Modeling sampling in tensor products of unitary invariant subspaces

The use of unitary invariant subspaces of a Hilbert space $\mathcal{H}$ is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of $L^2(\mathbb{R})$ and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature, it also allows to introduce the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both, finite/infinite dimensional cases.

In a recent paper [18] it was shown how to extend sampling reconstruction theorems to invariant subspaces of a separable Hilbert spaces under a unitary representation of finite groups which are semidirect products with an Abelian factor. This setting being appropriate for applications of the theory beyond the domain of classical telecommunications to quantum physics.
In this paper we go one step ahead by enlarging the class of target spaces for sampling: we deal with tensor products of different unitary invariant subspaces. This situation corresponds for instance to consider multichannel systems in classical telecommunications or composite systems in the case of quantum applications. Thus, in this setting, we are able to gather problems of diverse nature by means of a simple formalism involving tensor products and tensor operators in Hilbert spaces. Namely, we first consider an infinite U -unitary subspace A a = n∈Z a n U n a : {a n } ∈ ℓ 2 (Z) in a Hilbert space H 1 , and a finite V -unitary subspace in a Hilbert space H 2 , to finally obtain sampling formulas in its tensor product A a,b := A a ⊗ A b .
Apart from tensor products in Hilbert spaces, the paper involves the theory of frames. Concretely, in this situation, the generalized samples will be expressed as frame coefficients in an auxiliary Hilbert space L 2 (0, 1) ⊗ ℓ 2 N (Z), where ℓ 2 N (Z) denotes the space of all N -periodic complex sequences. Continuing the line of inquiry of [12,16], the problem reduces to find appropriate families of dual frames. By 'appropriate" we mean that these dual frames have a nice structure taking care of the unitary invariance of the involved sampling subspaces as it will be discussed in the sequel.
Later on, the infinite-infinite and finite-finite generator cases will be considered too. That is, the situation where both invariant subspaces are generated by a sequence of vectors U n a, V p b, n, p ∈ Z, and the simpler case where both subspaces are finite dimensional. Relevant examples of each situation will be discussed in detail.
The paper is organized as follows: For the sake of completeness we include in Section 2 the basics of frames and tensor products needed in the sequel. In Section 3 we focus on the above case that we call infinite-finite generators case; that is, we establish sampling formulas in the tensor product of two unitary invariant subspaces, one of them A a with an infinite generator a and the other one A b finitely generated. First, we obtain appropriate expressions for the samples of any x ∈ A a,b obtained from S = ss ′ systems L jj ′ acting on A a,b , where n ∈ Z, m = 0, 1, . . . , ℓ − 1, j = 1, 2, . . . , s and j ′ = 1, 2, . . . , s ′ . Here, h j1 , j = 1, 2, . . . , s, denote s fixed elements in H 1 and h j ′ 2 , j ′ = 1, 2, . . . , s ′ , denote s ′ fixed elements in H 2 , ; r andr the sampling periods, where r ∈ N andr is a divisor of N and ℓ := N/r (see Section 3 for the details). Then we state the suitable isomorphism T U V a,b between L 2 (0, 1) ⊗ ℓ 2 N (Z) and A a,b which allows to transform the derived frame expansions in L 2 (0, 1) ⊗ ℓ 2 N (Z) into stable sampling formulas in A a,b having the form where c j ⊗ d j ′ ∈ A a,b , j = 1, 2, . . . , s and j ′ = 1, 2, . . . , s ′ . We conclude the section giving a representative example arising from classical tomography (see, for instance, Refs. [24,27]). Sections 4 and 5 deal with the called infinite-infinite and finite-finite cases. They mimic the structure of Section 3 with auxiliary spaces L 2 (0, 1) ⊗ L 2 (0, 1) and Finally, it is worth to mention that for the sake of simplicity we only deal with the tensor product of two single generated unitary invariant subspaces; the same results apply for the tensor product of any finite number of multiple generated unitary invariant subspaces.

