On a Watson-like Uniqueness Theorem and Gevrey Expansions

TL;DR

This paper extends Watson's uniqueness theorem to classes of functions with Gevrey expansions in specific angular regions, providing conditions for uniqueness and improved accuracy estimates for solutions of differential equations.

Contribution

It introduces an extension of Watson's theorem for functions with Gevrey expansions in limited angular sectors, enhancing the understanding of solution uniqueness and approximation accuracy.

Findings

Extended Watson's theorem for Gevrey classes in angular regions

Provided conditions for solution uniqueness in differential equations

Improved estimates for deviation from Gevrey asymptotic sums

Abstract

We present a maximal class of analytic functions, elements of which are in one-to-one correspondence with their asymptotic expansions. In recent decades it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.), that the formal power series solutions of a wide range of systems of ordinary (even non-linear) analytic differential equations are in fact the Gevrey expansions for the regular solutions. Watson's uniqueness theorem belongs to the foundations of this new theory. This paper contains a discussion of an extension of Watson's uniqueness theorem for classes of functions which admit a Gevrey expansion in angular regions of the complex plane with opening less than or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We present conditions which ensure uniqueness and which suggest an extension of Watson's representation theorem. These results may be applied for solutions of certain classes of differential equations to obtain the best accuracy estimate for the deviation of a solution from a finite sum of the corresponding Gevrey expansion.