6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Volodymyr Dilnyi

Weighted Hardy spaces and signal processing

Let Hpσ(C+), σ ≥ 0, be a space of holomorphic in C+ functions such that

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

We denote a half-strip { z:|Im z|<σ, Re z<0 } by Dσ. The filter identification problem for the half-strip Dσ is to find, if possible, a test signal g belonging to the Hardy space E2 in D*σ = C\Dσ 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∈E2[Dσ]. More precisely, the question is whether there exists g ∈ E2*[Dσ] such that g∗f(τ) = 0 for all τ ≤ 0 implies f ≡ 0? The amplitude spectrum of g∈E2[D*σ] is defined by the equality

g(w)=1/2∏[0,∞) G(x)e-xwdx, Re w>0

and belongs to the H2σ(C+).

Theorem 1. Suppose the amplitude spectrum G of a signal g∈E2[D*σ] is continuous and zero-free in {z: Re z≥ 0}, f∈ E2[Dσ]. Then f∗g(τ ) = 0 for all τ ≤ 0 implies f ≡ 0 if and only if one of the following conditions holds:

  1. g admits a holomorphic continuation as an entire function and
    (∀ c∈R):g(w)exp(-cew∏/) ∉ E2[Dσ];
  2. g does not admit an analytic continuation to an entire function.
  1. Nikolski N.K., Operatos, functions and systems: an easy reading. V.1-2, AMS, 2002.
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