Characterization of function spaces via low regularity mollifiers

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Smoothness of a function f:Rⁿ→ R can be measured in terms of the rate of convergence of fρε to f, where ρ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that ρ is adapted to a given scale of spaces. Finally, we examine in detail the case where ρ is a characteristic function.