Borel Summation in a Martingale-Type Collar

Extrapolation of the asymptotic series with a cost functional imposed on iterated Borel summation is considered. The cost functionals are designed to determine the optimal control parameter, the role of which is performed by the number of iterations, which could be considered as fractional real or non-negative integers. New cost functional is inspired by a martingale with a penalty term written to penalize the solution to optimization with fractional number of iterations for deviations of the expected value of sought quantity from the results of a discrete iterated Borel summation. The optimization technique employed in the paper is unique since the penalty by itself is expressed through the sought quantity, such as critical amplitude dependent on the number of iterations. The solution to the extrapolation problem with the control parameter found by means of optimization with a new cost functional is accurate, robust and uniquely defined for a variety of extrapolation problems.