The final set and initial set for
a balanced 2π-periodic trigonometric polynomial with roots in a horizontal strip are
obtained. Assuming the limits have finite order, we show that only trigonometric
polynomials are uniform limits on compact sets of balanced 2π-periodic
trigonometric polynomials that have only real zeros. The final set result is
extended to balanced almost periodic trigonometric polynomials. Finally, we
give a final set result for balanced exponential sums whose exponents are
complex.