The Partial Inner Product Space Method: A Quick Overview

Many families of function spaces play a central role in analysis, in particular in signal processing (e.g. wavelet or Gabor analysis). Typical are L^{p} spaces, Besov spaces, amalgam spaces or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases of partial inner product spaces (PIP spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, frames on them should be defined globally, for the whole family, instead of individual spaces. In this paper, we shall give an overview of PIP spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. The interesting fact is that they allow a global definition of operators, and various operator classes on them have been defined.


Motivation
In the course of their curriculum, physics and mathematics students are usually taught the basics of Hilbert space, including operators of various types.The justification of this choice is twofold.On the mathematical side, Hilbert space is the example of an infinite-dimensional topological vector space that more closely resembles the familiar Euclidean space and thus it offers the student a smooth introduction into functional analysis.On the physics side, the fact is simply that Hilbert space is the daily language of quantum theory; therefore, mastering it is an essential tool for the quantum physicist.
However, the tool in question is actually insufficient.A pure Hilbert space formulation of quantum mechanics is both inconvenient and foreign to the daily behavior of most Advances in Mathematical Physics physicists, who stick to the more suggestive version of Dirac, although it lacks a rigorous formulation.On the other hand, the interesting solutions of most partial differential equations are seldom smooth or square integrable.Physically meaningful events correspond to changes of regime, which mean discontinuities and/or distributions.Shock waves are a typical example.Actually this state of affairs was recognized long ago by authors like Leray or Sobolev, whence they introduced the notion of weak solution.Thus it is no coincidence that many textbooks on PDEs begin with a thorough study of distribution theory 1-4 .All this naturally leads to the introduction of Rigged Hilbert Spaces RHS 5 .In a nutshell, a RHS is a triplet: where H is a Hilbert space, Φ is a dense subspace of the H, equipped with a locally convex topology, finer than the norm topology inherited from H, and Φ × is the space of continuous conjugate linear functionals on Φ, endowed with the strong dual topology.By duality, each space in 1.1 is dense in the next one and all embeddings are linear and continuous.In addition, the space Φ is in general required to be reflexive and nuclear.Standard examples of rigged Hilbert spaces are the Schwartz distribution spaces over R or R n , namely S ⊂ L 2 ⊂ S × or D ⊂ L 2 ⊂ D × 5-8 .
The problem with the RHS 1.1 is that, besides the Hilbert space vectors, it contains only two types of elements: "very good" functions in Φ and "very bad" ones in Φ × .If one wants a fine control on the behavior of individual elements, one has to interpolate somehow between the two extreme spaces.In the case of the Schwartz triplet, S ⊂ L 2 ⊂ S × , a wellknown solution is given by a chain of Hilbert spaces, the so-called Hermite representation of tempered distributions 9 .
In fact, this is not at all an isolated case.Indeed many function spaces that play a central role in analysis come in the form of families, indexed by one or several parameters that characterize the behavior of functions smoothness, behavior at infinity, . . . .The typical structure is a chain or a scale of Hilbert spaces, or a chain of reflexive Banach spaces a discrete chain of Hilbert spaces {H n } n∈Z is called a scale if there exists a self-adjoint operator B 1 such that H n D B n , for all n ∈ Z, with the graph norm f n B n f .A similar definition holds for a continuous chain {H α } α∈R . .Let us give two familiar examples.
i First, consider the Lebesgue the Lebesgue L p spaces on a finite interval, for example, I {L p 0, 1 , dx , 1 p ∞}: where 1 < q < r < 2.Here L q and L q are dual to each other 1/q 1/q 1 , and similarly are L r , L r 1/r 1/r 1 .By the Hölder inequality, the L 2 inner product is well defined if f ∈ L q , g ∈ L q .However, it is not well defined for two arbitrary functions f, g ∈ L 1 .Take, for instance, f x g x x −1/2 : f ∈ L 1 , but fg f 2 / ∈ L 1 .
