The notion of bi-inner product functionals P(f ,g) ＝ ∑ ＜ f ,f. ＞＜ g ,g. ＞ generated by two Besselrn rnseqnsences {fn} and {gn} of functions from L2 was introduced in our earlier work[5] as a vehicle to identify rndual frames and bi-orthogonal Riesz bases of L2. The objective was to find conditions under which P is a rnconstant multiple of the inner product ＜f ,g ＞ of L2. A necessary and sufficient condition derived in [5]isrnthat P is both spatial shift-invariant and phase shift-invariant. Although these two shift-invariance properrnties are, in general, unrelated, it could happen that one is a consequence of the other for certain clases of rnBessel sequences {fn} and {gn}. In this paper, we show that, indeed, for localized cosines with two-overrnlapping windows (i. e. , only adjacent window functions are allowed to overlap ) , spatial shift-inrvariance of rnP is already sufficient to guarantee that P is a constant multiple of the inner product, while phase shift-inrnvariance is not. Hence, phase shift-invariance of P for two-overlapping localized cosine Bessel sequences is rna consequence of spatial shift-invariance, but the eonverse is not valid. As an application, we also show thattwo families of localized cosines with uniformly bounded and two-overlapping windows are bi-orthogonalrnRiesz bases of L2, if and only if Pis spatial shi ft-invariant. In addition, we apply this result to generalize rna result on characterization of dual localized codne bases in our earlier work in [3] to the multivariate set-rnting. A method for computing the dual windows is also given in this paper.
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