A note on a basis problem

Abstract:

It is shown that the functions $\{ \exp - {\lambda _\nu }x\} _{\nu = 1}^\infty$ form a basis for the subspace of ${\mathcal {L}_2}(0,\infty )$ which they span if and only if \[ \inf \limits _\mu \prod \limits _{\nu = 1;\nu \ne \mu }^\infty {|\frac {{{\lambda _\nu } - {\lambda _\mu }}}{{{{\bar \lambda }_\nu } + {\lambda _\mu }}}| = \delta > 0.} \] The proof uses certain estimates concerning interpolation in ${H_2}$ due to Shapiro and Shields. The proof makes explicit a construction embedded in a paper of Binmore [1, Theorems 9-12.].