Beyond Frames: Semi-frames and Reproducing Pairs

Abstract

Frames are nowadays a standard tool in many areas of mathematics, physics, and engineering. However, there are situations where it is difficult, even impossible, to design an appropriate frame. Thus there is room for generalizations, obtained by relaxing the constraints. A first case is that of semi-frames, in which one frame bound only is satisfied. Accordingly, one has to distinguish between upper and lower semi-frames. We will summarize this construction. Even more, one may get rid of both bounds, but then one needs two basic functions and one is led to the notion of reproducing pair. It turns out that every reproducing pair generates two Hilbert spaces, conjugate dual of each other. We will discuss in detail their construction and provide a number of examples, both discrete and continuous. Next, we notice that, by their very definition, the natural environment of a reproducing pair is a partial inner product space (pip-space) with an L ² central Hilbert space. A first possibility is to work in a rigged Hilbert space. Then, after describing the general construction, we will discuss two characteristic examples, namely, we take for the partial inner product space a Hilbert scale or a lattice of L ^p spaces.

Appendix: Lattices of Banach or Hilbert Spaces

For the convenience of the reader, we summarize in this Appendix the basic facts concerning pip-spaces and operators on them. However, we will restrict the discussion to the simpler case of a lattice of Banach (LBS) or Hilbert spaces (LHS). Further information may be found in our monograph [6] or our review paper [7].

Let thus ${\mathcal J} = \{ V_p,\, p \in I \}$ be a family of Hilbert spaces or reflexive Banach spaces, partially ordered by inclusion. Then ${\mathcal I}$ generates an involutive lattice ${\mathcal J}$, indexed by J, through the operations (p, q, r ∈ I):

• involution:  V _r ↔ $V_{\overline {r}}= V_r ^\times$, the conjugate dual of V _r

• infimum:  V _p∧q :=  V _p ∧ V _q = V _p ∩ V _q

• supremum:  V _p∨q :=  V _p ∨ V _q = V _p + V _q.

It turns out that both V _p∧q and V _p∨q are Hilbert spaces, resp. reflexive Banach spaces, under appropriate norms (the so-called projective, resp. inductive norms). Assume that the following conditions are satisfied:

• (1) ${\mathcal I}$ contains a unique self-dual, Hilbert subspace $V_{o} =V_{\overline {o}}$.

• (2) for every $V_r\in {\mathcal I}$, the norm $\|\cdot \|{ }_{\overline {r}}$ on $V_{\overline {r}}=V_{r}^\times$ is the conjugate of the norm ∥⋅∥_r on V _r.

In addition to the family ${\mathcal J} =\{V_{r}, \,r\in J\}$, it is convenient to consider the two spaces V ^# and V  defined as

$$\displaystyle \begin{aligned} V = \sum_{q\in I}V_{q}, \quad V^{\#} = \bigcap_{q\in I}V_{q}. {} \end{aligned} $$

(A.1)

These two spaces themselves usually do not belong to ${\mathcal I}$. According to the general theory of pip-spaces [6], V  is the algebraic inductive limit of the V _p’s, and V ^# is the projective limit of the V _p’s.

We say that two vectors f, g ∈ V  are compatible if there exists $r \in J \mbox{ such that } f \in V_{r}, g \in V_{\overline {r}}$ . Then a partial inner product on V  is a Hermitian form 〈⋅|⋅〉 defined exactly on compatible pairs of vectors. In particular, the partial inner product 〈⋅|⋅〉 coincides with the inner product of V _o on the latter. A partial inner product space (pip-space) is a vector space V  equipped with a partial inner product. Clearly LBSs and LHSs are particular cases of pip-spaces.

We will assume that our pip-space (V, 〈⋅|⋅〉) is nondegenerate, that is, 〈f|g〉 = 0 for all f ∈ V ^# implies g = 0. As a consequence, (V ^#, V ) and every couple $(V_r , V_{\overline r} ), \, r\in J,$ are a dual pair in the sense of topological vector spaces [29]. In particular, the original norm topology on V _r coincides with its Mackey topology $\tau (V_{r},V_{\overline {r}})$, so that indeed its conjugate dual is $(V_r)^\times = V_{\overline {r}}, \; \forall \, r\in J$. Then, r < s implies V _r ⊂ V _s, and the embedding operator E _sr : V _r → V _s is continuous and has dense range. In particular, V ^# is dense in every V _r. In the sequel, we also assume the partial inner product to be positive definite, 〈f|f〉 > 0 whenever f ≠ 0.

