This work is unique as it provides a uniform treatment of the Fourier theories of functions (Fourier transforms and series, z-transforms), finite measures (characteristic functions, convergence in distribution), and stochastic processes (including arma series and point processes).

It emphasises the links between these three themes. The chapter on the Fourier theory of point processes and signals structured by point processes is a novel addition to the literature on Fourier analysis of stochastic processes. It also connects the theory with recent lines of research such as biological spike signals and ultrawide-band communications.

Although the treatment is mathematically rigorous, the convivial style makes the book accessible to a large audience. In particular, it will be interesting to anyone working in electrical engineering and communications, biology (point process signals) and econometrics (arma models).

A careful review of the prerequisites (integration and probability theory in the appendix, Hilbert spaces in the first chapter) make the book self-contained. Each chapter has an exercise section, which makes *Fourier Analysis and Stochastic Processes* suitable for a graduate course in applied mathematics, as well as for self-study.

Under the above conditions, the function (t) := f (t + nT ) n∈Z is T -periodic and locally integrable, and its formal Fourier series is S (t) := 1 T f n∈Z n 2iπ n t e T . T Recall once again that we speak of a “formal” Fourier series because nothing is said about the convergence of the sum of the right-hand side of the above expression. However: Corollary 1.1.10 Under the conditions of Theorem 1.1.20, and if moreover n n∈Z | f T | < ∞, then f (t + nT ) = n∈Z 1 T f n∈Z n 2iπ n t e T T

3.4.2 Linear Operations . . . . . . . . . . . . . . . . . . 3.4.3 Multivariate WSS Stochastic Processes. . . . . 3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(n/2B) is applied at time n/2B, the right-hand side of equation (1.12) is the response of the base-band (B) to the Dirac sampling comb: B, f (t) = 1 2B f n∈Z n n δ t− 2B 2B . (1.13) The adaptation of Theorem 1.1.23 to the spatial case is straightforward. Theorem 1.1.24 Let B1 , . . . , Bn ∈ R+ \{0} and let f : Rn → C be an integrable continuous function whose ft f vanishes outside [−B1 , +B1 ] × · · · × [−Bn , +Bn ], and assume that k1 kn < ∞. f ··· ,..., 2B1 2Bn k1 ∈Z kn ∈Z Then, we can

→ E. An element x ∈ E T is therefore a function from T to E: x = (x(t), t ∈ T). Let E ⊗T be the smallest sigma-field containing all the sets of the form {x ∈ E T ; x(t) ∈ C} where t ranges over T and C ranges over E. The measurable space (E T , E ⊗T ) so defined is called the canonical (measurable) space of stochastic processes indexed by T with values in (E, E) (we say: “with values in E” if the choice of the sigma-field on this space is clear in the given context). Denote by πt the coordinate

(dν). (3.34) R Proof (The case r = 1, s = 2). Let us consider the stochastic processes Y (t) = X 1 (t) + X 2 (t), Z (t) = i X 1 (t) + X 2 (t). These are wss stochastic processes with respective covariance functions CY (τ ) = C1 (τ ) + C2 (τ ) + C12 (τ ) + C21 (τ ), C Z (τ ) = −C1 (τ ) + C2 (τ ) + iC12 (τ ) − iC21 (τ ). From these two equalities we deduce C12 (τ ) = 1 {[CY (τ ) − C1 (τ ) − C2 (τ )] − i[C Z (τ ) − C1 (τ ) + C2 (τ )]} , 2 from which the result follows with μ12 = 1 {[μY − μ1