Permissible growth rates for Birkhoff type universal harmonic functions

A harmonic function H on Rn(n⩾2) is said to be universal (in the sense of Birkhoff) if its set of translates {x↦H(a+x):a∈Rn} is dense in the space of all harmonic functions on Rn with the topology of local uniform convergence. The main theorem includes the result that such functions, H, can have any prescribed order and type. The growth result is compared with a similar known theorem for G.D. Birkhoff's universal holomorphic functions and contrasted with known growth theorems for MacLane-type universal harmonic and holomorphic functions.