Restricted T-Universal Functions Original Research Article

We prove the existence of a function ϕ which is holomorphic exactly in the unit disk D and has universal translates with respect to a prescribed closed set E⊂∂D and satisfies ϕ∈C∞(∂D\E). If Q is a subsequence of N0 with upper density d(Q)=1 then the function ϕ can be constructed such that in additionϕ (z)=∑n=0∞ anzn with an=0 if n∉Q.