a continuous mapping between objects. Section 3 presents a method to compute 3D skeletons robustly and show how they can be used to perform statistical analyses describing the shape changes of the human hippocampus. Section 4 presents a method to approximate individual MS lesions' 3D geometry using spherical harmonics and its application for analyzing their changes over time by quantitatively characterizing the lesion's shape and depicting patterns of shape evolution.

The solutions for a class of Maxwell's Equations in matter are presented. These solutions describe the magnetic fields as generated by a hard ferromagnet of finite length with missing mass and are important in the area of nondestructive evaluation.

We study higher order Codazzi tensors on constant curvature spaces and show how they can be generated by functions. We give applications in Riemannian geometry and hypersurface theory; in particular we characterize ellipsoids in terms of second order sherical harmonics.

The singular value decomposition for the generalized transform of Radon type is derived when the generalized transform of Radon type is restricted to functions which are square integrable on Rⁿ with respect to the weight W_n. Furthermore, an approximation inversion formula about the measured data is also obtained.