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We prove spectral mapping theorem for linear hyperbolic systems of PDEs. The system is of the following form: For $0 < x < l$ and $t > 0$ $$ rm(H) quad left beginarrayl displaystyle partial over partial t beginpmatrix u(t,x) v(t,x) endpmatrix + K(x) partial over partial x beginpmatrix u(t,x) v(t,x) endpmatrix + C(x) beginpmatrix u(t,x) v(t,x) endpmatrix = 0, displaystyle d over dt left [ v(t,l) - D u(t,l) right ] = F u(t,cdot) + G v(t,cdot) , displaystyle u(t,0) = E v(t,0), endarray right . $$ where $u(t,x) in C^n_1$, $v(t,x) in C^n_2$, $K(x) = mathrmdiag , left( k_i(x) right )_1 le i le n$ is a diagonal matrix of functions $k_i in C^1left( [0,l], R right)$, $k_i(x) > 0$ for $i = 1, dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, dots, n=n_1+n_2$, and $D$,$E$ are matrices. We show high frequency estimates of spectra and resolvents in terms of reduced (block)diagonal systems. Let $A$ denote the infinitesimal generator for $mathrm(H)$ which generates $C_0$ semigroup $e^At$ on $L^2 times C^n_2$. Our main result is the following spectral mapping theorem $$sigma(e^At) setminus 0 = overlinee^sigma(A)t setminus 0 .$$

Sufficient conditions are given for a bounded positive measure μ \mu on R d {\mathbb {R}^d} to be the power spectrum of a function φ \varphi . Applications to spectrum estimation are made for the cases in which a signal φ \varphi is known or its autocorrelation P ϕ {P_\phi } is known. In the first case, it is shown that \[ ∫ | f ^ ( γ ) | 2 d μ ϕ ( γ ) = lim R → ∞ 1 | B ( R ) | ∫ B ( R ) | f ∗ φ ( t ) | 2 d t , \int {|\hat f(\gamma )|^2}d{\mu _\phi }(\gamma )= \lim \limits _{R \to \infty }\,\frac {1}{|B(R )|}\,\int _{B(R)} |f \ast \varphi (t)|^2\;dt, \] where P ^ φ = μ φ {\hat P}_{\varphi }= {\mu _\varphi } , B ( R ) B(R) is the d d -dimensional ball of radius R R , and f f ranges through a prescribed function space. In the second case, an example, which is a variant of the Tomas-Stein restriction theorem, is \[ ∀ f ∈ L 1 ( R d ) ∩ L p ( R d ) , ( ∫ ∑ d − 1 | f ^ ( θ ) | 2 d μ d − 1 ( θ ) ) 1 / 2 ≤ ( 1 2 ∥ μ d − 1 ∨ ∥ p ′ 1 / 2 ) ( ∥ f ∥ 1 + ∥ f ∥ p ) , \forall f \in {L^1}({\mathbb {R}^d})\, \cap \,{L^p}({\mathbb {R}^d}), \\ \left (\int _{\sum _{d - 1}} | \hat f(\theta )|^2 d {\mu _{d - 1}}(\theta ) \right )^{1/2} \leq \left (\frac {1}{2}\parallel \mu _{d - 1}^{\vee } \parallel _{p’}^{1/2} \right )\;\left (\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right ), \] where 1 ≤ p > 2 d / ( d + 1 ) 1 \leq p > 2d/(d + 1) and the power spectrum μ d − 1 {\mu _{d - 1}} is the compactly supported restriction of surface measure to the unit sphere ∑ d − 1 ⊆ R ^ d \sum \nolimits _{d - 1} { \subseteq } \;{{\hat {\mathbb {R}}}^d} .

Using the semigroup product formula of P \text {P} . Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.

It is proved that the geometric hypothesis of a theorem which generalizes Norbert Wiener's multiparameter ergodic theorem are satsified in the case of arbitrary convex regions, provided only that they form a substantial family as defined in the introduction.