A note on a theorem of Sparr

Abstract

We prove that, regardless of the choice of a positive, concave function $\psi$ on $\mathbf{R}_{+}$ and a "weight function" $\lambda$, the weighted $\ell_2$-space $\ell_2(\psi(\lambda))$ is $c$-interpolation with respect to the couple $(\ell_2,\ell_2(\lambda))$, where $c\leq\sqrt{2}$. Our main result is that $c=\sqrt{2}$ is best possible here; a fact which is implicit in the work of G. Sparr.