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Sampling of Operators Götz E. Pfander

Received: 9 March 2012 / Revised: 25 October 2012 / Published online: 3 April 2013 © Springer Science+Business Media New York 2013

Abstract Sampling and reconstruction of functions is a fundamental tool in science. We develop an analogous sampling theory for operators whose Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for in this sense bandlimited operators and show that our results generalize both, the classical sampling theorem, and the fact that a time-invariant operator is fully determined by its impulse response. Keywords Generalized sampling · Operator identification · Channel measurement · Paley-Wiener spaces · Modulation spaces Mathematics Subject Classification (2010) Primary 42B35 · 94A20 · Secondary 35S05 · 47B35 · 94A20 1 Introduction The classical sampling theorem for bandlimited functions states that a function whose Fourier transform is supported on an interval of length Ω is completely characterized by samples taken at rate at least 1/Ω per unit interval. That is, with F denoting the Fourier transform1 we have the following: Theorem 1.1 For f ∈L2 (R) with supp F f ⊆[−Ω/2, Ω/2], choose T with T Ω ≤1. Then    f (nT )  2 = T f  2 . L (R) l (Z) 1 See Sect. 2 for basic notation used throughout this paper.

Communicated by Chris Heil. G.E. Pfander () School of Engineering and Science, Jacobs University, 28759 Bremen, Germany e-mail: [email protected]

J Fourier Anal Appl (2013) 19:612–650

613

Moreover, f can be reconstructed by means of the uniformly and L2 -converging series f (x) =



f (nT )

n∈Z

sin(πT (x − n)) . πT (x − n)

Theorem 1.2 below describes sampling of operators in its simplest setting. We choose a Hilbert–Schmidt operator H on L2 (R) with kernel κH and Kohn-Nirenberg symbol σH , that is σH (x, D) = H in pseudodifferential operator notation [31, 62]. Recall that a Hilbert–Schmidt operator H on L2 (R) is a bounded operator with Hilbert–Schmidt norm H H S = κH L2 < ∞. Let F s denote the so-called symplectic Fourier transform on L2 (R2d ). Theorem 1.2 For H : L2 (R) −→ L2 (R) Hilbert–Schmidt with supp F s σH ⊆[0, T ]× [−Ω/2, Ω/2] and T Ω≤1, we have      H δkT  = T H H S ,   k∈Z

L2 (R)

and H can be reconstructed by means of    sin(πT (x − n)) H δkT (t + nT ) κH (x + t, x) = χ[0,T ] (t) πT (x − n) n∈Z

k∈Z

where χ[0,T ] (t) = 1 for t ∈ [0, T ] and 0 else and with convergence in Hilbert–Schmidt norm. As shown in Sect. 4, Theorems 1.1 and 1.2 are special cases of Theorem 4.4, one of the key results presented in this paper. The appearance of the sampling rate T in the description of the bandlimitation of the operator’s Kohn–Nirenberg symbol reflects a fundamental difference between sampling of operators and sampling of functions. This fact is illuminated in terms of operator identification in [41, Theorem 3.6] and [53, Theorem 1.1], results which are extended in Theorems 5.6 and 5.7. In fact, in classical sampling theory, the bandlimitation of a function to a large interval can be compensat