The notion of bi-inner product functionals P(f,g)=Σgenerated by two Bessel sequences {fn} and {gn} of functions from L~2 was introduced in our earlier work~[5] as a vehicle to identify dual frames and bi-orthogonal Resz bases of L~2. The objective was to find conditions under which P is a constant multiple of the inner produceof L~2. A necessary and sufficient condition derived in [5] is that P is both spatial shift-invariant and phase shift-invariant. Although these two shift-invariance proper- ties are, in general, unrelated, it could happen that one is a consequence of the other for certain classes of Bessel sequences {fn} and {gn}. In this paper, we show that, indeed, for localized cosines with two-over- Lapping windows (i.e., only adjacent window functions are allowed to overlap), spatial shift-invariance of P is already sufficient to guarantee that P is a constant multiple of the inner product, while phase shift-in- Variance is not. Hence, phase shift-invariance of P for two-overlapping localized cosine Bessel sequences is A consequence of spatial shift-invariance, but the converse is not valid. As an application, we also show that Two families of localized cosines with uniformly bounded and two-overlapping windows are bi-orthogonal Riesz bases of L~2, if and only if P is spatial shift-invariant. In addition, we apply this result to generalize A result on characterization of dual localized cosine bases on our earlier work in [3] to the multivariate set- Ting. A method for computing the dual windows is also given in this paper.
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