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Analytic continuation into the future

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  • A class of analytic advanced and delayed differential equations, which are defined in a neighborhood of an initial point, and which are assumed to have formal solutions in terms of power series, is studied. We provide growth conditions whereby the (perhaps non-convergent) formal series solutions can be extended to analytic solutions defined on a sectorial domain with vertex at the initial point. By introducing a new Laplace-Borel kernel, and obtaining estimates on its decay rate, the concept of a Gevrey series is generalized. The class of equations studied includes advanced and delayed initial value problems with polynomial coefficients. Key estimates are shown and an example of a new application is given.
    Mathematics Subject Classification: Primary: 34M25, 34M30; Secondary: 40G10, 44A10.


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