A note on a basis problem

Abstract

It is shown that the functions { exp − λ ν x } ν = 1 ∞ \{ \exp - {\lambda _\nu }x\} _{\nu = 1}^\infty form a basis for the subspace of L 2 ( 0 , ∞ ) {\mathcal {L}_2}(0,\infty ) which they span if and only if \[ inf μ ∏ ν = 1 ; ν ≠ μ ∞ | λ ν − λ μ λ ¯ ν + λ μ | = δ > 0. \inf \limits _\mu \prod \limits _{\nu = 1;\nu \ne \mu }^\infty {|\frac {{{\lambda _\nu } - {\lambda _\mu }}}{{{{\bar \lambda }_\nu } + {\lambda _\mu }}}| = \delta > 0.} \] The proof uses certain estimates concerning interpolation in H 2 {H_2} due to Shapiro and Shields. The proof makes explicit a construction embedded in a paper of Binmore [1, Theorems 9-12.].