Let { a n } n = 0 ∞ \{ {a_n}\} _{n = 0}^\infty be the cosine Fourier-Stieltjes coefficients of the Borel measure μ \mu and { a 0 , ( a 1 + ⋯ + a n ) / n } n = 1 ∞ = { ( T a ) n } n = 0 ∞ \{ {a_0},({a_1} + \cdots + {a_n})/n\} _{n = 1}^\infty = \{ {(Ta)_n}\} _{n = 0}^\infty be the sequence of their arithmetic means. Then ∑ n = 0 ∞ ( T a ) n cos n x \sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx} is a Fourier-Stieltjes series. Moreover, (a) ∑ n = 0 ∞ ( T a ) n cos n x \sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx} is a Fourier series if and only if ( T a ) n → 0 {(Ta)_n} \to 0 at infinity or, equivalently, the measure μ \mu is continuous at the origin, (b) ∑ n = 1 ∞ ( T a ) n sin n x \sum \nolimits _{n = 1}^\infty {{{(Ta)}_n}\sin nx} is a Fourier series if and only if the function x − 1 μ ( [ 0 , x ) ) {x^{ - 1}}\mu ([0,x)) is in L 1 [ 0 , π ] {L^1}[0,\pi ] . These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients.