On Various R-duals and the Duality Principle

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.


Introduction
A countable collection of vectors {f i } i∈I in a separable Hilbert space H is a frame for H with (frame) bounds A, B if A and B are strictly positive constants and the inequalities hold for all f ∈ H. Frames play an increasing role in analysis and applications, mainly due to the fact that frames yield expansions of the elements in the Hilbert space of a similar type as the one that is known for orthonormal bases. In fact, if {f i } i∈I is a frame for H, the frame operator S : H → H, Sf := i∈I f, f i f i , is known to be invertible, and  (1.1). For so-called Gabor systems in L 2 (R) (see the description below), the duality principle [5,11,12] states that the frame condition is equivalent to a Riesz basis condition on an associated sequence (the adjoint Gabor system), see Theorem 1.4; this leads to a method to check the frame condition for Gabor systems in an (at least conceptually) easier way.
In an attempt to extend this to general sequences in arbitrary Hilbert spaces, Casazza, Kutyniok and Lammers introduced the R-duals in the paper [1]. The R-duals also yield a method for checking the frame condition for a sequence of vectors by checking the Riesz basis condition for a related sequence. At present it is not known whether the theory for R-duals yields a generalization of the duality principle. In [13] the authors introduced certain variations of the R-duals (see Definition 1.1) and showed that R-duals of type II cover the duality principle for integer-oversampled Gabor systems leaving open the general case, while R-duals of type III generalize the duality principle and keep some of the attractive properties of the R-duals, but not all. In the current paper we show that R-duals of type II in fact do not generalize the duality principle for arbitrary Gabor frames. This brings the attention to the R-duals of type III and we determine a sub-class of the R-duals of type III, which possesses the missing properties. We also provide further insight into the various R-duals by providing characterizations in the special case where the given frame is a basis.
In the rest of this introduction we state the key definitions and results from the literature concerning the R-duals. In Sect. 3 we introduce the modified R-duals of type III and prove that they generalize the duality principle and keep the main properties known from the Gabor case. The special case of Riesz bases is analysed in Sect. 4. We refer to the monographs [2,9,10] for detailed treatments of frames and further references.
For a sequence {f i } i∈I satisfying at least the upper frame condition, the analysis operator is defined by and the synthesis operator is In this case S is a bijection on span{f i } i∈I and the sequence {S −1 f i } i∈I is a frame for span{f i } i∈I satisfying the representation formula (1.5) (iv) [14] When {e i } i∈I and {h i } i∈I are Riesz bases for H, the R-dual of type and there exists an antiunitary transformation Λ : In the discussion of the duality principle we need the definition of a Gabor system. Consider the Hilbert space L 2 (R). For p, q ∈ R, let T p : . Given parameters a > 0, b > 0 and g ∈ L 2 (R), the associated Gabor system is the sequence {E mb T na g} m,n∈Z ; the adjoint Gabor system is the sequence The duality principle, due to Janssen [11], Daubechies et al. [5], and Ron and Shen [12], states the following: Theorem 1.4. [5,11,12] Let g ∈ L 2 (R) and a, b > 0 be given. Then the Gabor It is well known that if a Gabor system {E mb T na g} m,n∈Z is a Riesz basis, then ab = 1; via the duality principle this implies that the Gabor system  [1], the answer is affirmative for tight Gabor frames and Gabor Riesz bases). This is the motivation for the introduction of the other types of R-duals in [13]. In fact, in [13] it was shown that the R-duals of type III generalize the duality principle for all Gabor systems. However, R-duals of type III do not enjoy all of the attractive properties of the duality principle, so it is natural and necessary to search for subclasses which are in closer correspondence with the duality principle. In the present paper we determine a relevant subclass of the R-duals of type III which both extends the duality principle and has the desired properties as in Theorem 1.2.

R-duals of Type II
In this section we solve one of the remaining problems in [13], by showing that the R-duals of type II do not generalize the duality principle. This will motivate the analysis in the rest of the paper, where we focus on a subclass of the R-duals of type III having particular properties.
Example. Consider the B-spline B 2 determined by Denote ω m,n := 1 √ ab E m/a T n/b B 2 , m, n ∈ Z. We will show that {S −1/2 ω m,n } m,n∈Z is not an orthonormal sequence, which by Lemma 1.3(iv) will imply that {ω m,n } m,n∈Z is not an R-dual of type II of {E mb T na g} m,n∈Z . For m = 0 and n = 1, we have Since  7 2 , it follows that Therefore, {ω m,n } m,n∈Z is not an R-dual of type II of {E mb T na g} m,n∈Z .

