Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

[摘要] Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean andscalarcovariance matrices. The resultinganisotropyfunctional is defined forfinite powerrandom vectors. Originally, anisotropy was introduced fordirectionally genericrandom vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associateda-anisotropicnorm of a matrix is then its maximumroot mean squareoraverage energygain with respect to finite power or directionally generic inputs whose anisotropy is bounded above bya≥0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yieldmean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

[学科分类] 应用数学