Uniform convergence of translation operators

TL;DR

The paper investigates the limitations of uniform convergence in the context of translation operators acting on entire functions, showing that while pointwise approximation is possible, uniform convergence does not hold.

Contribution

It proves that the known pointwise universality of translation operators cannot be extended to uniform convergence.

Findings

Pointwise universality of translation operators is established.

Uniform convergence of translation operators does not occur.

The result clarifies the boundaries of approximation capabilities of translation operators.

Abstract

It is well known that there are entire functions whose orbit approximates any other entire function under the action of a sequence of translation operators . This result also holds for an uncountable family of sequences of translation operators when the convergence is pointwise .We prove that this result does not hold uniformly .