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Comment: 34 pages. This revision contains new content and some corrections
Comment: 10 pages, 2 figures
Comment: 19 pages, LaTeX
Using the integral representations of the solutions of Schr\"odinger equation, which are the essential ingredients of the Gel'fand-Levitan and Marchenko integral equations of inverse scattering theory, we obtain a general theorem on the positivity of some integral transforms, and extend the theorem of Bochner on Fourier transforms of functions of positive type to more general transforms. The present study is restricted to the positive half-axis. We then obtain a theorem on the positivity of Fourier cosine transform of the phase-shifts.
Comment: 38 pages, some typos corrected, one reference added, one remark added
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