Pre-asymptotic Expansions

Asymptotic Expansions

Asymptotic expansions of functions are useful in statistics in three main ways. Firstly, conventional asymptotic expansions of special functions are useful for approximate computation of integrals arising in statistical calculations. An example given below is the use of Stirling's approximation to the gamma function. Second, asymptotic expansions of density or distribution functions of estimato...

Asymptotic Expansions Methodsand Applications

Feynman diagrams are the most important theoretical tool for particle physicists. They are an efficient link between theory and experiment. However, their translation into actual numerical predictions is often very tedious if not impossible. Huge efforts have been devoted to their evaluation, and several powerful methods have been developed to systemize their treatment. The more complex a Feynm...

Asymptotic Expansions for Integrals

Asymptotic hyperfunctions, tempered hyperfunctions, and asymptotic expansions

We introduce new subclasses of Fourier hyperfunctions of mixed type, satisfying polynomial growth conditions at infinity, and develop their sheaf and duality theory. We use Fourier transformation and duality to examine relations of these asymptotic and tempered hyperfunctions to known classes of test functions and distributions, especially the Gel’fand-Shilov spaces. Further it is shown that th...

Asymptotic expansions by Γ-convergence

We propose a new framework aimed at constructing approximations of different order to static mechanical models with variational structure. Our starting point is a parameterized family of functionals (a ‘theory’) and we are interested in approximating the global minimuma of the energy when a secondary small parameter goes to zero. The goal is to develop a set of increasingly accurate asymptotic ...