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A local lattice point counting formula, and more generally a localEuler-Maclaurin formula follow by comparing two natural families of meromorphicfunctions on the dual of a rational vector space $V$, namely the family ofexponential sums (S) and the family of exponential integrals (I) parametrizedby the set of rational polytopes in $V$. The paper introduces the notion of aninterpolator between these two families of meromorphic functions. We prove thatevery rigid complement map in $V$ gives rise to an effectively computable\SI-interpolator (and a local Euler-MacLaurin formula), an \IS-interpolator(and a reverse local Euler-MacLaurin formula) and an \ISo-interpolator. Rigidcomplement maps can be constructed by choosing an inner product on $V$ or bychoosing a complete flag in $V$. The corresponding interpolators generalize andunify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.

Sum-integral interpolators and the Euler-Maclaurin formula for polytopes