Sharp inequalities for convolution-type operators

Abstract

Let μ be a probability measure on [− a, a], a > 0, and let x 0 ϵ[− a, a], f ϵ C n ([−2 a, 2 a]), n ⩾ 0 even. Using moment methods we derive best upper bounds to ¦∫ −a a ( [f(x 0 + y) + f(x 0 − y)] 2 ) μ(dy) − f(x 0)¦ , leading to sharp inequalities that are attainable and involve the second modulus of continuity of f ( n) or an upper bound of it.