In this paper a multiplier theorem in the Hardy space
H
1
(
T
)
H^1(\mathbb {T})
associated with Jacobi expansions of exponential type is proved, that is, a bilateral sequence
{
λ
n
}
n
=
−
∞
∞
\left \{\lambda _n\right \}_{n=-\infty }^{\infty }
is a multiplier from
H
1
(
T
)
H^1(\mathbb {T})
into the sequence space
ℓ
1
(
Z
)
\ell ^1(\mathbb {Z})
associated with Jacobi expansions of exponential type, if
\[
sup
N
∑
k
=
1
∞
(
∑
k
N
>
|
j
|
≤
(
k
+
1
)
N
|
λ
j
|
)
2
>
∞
.
\sup _N\sum _{k=1}^{\infty }\left (\sum _{kN>|j|\le (k+1)N}|\lambda _j|\right )^2>\infty .
\]
This is a generalization of a multiplier theorem on usual Fourier expansions in the Hardy space
H
1
(
T
)
H^1(\mathbb {T})
, and for
λ
n
=
(
|
n
|
+
1
)
−
1
\lambda _n=(|n|+1)^{-1}
, a Hardy type inequality for Jacobi expansions is immediate which has ever been proved by Kanjin and Sato [Math. Inequal. Appl. 7 (2004), pp. 551–555].