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On the abs-polynomial expansion of piecewise smooth functions

Abstract:

Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order (Formula presented.) can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function (Formula presented.) apart from arithmetic operations and (Formula presented.) times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of (Formula presented.), provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in (Formula presented.). As already conjectured, we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.