R-duals of Type III
The key motivation behind the definition of R-duals of type III is that they generalize the duality principle [13], in the sense that whenever {E mb T na g} m,n∈Z is a frame for L 2 (R), the system { 1 √ ab E m/a T n/b g} m,n∈Z can be realized as an R-dual of type III of {E mb T na g} m,n∈Z . However, not all R-duals of type III have exactly the same properties as encountered in the duality principle. For example, for a frame {f i } i∈I with frame operator S, the optimal frame bounds are 1 S −1 , S ; these numbers are also bounds for the R-duals of type III, but not necessarily the optimal bounds. This calls for an identification of a subclass of the R-duals of type III with properties that better match what we know from the duality principle.
As starting point we will now determine conditions on the operator Q in (1.5) which are necessary and sufficient for an R-dual of type III of {f i } i∈I to keep the optimal bounds of {f i } i∈I . Proposition 3.1. Let {f i } i∈I be a frame for H (resp. Riesz sequence in H) with frame operator S and analysis operator U, both considered as operators on span{f i } i∈I , and let {ω j } j∈I be an R-dual of type III of {f i } i∈I with respect to the triplet ({e i } i∈I , {h i } i∈I , Q). Then the following are equivalent: (i) The Riesz sequence (resp. the frame) {ω j } j∈I has the same optimal bounds as {f i } i∈I . (ii) The operator Q has the property ⎧ ⎨ Proof. Notice that when {f i } i∈I is a frame for H (resp. Riesz sequence), [13,Prop. 4.3] shows that {ω j } j∈I is a Riesz sequence (resp. frame for H), with bounds 1/ S −1 , S . We first consider the case where {f i } i∈I is assumed to be a frame for H. (i) ⇒ (ii) Assume that 1 S −1 and S are the optimal bounds of {ω j } j∈I . We will prove that (3.1) holds. Since Q ≤ ||S|| and Q −1 ≤ ||S −1 ||, it follows that the inequalities of (3.1) hold for all {d i } i∈I ∈ 2 and in particular Vol. 84 (2016) On Various R-duals and the Duality Principle 583 whenever {d i } i∈I ∈ R(U ); it remains to prove the optimality of the bounds of (3.1). Assume that there exists B 1 < S so that Then for every finite scalar sequence {c j }, taking This implies that B 2 1 is an upper bound of the Riesz sequence {ω j } j∈I , which contradicts the assumptions. Therefore, S is the optimal upper bound in (3.1). In a similar way, it follows that 1/ S −1 is the optimal lower bound in (3.1). (ii) ⇒ (i) Now assume that (3.1) holds; we will prove that 1/ S −1 and S are the optimal bounds of the Riesz sequence {ω j } j∈I . As already mentioned, 1/ S −1 and S are bounds of the Riesz sequence {ω j } j∈I , so it remains to prove their optimality. Assume that the optimal upper bound of {ω j } j∈I is B 2 with B 2 < S . Then for every finite scalar sequence {c j }, we have Since the set of (finite) linear combinations j c j e j is dense in H, it follows that for every y ∈ H. Since S −1/2 is bijective and self-adjoint, it follows that  1). Therefore, S is the optimal upper bound of {ω j } j∈I . In a similar way, it follows that 1/ S −1 is the optimal lower bound of {ω j } j∈I . Now let {f i } i∈I be a Riesz sequence in H. In this case R(U ) = 2 ; since the optimal bounds in the inequalities C||x|| ≤ ||Qx|| ≤ D||x||, x ∈ H, are C = ||Q −1 || −1 , D = ||Q||, the condition (3.1) means precisely that Q = ||S|| and Q −1 = ||S −1 ||. An argument as in the proof of [13,Prop. 4.3(ii)] shows that {ω j } j∈I is a frame with optimal bounds (ii) Assume that {f i } i∈I is tight. Then the classes of R-duals of type I and type III of {f i } i∈I coincide [13,Prop. 4.2]. Now the statement follows from (i).
(iii) Assume that {f i } i∈I is not tight and let A and B denote the optimal bounds of {f i } i∈I , A < B. Take any constant C ∈ ( √ A, √ B) and let Q := C Id H . Let {ω j } j∈I be an R-dual of type III with respect to some orthonormal bases {e i } i∈I , {h i } i∈I and the operator Q. Then {C −1 ω j } j∈I is an R-dual of type I of {S −1/2 f i } i∈I , which by Theorem 1.2 implies that {C −1 ω j } j∈I is an orthonormal sequence. Therefore, {ω j } j∈I is a tight Riesz sequence with bound C 2 ∈ (A, B), which by Theorem 3.3(i) implies that {ω j } j∈I can not be written as an R-dual of type III with property (3.1).
In [13] we have proved that canonical dual frames lead to biorthogonality of appropriately determined R-duals of type III. Here we provide further insight in the relations considering the converse situation, namely, biortogonality of appropriate R-duals of type III leading to canonical dual frames.
whenever {d i } i∈I ∈ R(U ), with optimality of the bounds.