Let f be an entire function of exponential type σ bounded on the real line. We associate with f an explicitly given function fL,0 which coincides with a trigonometric polynomial of period L and exponential type σ + 2π/L on the strip ℜz ≤ L/2 and which is zero outside. When f (x) → 0 as x → ± ∞, we show that fL,0 converges to f as L → ∞ uniformly on horizontal strips. We also prove convergence with respect to Lp} norms on ℝ and on lines parallel to ℝ and convergence with respect to lp} norms. These results include a simple and more general alternative answer to a question which was recently raised and answered by R. Martin.