Thus, on L 1 , 1.3 defines only a partial inner product.The same result holds for any compact subset of R instead of 0,1 .
ii As a second example, take the scale of Hilbert spaces built on the powers of a positive self-adjoint operator A 1 in a Hilbert space H 0 .Let 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 ∈ N or n ∈ R , and H n : H −n H × n conjugate dual Note that, in the second example ii , the index n could also be taken as real, the link between the two cases being established by the spectral theorem for self-adjoint operators.Here again the inner product of H 0 extends to each pair H n , H −n , but on D ∞ A it yields only a partial inner product.The following examples are standard: The notation is suggested by the operators of position, momentum and harmonic oscillator energy in quantum mechanics, resp. .Note that both D ∞ A p ∩ D ∞ A m and D ∞ A osc coincide with the Schwartz space S R of smooth functions of fast decay, and D ∞ A osc with the space S × R of tempered distributions considered here as continuous conjugate linear functionals on S .As for the operator A m , it generates the scale of Sobolev spaces H s R , s ∈ Z or R.However, a moment's reflection shows that the total-order relation inherent in a chain is in fact an unnecessary restriction; partially ordered structures are sufficient, and indeed necessary in practice.For instance, in order to get a better control on the behavior of individual functions, one may consider the lattice built on the powers of A p and A m simultaneously.Then the extreme spaces are still S R and S × R .Similarly, in the case of several variables, controlling the behavior of a function in each variable separately requires a nonordered set of spaces.This is in fact a statement about tensor products remember that L 2 X × Y L 2 X ⊗ L 2 Y .Indeed the tensor product of two chains of Hilbert spaces, {H n } ⊗ {K m }, is naturally a lattice {H n ⊗ K m } of Hilbert spaces.For instance, in the example above, for two variables x, y, that would mean considering intermediate Hilbert spaces corresponding to the product of two operators, A m x n A m y m .
Thus the structure to analyze is that of lattices of Hilbert or Banach spaces, interpolating between the extreme spaces of an RHS, as in 1.1 .Many examples can be given, for instance, the lattice generated by the spaces L p R, dx , the amalgam spaces W L p , q , the mixed-norm spaces L p,q m R, dx , and many more.In all these cases, which contain most families of function spaces of interest in analysis and in signal processing, a common structure emerges for the "large" space V , defined as the union of all individual spaces.There is a lattice of Hilbert or reflexive Banach spaces V r , with an order-reversing involution V r ↔ V r , where V r V × r the space of continuous conjugate linear functionals on V r , a central Hilbert space V o V o , and a partial inner product on V that extends the inner product of V o to pairs of dual spaces V r , V r .
Moreover, many operators should be considered globally, for the whole scale or lattice, instead of on individual spaces.In the case of the spaces L p R , such are, for instance, operators implementing translations x → x − y or dilations x → x/a , convolution operators, Fourier transform, and so forth.In the same spirit, it is often useful to have a common basis for the whole family of spaces, such as the Haar basis for the spaces L p R , 1 < p < ∞.Thus we need a notion of operator and basis defined globally for the scale or lattice itself.
This state of affairs prompted A. Grossmann and one of us the first author to systematize this approach, and this led to the concept of partial inner product space or PIPspace 10-13 .After many years and various developments, we devoted a full monograph 14 to a detailed survey of the theory.The aim of this paper is to present the formalism of PIP-spaces, which indeed answers these questions.In a first part, the structure of PIP-space is derived systematically from the abstract notion of compatibility and then particularized to the examples listed above.In a second part, operators on PIP-spaces are introduced and illustrated by several operators commonly used in Gabor or wavelet analysis.Finally we describe a number of applications of PIP-spaces in mathematical physics and in signal processing.Of course, the treatment is sketchy, for lack of space.For a complete information, we refer the reader to our monograph 14 .

