Let L 1 {L^1} be the space of all complex-valued 2 π 2\pi -periodic integrable functions f f and let L 1 ^ \widehat {{L^1}} be the space of sequences of Fourier coefficients f ^ \hat f . A sequence λ \lambda is an ( L 1 ^ → L 1 ^ ) \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right ) multiplier if λ ⋅ f ^ = ( λ ( n ) f ^ ( n ) ) \lambda \cdot \hat f = \left ( {\lambda \left ( n \right )\hat f\left ( n \right )} \right ) belongs to L 1 ^ \widehat {{L^1}} for every f f in L 1 {L^1} . The space of even sequences of bounded variation is defined by b υ = { λ | λ n = λ − n , ∑ k = 0 ∞ | Δ λ k | + sup n | λ n | > ∞ } b\upsilon = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 0}^\infty {\left | {\Delta {\lambda _k}} \right |} + {{\sup }_n}\left | {{\lambda _n}} \right | > \infty } \right .} \right \} , where Δ λ k = λ k − λ k + 1 \Delta {\lambda _k} = {\lambda _k} - {\lambda _{k + 1}} and the space of even bounded quasiconvex sequences is defined by q = { λ | λ n = λ − n , ∑ k = 1 ∞ k | Δ 2 λ k | + sup n | λ n | > ∞ } q = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 1}^\infty {k\left | {{\Delta ^2}{\lambda _k}} \right | + {{\sup }_n}\left | {{\lambda _n}} \right | > \infty } } \right .} \right \} , where Δ 2 λ k = Δ λ k − Δ λ k + 1 {\Delta ^2}{\lambda _k} = \Delta {\lambda _k} - \Delta {\lambda _{k + 1}} . It is well known that q ⊂ b υ q \subset b\upsilon and q ⊂ ( L 1 ^ → L 1 ^ ) q \subset \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right ) but b υ ⊄ ( L 1 ^ → L 1 ^ ) b\upsilon \not \subset \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right ) . This result is significantly improved by finding an increasing family of sequence spaces d υ p d{\upsilon _p} between q q and b υ b\upsilon which are ( L 1 ^ → L 1 ^ ) \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right ) multipliers. Since the ( L 1 ^ → L 1 ^ ) \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right ) multipliers are the 2 π 2\pi -periodic measures, this result gives sufficient conditions for a sequence to be the Fourier coefficients of a measure.