A brief on frames and tensor products
The frame concept was introduced by Duffin and Shaeffer in [8] while studying some problems in nonharmonic Fourier series; some years later it was revived by Daubechies, Grossman and Meyer in [7]. Nowadays, frames have become a tool in pure and applied mathematics, computer science, physics and engineering used to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. Recall that a sequence {x n } is a frame for a separable Hilbert space H if there exist two constants A, B > 0 (frame bounds) such that A sequence satisfying only the right-hand inequality is said to be a Bessel sequence for H. Given a frame {x n } for H the representation property of any vector x ∈ H as a series x = n c n x n is retained, but, unlike the case of Riesz bases, the uniqueness of this representation (for overcomplete frames) is sacrificed. Suitable frame coefficients c n which depend continuously and linearly on x are obtained by using dual frames {y n } of {x n }, i.e., {y n } is another frame for H such that for each Recall that a Riesz basis in a separable Hilbert space H is the image of an orthonormal basis by means of a boundedly invertible operator; it is a particular case of frame: the so called exact frame. Any Riesz basis {x n } has a unique biorthonormal (dual) Riesz basis {y n }, i.e., x n , y m H = δ n,m , such that expansion (1) holds for every x ∈ H. A Riesz sequence is a Riesz basis for its closed span. For more details on frames and Riesz bases theory see, for instance, the monograph [5] and references therein; see also Ref. [4] for finite frames.
Traditionally, frames were used in signal and image processing, nonharmonic analysis, data compression, and sampling theory, but nowadays frame theory plays also a fundamental role in a wide variety of problems in both pure and applied mathematics, computer science, physics and engineering. The redundancy of frames, which gives flexibility and robustness, is the key to their significance for applications; see, for instance, the nice introduction in Chapter 1 of Ref. [4] and the references therein.
Next we briefly recall some basic facts about tensor products of Hilbert spaces which will be useful in the current work. Let H 1 , H 2 be two Hilbert spaces. Among the different ways of constructing tensor product spaces we adopt the model proposed in [9]. There, the tensor product H 1 ⊗ H 2 is defined as the space of all antilinear maps A : H 2 −→ H 1 such that i Ae i 2 < ∞ for some orthonormal basis of {e i } i of H 2 . As for every A, B ∈ H 1 ⊗H 2 the series i Ae i 2 and i Ae i , Be i are independent of the orthonormal basis {e i } i for H 2 , then H 1 ⊗ H 2 can be turned into an inner product space by defining the norm A 2 = i Ae i 2 (and the associated inner product A, B = i Ae i , Be i ). Indeed H 1 ⊗ H 2 endowed with this inner product becomes a Hilbert space.
If u ∈ H 1 and v ∈ H 2 the tensor product u ⊗ v ∈ H 1 ⊗ H 2 is defined to be the rank one map such that (u ⊗ v)(w) = v, w u for every w ∈ H 2 .
Let B(H 1 ), B(H 2 ) and B(H 1 ⊗H 2 ) denote the spaces of all bounded linear operators on H 1 , H 2 and H 1 ⊗ H 2 , respectively. If S ∈ B(H 1 ) and T ∈ B(H 2 ) the tensor product For further information on tensor products of Hilbert spaces see [3,9,20,22]. In these sources are to be found the following results for tensor products needed in the sequel: • u⊗v = u v and u⊗v, u ′ ⊗v ′ = u, u ′ v, v ′ for any u, u ′ ∈ H 1 and v, v ′ ∈ H 2 .
• The tensor product of two orthonormal bases is an orthonormal basis.
• The operator S ⊗ T is invertible in B(H 1 ⊗ H 2 ) if and only each operator, S and T , is invertible in B(H 1 ) and B(H 2 ) respectively.
• The tensor product of two sequences is a Riesz basis for H 1 ⊗ H 2 if and only if each sequence is a Riesz basis for its corresponding Hilbert space.
• The tensor product of two Bessel sequences is a Bessel sequence for the corresponding Hilbert space.
• The tensor product of two sequences is a frame for H 1 ⊗H 2 if and only if each sequence is a frame for its corresponding Hilbert space.
Note that the fact that H 1 ⊗ H 2 is the completion of the linear span of D := {u ⊗ v : u ∈ H 1 , v ∈ H 2 } yields that any operator of B(H 1 ⊗ H 2 ), in particular the tensor product S ⊗ T ∈ B(H 1 ⊗ H 2 ), is determined by its values on D.

Infinite-finite generators case
Let H 1 , H 2 be two separable Hilbert spaces, and U : H 1 −→ H 1 , V : H 2 −→ H 2 two unitary operators. Consider two elements a ∈ H 1 and b ∈ H 2 such that the sequence {U n a} n∈Z forms a Riesz sequence in H 1 (see [12,Theorem 2.1] for a necessary and sufficient condition), and there exists an N ∈ N such that [16,Proposition 1] for a necessary and sufficient condition). In the tensor product Hilbert space H 1 ⊗ H 2 we consider the closed subspace Since the sequence U n a ⊗ V p b n∈Z; p=0,1,...,N −1 is a Riesz basis for the tensor product , and it can be described as We will refer to the vectors {a, b} as the infinite-finite generators of the subspace A a,b in H 1 ⊗ H 2 .
Let ℓ 2 N (Z) denote the space of all N -periodic sequences with inner product x, y ℓ 2 N = N −1 p=0 x(p) y(p), and its canonical basis as {e p } N −1 p=0 . This space is isomorphic to the euclidean space C N ; along the paper we will identify sequences in ℓ 2 N (Z) with vectors in C N : any vector in C N defines the terms from 0 to N − 1 of the corresponding sequence in ℓ 2 N (Z).
The following shifting property will be crucial later in obtaining our sampling formulas: Proof. Indeed, having in mind the shifting properties: , and the properties of the tensor product of operators we have

An expression for the samples
A suitable expression for the samples given in (2) will allow us to obtain the reconstruction conditions of any x in the subspace A a,b from the sequence of its samples (2).
Thus, Plancherel identity for orthonormal bases gives That is, for n ∈ Z, m = 0, 1, . . . , ℓ − 1, j = 1, 2, . . . , s and j ′ = 1, 2, . . . , s ′ we have got the following expression for the samples: belong to L 2 (0, 1) (recall that {U n a} n∈Z is a Riesz sequence for H 1 ), and denotes the (N -periodic) cross-covariance sequence between the sequences {V q b} q∈Z and {V q h j ′ 2 } q∈Z in H 2 (see [21]). Thus, we deduce the following result: . Having in mind that a tensor product of two sequences is a frame in the tensor product if and only if the respective factors are frames, we only need a characterization of the sequences g j (x) e 2πirnx n∈Z; j=1,2,...,s and G j ′ ,m m=0,1,...,ℓ−1 j ′ =1,2,...,s ′ as frames for L 2 (0, 1) and ℓ 2 N (Z) respectively. This study has been done in Refs. [14,16] respectively. • For the first one, consider the s × r matrix of functions in L 2 (0, 1) and its related constants where G * (x) denotes the transpose conjugate of the matrix G(x), and λ min (respectively λ max ) the smallest (respectively the largest) eigenvalue of the positive semidefinite matrix Notice that in the definition of the matrix G(x) we are considering 1-periodic extensions of the involved functions g j , j = 1, 2, . . . , s.
A complete characterization of the sequence g j (x) e 2πirnx n∈Z; j=1,2,...,s as a frame or a Riesz basis for L 2 (0, 1) is given in next lemma (see [14,Lemma 3] or [15,Lemma 2] for the proof): Lemma 2. For the functions g j ∈ L 2 (0, 1), j = 1, 2, . . . , s, consider the associated matrix G(x) given in (5). Then, the following results hold: In this case, the optimal frame bounds are α G /r and β G /r. • For the second one, consider the sℓ × N matrix of cross-covariances In Ref. [16] it was proved that: spanning set since we are in finite dimension) for ℓ 2 N (Z) if and only rank R b,h 2 = N . Notice that, necessarily, s ≥ r and s ′ ≥r and, consequently, the number S = ss ′ of needed U V -systems L jj ′ must be S ≥ rr.
All the possible choices in (7) are given by the first row of the r × s matrices given by U(x) is any r × s matrix with entries in L ∞ (0, 1), and I s is the identity matrix of order s (see [25]). Notice that the entries of G † (x) are essentially bounded in (0, 1) since the functions g j , j = 1, 2, . . . , s, and det −1 G * (x) G(x) are essentially bounded in (0, 1).
• For the second case, the N -periodic extensions of the columns H j ′ ,m m=0,1,...,ℓ−1  [16] for the details about the construction of H S ).
All the possibler × s ′ ℓ matrices S satisfying (8) are given by the firstr rows of any any left-inverse H of the matrix R b,h 2 . All these left-inverses can be expressed as Proof. Let T k N and G N , k = 1, 2 and N ∈ N, be the bounded operators defined by for every N ∈ N and every h k ∈ H k , k = 1, 2, then, (T 1 As a consequence, for each x ∈ A a,b there exists a unique F ∈ L 2 (0, 1) This F can be expressed as the frame expansion Then, applying the isomorphism T U V a,b and the shifting property in Lemma 1 (here it is the point where we are using that the proposed dual frames are convenient for sampling purposes) one gets . . , s ′ . Collecting the pieces we have obtained until now we prove the following result: . . , s, j ′ = 1, 2, . . . , s ′ , and let L jj ′ be the associated U V -system giving the samples of any x ∈ A a,b as in (2), j = 1, 2, . . . , s, j ′ = 1, 2, . . . , s ′ . Assume that the function g j , j = 1, 2, . . . , s, given in (4) belongs to L ∞ (0, 1); or equivalently, that β G < ∞ for the associated s × r matrix G(x) defined in (5). The following statements are equivalent: . . , s ′ , such that the sequence U rn ⊗ Vr m (c j ⊗ d j ′ ) n∈Z; m=0,1,...,ℓ−1 j=1,2,...,s; j ′ =1,2,...,s ′ is a frame for A a,b , and for any x ∈ A a,b the expansion holds.
In case s = r and s ′ =r we are in the Riesz bases setting: see statement (d) in Lemma 2; moreover, the square matrix R b,h 2 must be invertible (see [16,Corollary 4]). In fact, the following corollary holds: Corollary 6. In addition to the hypotheses of theorem above, assume that s = r and s ′ =r. The following statements are equivalent: 1. α G > 0 and the square matrix R b,h 2 is invertible.
Proof. The uniqueness of the expansion with respect to a Riesz basis gives the stated interpolation property (10).
Under the hypotheses in Theorem 5 there will exist functions S j ∈ A ϕ , j = 1, 2, . . . , s, and Θ j ′ ∈ A ψ , j ′ = 1, 2, . . . , s ′ , such that for any f ∈ A ϕ,ψ the sampling expansion (9) reads: In this particular example, adding some mild hypotheses we can also derive pointwise convergence in the above sampling formula. Indeed, assuming that the generators ϕ, ψ are continuous functions on R such that n∈Z |ϕ(x − n)| 2 is bounded on [0, 1] and since N −1 m=0 |ψ(y − m/N )| 2 is bounded on [0, 1/N ], it is easy to deduce that any function f in A ϕ,ψ is a continuous function defined by the pointwise sum Besides, the subspace A ϕ,ψ is a reproducing kernel Hilbert space (RKHS) since the evaluation functionals are bounded in A ϕ,ψ . Namely, using Cauchy-Schwarz's inequality in (11) and Riesz basis definition, for each (x, y) ∈ R × [0, 1) we have where A denotes the lower Riesz bound of the Riesz basis ϕ(x − n) ψ(y − m/N ) n,m for A ϕ,ψ . The above inequality shows that convergence in L 2 R × (0, 1) implies pointwise convergence which is uniform on R × [0, 1). See, for instance, Ref. [17] for the relationship between sampling theory and RKHS's.

Infinite-infinite generators case
Since the sequence U n a ⊗ V m b n,m∈Z is a Riesz basis for the tensor product A a ⊗ A b of the U -invariant subspace A a = n∈Z a n U n a : {a n } ∈ ℓ 2 (Z) in H 1 and the We will refer to the vectors {a, b} as the infinite-infinite generators of the subspace A a,b in H 1 ⊗ H 2 .
In this case we introduce the isomorphism T U V a,b which maps the orthonormal basis e 2πinx ⊗ e 2πimx n,m∈Z for the Hilbert space L 2 (0, 1) ⊗ L 2 (0, 1) onto the Riesz basis U n a ⊗ V m b n,m∈Z for A a,b . That is, Here, the shifting property reads: where g, g ∈ L 2 (0, 1) and N, M ∈ Z. The proof of (13) goes in the same manner as in Lemma 1.

holds.
In case s = r and s ′ =r we are in the Riesz bases setting and theorem above admits the following corollary: Corollary 8. In addition to the hypotheses of theorem above, assume that s = r and s ′ =r. The following statements are equivalent: 2. There exist rr unique elements c j ⊗ d j ′ in the subspace A a,b , j = 1, 2, . . . , r and j ′ = 1, 2, . . . ,r, such that the sequence U rn ⊗ V rm (c j ⊗ d j ′ ) n∈Z; j=1,2,...,r m∈Z; j ′ =1,2,...,r is a Riesz basis for A a,b , and the expansion of any x ∈ A a,b with respect to this basis is In case the equivalent conditions are satisfied, the vectors c j ⊗ d j ′ , j = 1, 2, . . . , r and j ′ = 1, 2, . . . ,r, satisfy the interpolation property whenever n, m ∈ Z, j, k = 1, 2, . . . , r and j ′ , k ′ = 1, 2, . . . ,r.

