Generalization of Bochner's theorem for functions of the positive type

Abstract

Using the integral representations of the solutions of Schrödinger 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.

Consistent time‐homogeneous modeling of SPX and VIX derivatives

“…However, non‐negativity of the solution is difficult to prove; this issue is explored further in the examples of Section 3. Similar non‐negativity issues arise elsewhere in the literature, for example in quantum inverse scattering theory (Chadan, 2007; Chadan & Sabatier, 2012) and in Fourier analysis Tuck (2006). For the most part, results in the literature proving positivity are less general than theory for existence of solutions.…”

Low-energy potential scattering in two and three dimensions

“…This investigation gave us the opportunity to revisit the same questions in three dimensions, and to clarify and expand the existing results. An alternative derivation of our conditions (displayed in the first preprint version of this article) have been proposed recently by Chadan [20].…”

The theorem of Bochner for adjointable operators valued maps

“…Bochner's theorem has many generalizations. Examples include references [5][6][7][8][12][13][14], this list is of course non exhaustive.…”