相关论文: On E-Discretization of Tori of Compact Simple Lie …

Three types of numerical data are provided for simple Lie groups of any type and rank. This data is indispensable for Fourier-like expansions of multidimensional digital data into finite series of $C-$ or $S-$functions on the fundamental…

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-,…

We develop and describe continuous and discrete transforms of class functions on compact simple Lie group $G$ as their expansions into series of uncommon special functions, called here $\E$-functions in recognition of the fact that the…

Ten types of discrete Fourier transforms of Weyl orbit functions are developed. Generalizing one-dimensional cosine, sine and exponential, each type of the Weyl orbit function represents an exponential symmetrized with respect to a subgroup…

New continuous group transforms, together with their discretization over a lattice of any density and admissible symmetry, are defined for a general compact simple Lie groups of rank $2\leq n<\infty$. Rank 1 transforms are known. Rank 2…

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even…

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group $G$. Such functions can…

New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently…

Let $G$ be a compact, connected, simply-connected Lie group. We use the Fourier-Mukai transform in twisted $K$-theory to give a new proof of the ring structure of the $K$-theory of $G$.

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…

Given a real closed polytope $P$, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula,…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

The discrete orthogonality of special function families, called $C$- and $S$-functions, which are derived from the characters of compact simple Lie groups, is described in Hrivn\'ak and Patera (2009 J. Phys. A: Math. Theor. 42 385208).…

New class of special functions of three real variables, based on the alternating subgroup of the permutation group $S_3$, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and…

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$,…

For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^{\omega}. There exists a natural algebra of…

Let $\gg$ be the Lie algebra of a compact Lie group and let $\theta$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^\theta$. In this paper we present LiE programs that, for any finite dimensional complex…

Consider a simple algebraic group $G$ of classical type and its Lie algebra $\mathfrak{g}$. Let $(e,h,f) \subset \mathfrak{g}$ be an $\mathfrak{sl}_2$-triple and $Q_e= C_G(e,h,f)$. The torus $T_e$ that comes from the…

We establish a "matrix simultaneous diagonalization theorem" for disconnected reductive groups which relaxes both the semisimplicity condition and the commutativity condition. As an application, we prove the following basic results…

We give two algebro-geometric inspired approaches to fast algorithms for Fourier transforms in algebraic signal processing theory based on polynomial algebras in several variables. One is based on module induction and one is based on a…