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Register a new userFourier coefficients of invariant random fields on homogeneous spaces of compact groups
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Let $T$ be a random field invariant under the action of a compact group $G$. In the line of previous work we investigate properties of the Fourier coefficients as orthogonality and Gaussianity. In particular we give conditions ensuring that independence of the random Fourier coefficients implies Gaussianity. As a consequence, in general, it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients.
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Given a compact (Hausdorff) group $G$ and a closed subgroup $H$ of $G,$ in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space $G/H$ characterizing the Schatten-von Neumann classes $S_r(L^2(G/H))$ for all $0<r leq infty.$ We go on to provide a symbolic characterization for $r$-nuclear, $0< r leq 1,$ pseudo-differential operators on $L^{p}(G/H)$-space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space $G/H times widehat{G/H}.$ Finally, we present applications of aforementioned results in the context of heat kernels.
We establish spectral expansions of homogeneous and isotropic random fields taking values in the $3$-dimensional Euclidean space $E^3$ and in the space $mathsf{S}^2(E^3)$ of symmetric rank $2$ tensors over $E^3$. The former is a model of turbulent fluid velocity, while the latter is a model for the random stress tensor or the random conductivity tensor. We found a link between the theory of random fields and the theory of finite-dimensional convex compacta.
Let $G$ be a compact group. For $1leq pleqinfty$ we introduce a class of Banach function algebras $mathrm{A}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in cite{forrestss1}. In the case $p ot=2$ we find that $mathrm{A}^p(G)cong mathrm{A}^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups. These algebras admit natural operator space structures, and also weighte
Many examples of natural systems can be described by random Gaussian surfaces. Much can be learned by analyzing the Fourier expansion of the surfaces, from which it is possible to determine the corresponding Hurst exponent and consequently establish the presence of scale invariance. We show that this symmetry is not affected by the distribution of the modulus of the Fourier coefficients. Furthermore, we investigate the role of the Fourier phases of random surfaces. In particular, we show how the surface is affected by a non-uniform distribution of phases.
Let $G$ be one of the classical compact, simple, centre-less, connected Lie groups or rank $n$ with a maximal torus $T$, the Lie algebra $clg$ and let ${ E_i, F_i, H_i, i=1, ldots, n }$ be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M={ {rm Ad}_g(H_1),~g in G }$, identified with the homogeneous space $G/L$ where $L={ g in G:~{rm Ad}_g(H_1)=H_1}$. We prove that the `coordinate functions ${ f_i, i=1, ldots, n }$, (where $f_i(g):=lambda_i({rm Ad}_g(H_1))$, ${ lambda_1, ldots, lambda_n}$ is basis of $clg^prime$) are `quadratically independent in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithtully on $C(M)$ such that the action leaves invariant the linear span of the above cordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of $M$ satisfying a similar `linearity condition must be a Rieffel-Wang type deformation of some compact group.
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