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A new lemma relates the analytic extensions of two time functions and to the Laplace transform of their ambiguity function . This lemma is used to derive necessary conditions for and from two bounds on the behavior of , at infinity. In particular, if the first bound is fulfilled, then and must be integral analytic functions. If both bounds are fulfilled, then and are each equal to times a polynomial in , and the two polynomials can be found from by comparing coefficients.