On possible deterioration of smoothness under the operation of convolution

Abstract

Let μ be a finite non-negative Borel measure on the real line R . We give a condition which is necessary and sufficient for the existence of an entire function p satisfying the following conditions: (i) p(x)⩾0 for x∈ R , (ii) p∈ L 1( R) , (iii) ess sup {(p∗μ)(x):x∈I}=∞ for every non-empty interval I⊂ R . We give also a sufficient condition for the existence of an entire function p of finite order ρ>1 and normal type satisfying conditions (i)–(iii).