On the behavior at infinity of an integrable function

Emmanuel Lesigne (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

We prove that, in a weak sense, any integrable function on the real line tends to zero at infinity : if f is an integrable function on R, then for almost all real number x, the sequence (f(nx)) tends to zero when n goes to infinity. Using Khinchin's metric theorem on Diophantine approximation, we establish that this convergence to zero can be arbitrarily slow.

Domaines: Analyse classique [math.CA]

Emmanuel Lesigne. On the behavior at infinity of an integrable function. The American Mathematical Monthly, 2010, 117 (2), pp.175-181. ⟨hal-00276738v3⟩