Shift-invariant bi-inner product functionals are inner products

Bi-inner product functionals generated by a pair of Bessel sequences of L² functions are introduced. It is shown that these functionals are constant multiples of the inner products of L² and l², if and only if they are shift-invariant both in space (or time) and in phase. This result is then applied to characterize dual frames and bi-orthogonal Riesz bases of L². The research of this author was supported by NSF Grant # DMS-95-05460 and ARO Contract #DAAH-04-95-10193, This author is currently visiting the Department of statistics, Stanford University, Stanford, CA 94305. The research of this author was supported by the Texas Coordinating Board of Higher Education under Grant # 999903-109.