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.