Dual Gramian analysis: Duality principle and unitary extension principle

Abstract

Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames [40] and the unitary extension principle for wavelet frames [38]. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in [38], e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.

Duality for Frames

“…Dual Gramian analysis and the duality principle therefore are available to study frames also on the abstract infinite dimensional level. In [52] systems in abstract Hilbert spaces are studied by means of the pre-Gramian matrix, representing the synthesis operator, and by the Gramian and dual Gramian matrices, which represent the composition of the synthesis operator and the analysis operator in different orders. Adjoint systems are introduced for a given system by considering an adjoint (columnrow) relationship of the respective pre-Gramian matrices and complement the study of the original system through the duality principle in the exact spirit of the results for shift-invariant systems in the function spaces: The dual Gramian of the system is the Gramian of its adjoint system.…”

“…This can be considered the core statement of the duality principle of abstract frame theory. It is underlying the abstract duality principle previoulsy formulated in [19] and offers applications well beyond Gabor analysis, for example a new perspective on the unitary extension principle, which leads to a simple construction scheme for MRA-based multivariate tight wavelet frames proposed in [52]. In this paper, we move on from the study of the frame properties of a single system as in [52], to the study of dual pairs of systems in separable Hilbert spaces.…”

“…It is underlying the abstract duality principle previoulsy formulated in [19] and offers applications well beyond Gabor analysis, for example a new perspective on the unitary extension principle, which leads to a simple construction scheme for MRA-based multivariate tight wavelet frames proposed in [52]. In this paper, we move on from the study of the frame properties of a single system as in [52], to the study of dual pairs of systems in separable Hilbert spaces. While for a given frame one always has its pre-image under its frame operator as a canonical dual frame at one's disposal, in concrete situations it can be desirable to consider other dual frame pairs.…”

“…The analysis of the mixed frame operator of two systems, which we therefore turn to, differs from that of a single system, for instance, in that the mixed frame operator is no longer self-adjoint. We extend the matrix notions and results of [52] to cover the case of pairs of systems by introducing the mixed dual Gramian matrices. Our most central result remains the abstract duality principle, part of which characterizes dual frame pairs in terms of biorthogonality relations of the adjoint system counterparts.…”

“…In particular, the mask may contain negative entries. This flexibility cannot be facilitated by the related tight wavelet frame construction proposed in [52]. We outline several examples of multivariate dual wavelet frames derived from our construction.…”