Then we have the familiar (Schwarz) inequality

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \xi \in V_p, \;\eta \in V_{\overline p}\quad \mathrm{implies} \quad \xi\overline{\eta} \in L^1 (X, \mu)\quad \mathrm{and} \\ \quad \left| \int_X \xi(x) \overline{\eta (x)} \,\mathrm{d}\mu (x)\right| {\leqslant} \|\xi\|{}_p \, \|\eta\|{}_{\overline p}. \end{array} \end{aligned} $$

(A.2)

A standard, albeit trivial, example is that of a rigged Hilbert space (RHS) $\Phi \subset {{\mathcal H}} \subset \Phi ^{\#}$ (it is trivial because the lattice ${\mathcal I}$ contains only three elements).

Familiar concrete examples of pip-spaces are sequence spaces, with V = ω the space of all complex sequences x = (x _n), and spaces of locally integrable functions with $V =L^1_{\mathrm {loc}}(\mathbb {R}, \,\mathrm {d} x)$, the space of Lebesgue measurable functions, integrable over compact subsets.

Among LBSs, the simplest example is that of a chain of reflexive Banach spaces. The prototype is the chain ${\mathcal I} = \{L^p := L ^p ( [0,1 ];dx),\; 1 < p < \infty \}$ of Lebesgue spaces over the interval [0, 1].

$$\displaystyle \begin{aligned} L^{\infty}\;\subset \;\ldots\;\subset \; L^{\overline{q}}\;\subset\; L^{\overline{r}}\;\subset \; \ldots \; {\subset \;L^{2}\;\subset} \; \ldots \subset \; L^{r} \subset \; L^{q} \;\subset \;\ldots\;\subset \;L^{1} , \end{aligned} $$

(A.3)

where 1 < q < r < 2 (of course, L ^∞ and L ¹ are not reflexive). Here L ^q and $L^{\overline {q}}$ are dual to each other $(1/q + 1/\overline { q} = 1)$, and similarly $L^{r}, L^{\overline {r}}\; (1/r + 1/\overline {r} = 1)$.

As for an LHS, the simplest example is the Hilbert scale generated by a self-adjoint operator A > I in a Hilbert space ${{\mathcal H}}_o$. Let ${{\mathcal H}}_{n}$ be D(A ⁿ), the domain of A ⁿ, equipped with the graph norm ∥f∥_n = ∥A ⁿ f∥, f ∈ D(A ⁿ), for $n \in \mathbb {N}$ or $n\in \mathbb {R}^+$, and ${{\mathcal H}}_{\overline n}:= {{\mathcal H}}_{-n} ={{\mathcal H}}_{n}^{\times }$ (conjugate dual):

$$\displaystyle \begin{aligned} {\mathcal D}^{\infty}(A):=\bigcap_{n} {{\mathcal H}}_n \subset \ldots \subset {{\mathcal H}}_2 \subset {{\mathcal H}}_1 \subset {{\mathcal H}}_0 \ \subset {{\mathcal H}}_{\overline 1} \subset {{\mathcal H}}_{ \overline 2} \ldots \subset {\mathcal D}_{\overline \infty}(A):=\bigcup_{n} {{\mathcal H}}_{n}. \end{aligned} $$

(A.4)

Note that here the index n may be integer or real, the link between the two cases being established by the spectral theorem for self-adjoint operators. Here again the inner product of ${{\mathcal H}}_0$ extends to each pair ${{\mathcal H}}_n ,{{\mathcal H}}_{-n}$, but on ${\mathcal D}_{\overline \infty }(A)$ it yields only a partial inner product. A standard example is the scale of Sobolev spaces $H^s(\mathbb {R}), \, s\in \mathbb {Z}$, in ${{\mathcal H}}_0 = L^2(\mathbb {R}, dx)$.