It is proved that every bounded measurable function on ( − ∞ , ∞ ) ( - \infty ,\infty ) which for some constant a > 0 a > 0 is of bounded variation on ( − ∞ , − a ) ( - \infty , - a) and on ( a , ∞ ) (a,\infty ) , admits spectral synthesis in the weak-star topology of L ∞ ( − ∞ , ∞ ) {L^\infty }( - \infty ,\infty ) .