a b s t r a c tThe synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2 n (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration 'complete' for the standard complexity classes NP and #P -even when restricted to smooth (=C ∞ ) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n.The present work investigates the uniform parameterized complexity of natural operators Λ on subclasses of smooth functions:evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural ✩ Supported in part by JSPS Kakenhi projects 23700009 and 24106002, by 7th EU IRSES project 294962, and by DFG project Zi 1009/4-1. We gratefully acknowledge seminal discussions with Robert Rettinger, Matthias Schröder, and FlorianSteinberg -and the helpful feedback of thorough reviewers!.

Abstract. The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h (t) = g(t, h(t)) under various assumptions on the function g has been investigated. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C 1 ; for each k ≥ 2, the solution h can be hard for the counting hierarchy even if g is of class C k .

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