The authors develop the Figa-Talamanca-Herz algebras and the space of \(p\)-pseudomeasures to inverse semigroups with restricted semigroup algebras. Let \(1 < p, q < \infty\) be such that \(\frac{1}{p}+\frac{1}{q}=1\). The Banach algebra of \(p\)-pseudomeasures \(PM_{p} (S)\) and the Figa-Talamanca-Herz algebras \(A_{q} (S)\) are defined and it is shown that \(A_{q} (S)^*= PM_{p} (S)\). Moreover, \(PM_{p} (S)\), \(A_{q} (S)\) are characterized for the Clifford semigroup in the sense of \(p\)-pseudomeasures and Figa-Talamanca-Herz algebras of maximal subgroups of \(S\), respectively. Also, it is shown that the character space of \(A_{q} (S)\) is equal to \(S\) for a Clifford semigroup \(S\). As an example of these Banach algebras and restricted semigroup algebras, the authors discuss locally finite inverse semigroups.