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Uncertainty principles for the ambiguity function. (English) Zbl 1090.42004

Let \(u, v \in L^2(R^d)\). The radar ambiguity function associated with \(u\) and \(v\) is defined for \(x, y \in R^d\) by \[ A(u, v)(x,y) = \int_{R^d} u(t+x/2) \overline{v(t-x/2)} e^{2 i \pi \langle t, x \rangle} \, dt. \] The author extends uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. The main result of this paper is the following version of Beurling’s theorem:
Theorem. Let \(u, v \in L^2(R^d)\). If there exists an \(N \geq 0\) such that \[ \int_{R^d \times R^d} {{| A(u, v)(x,y)| } \over {(1 + \| x\| + \| y\| )^N}} e^{\pi | \langle x, y \rangle| } \, dx \, dy < \infty, \] then either \(u\) or \(v\) is identically zero, or both can be written as \[ u(x) = P(x) e^{- \langle Ax, x \rangle - 2 i \pi \langle w, x \rangle}, \qquad v(x) = Q(x) e^{- \langle Ax, x \rangle - 2 i \pi \langle w, x \rangle}, \] with \(P\) and \(Q\) polynomials whose degrees satisfy \(\text{ deg} (P) + \text{ deg} (Q) < N - d\), \(w \in C^d\), and \(A\) a real positive symmetric matrix. The converse is also true. In particular, for \(N \leq d\), \(u\) or \(v\) are identically vanishing. This theorem improves a result of A. Bonami, B. Demange and P. Jaming [Rev. Mat. Iberoam. 19, No. 1, 23–55 (2003; Zbl 1037.42010)].

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Citations:

Zbl 1037.42010
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