A representative example
In . Let ϕ, ψ ∈ L 2 (R) be two functions such that the sequences ϕ(x − n) n∈Z and ψ(y − m) m∈Z are Riesz sequences for L 2 (R) (for instance, the functions ϕ, ψ may be two B-splines).
In the tensor product L 2 (R)⊗L 2 (R) = L 2 (R 2 ) consider the closed subspace A ϕ,ψ := A ϕ ⊗ A ψ , i.e., the tensor product of the shift-invariant subspaces A ϕ and A ψ of L 2 (R). Given the functions h j1 , h j ′ 2 ∈ L 2 (R), j = 1, 2, . . . , s and j ′ = 1, 2, . . . , s ′ , and fixing the sampling periods r andr in N, for each f in A ϕ,ψ we consider its samples defined by Under the hypotheses in Theorem 7 there will exist functions S j ∈ A ϕ , j = 1, 2, . . . , s and S j ′ ∈ A ψ , j ′ = 1, 2, . . . , s ′ , such that for any f ∈ A ϕ,ψ the sampling expansion (16) reads: As in the example of the infinite-finite case, assuming that the generators ϕ, ψ are continuous functions on R such that the sums n∈Z |ϕ(x − n)| 2 and m∈Z |ψ(y − m)| 2 are bounded on [0, 1], it is easy to deduce that A ϕ,ψ is a RKHS of continuous functions on R 2 . Furthermore, the convergence in L 2 R 2 implies pointwise convergence which is uniform on R 2 .
The subspace A a,b coincides with the tensor product A a ⊗ A b of the finite subspaces We will refer to the vectors {a, b} as the finite-finite generators of the subspace A a,b in
As in the former cases, now we introduce the isomorphism T U V a,b which maps the orthonormal basis e p ⊗ e q p=0,1,...,N −1 Here, the shifting property reads: belong to ℓ 2 M (Z) and 1 ≤ q ≤ M − 1. The proof of (18) goes in the same manner as in Lemma 1.

An expression for the samples
Thus, Plancherel identity for orthonormal bases gives That is, for n = 0, 1, . . . , ℓ − 1, m = 0, 1, . . . ,l − 1, j = 1, 2, . . . , s and j ′ = 1, 2, . . . , s ′ we have got the following expression for the samples: where G 1 j,n := and Here, R a,h j1 (k) := U k a, h j1 H 1 , k ∈ Z, denotes the (N -periodic) cross-covariance sequence between the sequences {U k a} k∈Z and The thesis of the above proposition is true if and only if the sℓ × N matrix R a,h 1 (defined in (6) for G 1 j,n ) has rank N and the s ′l × M matrix R b,h 2 (defined in (6) for G 2 j ′ ,m ) has rank M . Notice that, necessarily, s ≥ r and s ′ ≥ r ′ and ss ′ ℓl ≥ N M = dim ℓ 2 N (Z) ⊗ ℓ 2 M (Z) .

The sampling result
In case the sequence G 1 j,n ⊗G 2 Thus, for each x ∈ A a,b there exists a unique F ∈ ℓ 2 This F can be expressed as the frame expansion Then, applying the isomorphism T U V a,b and (18) one gets where c j = T U a,N H 1 j,0 ∈ A a , j = 1, 2, . . . , s, and . . , s ′ . Next, we state the equivalent result to Theorem 5: Theorem 9. Let h jj ′ := h j1 ⊗ h j ′ 2 ∈ H 1 ⊗ H 2 , j = 1, 2, . . . , s, j ′ = 1, 2, . . . , s ′ , and let L jj ′ be the associated U V -system giving the samples of any x ∈ A a,b as in (17), j = 1, 2, . . . , s, j ′ = 1, 2, . . . , s ′ . The following statements are equivalent: (a) rank R a,h 1 = N and rank R b,h 2 = M .
As in the previous cases, whenever s = r and s ′ =r in the above theorem we are in the Riesz bases setting necessarily, and condition (a) says that both square matrices R a,h 1 and R b,h 2 are invertible.

Discussion and conclusions
A sampling theory for tensor products of unitary invariant subspaces that allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature is derived. The involved samples are identified as frame coefficients in suitable tensor product spaces, thus it also allows to introduce the several variables case in the formalism.
Alternatively, the theory developed here can also be considered as a theory of invariant subspaces in the tensor product of the corresponding Hilbert space with respect to the canonical unitary tensor product representation of the product group Z ⊗ Z p defined from the corresponding factors (in the case of infinite-finite generators).
In this sense, the results exhibited here point again in the direction recently started in the paper [18] and natural generalizations of them, like sampling theorems for tensor products of invariant subspaces with respect to unitary representations of finite or infinite groups on each factor, can be addressed using similar ideas.
Reduction techniques corresponding to the decomposition of tensor products of unitary representations into their irreducible components with respect to proper subgroups of the product group, could also be used to simplify the construction of samples, i.e., to find adapted frames. This, as well as other applications beyond classical telecommunications all involving sampling of states of quantum systems, will be the subject of further research.