The Structures of the Spaces H(T) and LT(F)
We have already seen (in Section 1.6) that the Hilbert space of random variables H(T) generated by a stationary process ξ (t), t ∈ T (with spectral measure F(dλ)), is isometric to a space of functions L T (F), which is the closed linear hull of the functions e iλt for λ ∈ [-π, π] in the case of discrete time t and for λ ∈ [-∞,∞] in the case of continuous time t. This fact enables us to investigate stationary processes using analytic tools. To do this, it is useful first to study in detail the analytic structure of spaces L T (F), as we shall do in this chapter, restricting ourselves to the case where T is a finite interval or a half-line.
KeywordsSpectral Density Entire Function Continuous Time Spectral Measure Zero Degree
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