Sequence space representations for spaces of entire functions with rapid decay on strips

We obtain sequence space representations for a class of Fr'echet spaces of entire functions with rapid decay on horizontal strips. In particular, we show that the projective Gelfand-Shilov spaces ${\mathrm{\Sigma }}_{\nu }^1$ and ${\mathrm{\Sigma }}_1^{\nu }$ are isomorphic to ${\mathrm{\Lambda }}_{\mathrm{\infty }}(n^{1\mathrm{/}(\nu +1)})$ for $\nu >0$.