**Weighted Hardy spaces and signal processing**

Let H^{p}_{σ}(C_{+}), σ ≥ 0, be a space of holomorphic in C_{+} functions such that

∥f∥^{p}:=sup_{φ ∈ (-∏/2};_{∏/2)}{∫_{[0,∞)} |f(re^{iφ})|^{p} e^{-pσr|sin φ|}dr}<+∞.

We denote a half-strip { z:|Im z|<σ, Re z<0 } by D_{σ}. The *filter identification problem* for the half-strip D_{σ} is to find, if possible, a test signal g belonging to the Hardy space E^{2} in D*_{σ} = C\D_{σ} whose output

g∗f(τ)=∫_{∂Dσ} g(w)f(w + τ)dw

measured at all time moments τ ≤ 0 defines uniquely an unknown filter f∈E^{2}[D_{σ}]. More precisely, the question is whether there exists g ∈ E^{2}_{*}[D_{σ}] such that g∗f(τ) = 0 for all τ ≤ 0 implies f ≡ 0? The amplitude spectrum of g∈E^{2}[D*_{σ}] is defined by the equality

g(w)=^{1}/_{√2∏∫[0,∞) G(x)e-xwdx, Re w>0}

and belongs to the H^{2}_{σ}(C_{+}).

**Theorem 1.** Suppose the amplitude spectrum G of a signal g∈E^{2}[D*_{σ}] is continuous and zero-free in {z: Re z≥ 0}, f∈ E^{2}[D_{σ}]. Then f∗g(τ ) = 0 for all τ ≤ 0 implies f ≡ 0 if and only if one of the following conditions holds:

- g admits a holomorphic continuation as an entire function and

(∀ c∈R):g(w)exp(-ce^{–w∏/2σ}) ∉ E^{2}[D_{σ}]; - g does not admit an analytic continuation to an entire function.

- Nikolski N.K.,
*Operatos, functions and systems: an easy reading*. V.1-2, AMS, 2002.