Functions and Charges with Semibounded Spectra

Abstract

Let us imagine a device transforming an input f into the output L(f); both are some functions of time (“signals”). The domain χ of the operator L (i.e., the set of all admissible inputs) and the range of L are vector spaces consisting of (generalized) functions defined on ℝ. We assume L to be linear. Set τ h (f)(t) := f(t - h). The signal τ h (f) is a shift of f; if h > 0, then τ h (f) is h time units later than f. Suppose that our device is indifferent to the choice of the origin of the time axis. In other words, τ h (χ) ⊂ χ, L(τ h (f)) = τ h L(f) (h ∈ ℝ, f ∈ χ). Such operators L are called shift invariant. It is well known that (under some natural restrictions) a shift invariant operator L can be represented as a convolution: $$L\left( f \right) = a*f\quad \left( {f \in X} \right)$$ (1) where a is a (generalized) function. It coincides with the output corresponding to the δ-input: a = L(δ). Real devices obey the causality principle (“no output without an input”). This means that if f ∈ χ, t0 ∈ ℝ and f | (-∞, t0) =0, then L(f) | (-∞, t0) = 0. At any given moment to such a device takes into account the past of the signal f and ignores its future; L(f) | (-∞, t0) is completely determined by f | (-∞, t0). It is easy to see that a device described by a shift invariant operator L is causal iff a | (-∞, 0) = 0.