Basic Definitions
The basic question is how to generate PIP-spaces in a systematic fashion.In order to answer, we may reformulate it as follows: given a vector space V and two vectors f, g ∈ V , when does their inner product make sense?A way of formalizing the answer is given by the idea of compatibility.Definition 2.1.A linear compatibility relation on a vector space V is a symmetric binary relation f#g which preserves linearity:

2.1
As a consequence, for every subset S ⊂ V , the set S # {g ∈ V : g#f, for all f ∈ S} is a vector subspace of V and one has Thus one gets the following equivalences: From now on, we will call assaying subspace of V a subspace S such that S ## S and denote by F V, # the family of all assaying subsets of V , ordered by inclusion.Let F be the isomorphy class of F, that is, F is considered as an abstract partially ordered set.Elements of F will be denoted by r, q, . .., and the corresponding assaying subsets by V r , V q , . ... By definition, q r if and only if V q V r .We also write V r V # r , r ∈ F. Thus the relations 2.3 mean that f#g if and only if there is an index r ∈ F such that f ∈ V r , g ∈ V r .In other words, vectors should not be considered individually, but only in terms of assaying subspaces, which are the building blocks of the whole structure.
It is easy to see that the map S → S ## is a closure, in the sense of universal algebra, so that the assaying subspaces are precisely the "closed" subsets.Therefore one has the following standard result.
Theorem 2.2.The family F V, # ≡ {V r , r ∈ F}, ordered by inclusion, is a complete involutive lattice, that is, it is stable under the following operations, arbitrarily iterated: ii infimum: The smallest element of F V, # is V # r V r and the greatest element is V r V r .By definition, the index set F is also a complete involutive lattice; for instance, 2.4 Definition 2.3.A partial inner product on V, # is a Hermitian form • | • defined exactly on compatible pairs of vectors.A partial inner product space PIP-space is a vector space V equipped with a linear compatibility and a partial inner product.
Note that the partial inner product is not required to be positive definite.
The partial inner product clearly defines a notion of orthogonality: f ⊥ g if and only if f#g and f | g 0.
Definition 2.4.The PIP-space V, #, We will assume henceforth that our PIP-space V, #, • | • is nondegenerate.As a consequence, V # , V and every couple V r , V r , r ∈ F, are dual pairs in the sense of topological vector spaces 15 .We also assume that the partial inner product is positive definite.Now one wants the topological structure to match the algebraic structure, in particular, the topology τ r on V r should be such that its conjugate dual be V r : V r τ r × V r , for all r ∈ F. This implies that the topology τ r must be finer than the weak topology σ V r , V r and coarser than the Mackey topology τ V r , V r : From here on, we will assume that every V r carries its Mackey topology τ V r , V r .This choice has two interesting consequences.First, if V r τ r is a Hilbert space or a reflexive Banach space, then τ V r , V r coincides with the norm topology.Next, r < s implies that V r ⊂ V s , and the

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embedding operator E sr : V r → V s is continuous and has dense range.In particular, V # is dense in every V r .

Sequence Spaces
Let V be the space ω of all complex sequences x x n and define on it i a compatibility relation by x n y n .Then ω # ϕ, the space of finite sequences, and the complete lattice F ω, # consists of K öthe's perfect sequence spaces 15, § 30 .Among these, typical assaying subspaces are the weighted Hilbert spaces 2 r x n : where r r n , r n > 0, is a sequence of positive numbers.The involution is 2 r ↔ 2 r 2 r × , where r n 1/r n .In addition, there is a central, self-dual Hilbert space, namely, 2 1 2 1 2 , where 1 1 .

Spaces of Locally Integrable Functions
Let now V be L 1 loc R, dx , the space of Lebesgue measurable functions, integrable over compact subsets, and define a compatibility relation on it by f#g ⇔ R |f x g x |dx < ∞ and a partial inner product f | g R f x g x dx.Then V # L ∞ c R , the space of bounded measurable functions of compact support.The complete lattice F L 1 loc , # consists of K öthe function spaces 16, 17 .Here again, typical assaying subspaces are weighted Hilbert spaces with r, r −1 ∈ L 2 loc R, dx , r x > 0 a.e.The involution is L 2 r ↔ L 2 r , with r r −1 , and the central, self-dual Hilbert space is L 2 R, dx .

Nested Hilbert Spaces
This is the original construction of Grossmann 18 for finding an "easy" substitute to distributions, and actually one of the motivations for introducing PIP-spaces.And indeed the two are closely related; see 14, Section 2.4.1

Rigged Hilbert Spaces
This is the simplest example of PIP-space, but it is a rather poor one.Indeed, in the RHS 1.1 , two elements are compatible if both belong to H, or one of them belongs to Φ. Thus the three defining spaces are the only assaying subspaces.The partial inner product is, of course, simply that of H, provided the sesquilinear form that puts Φ and Φ × in duality has been correctly normalized.

Lattices of Hilbert or Banach Spaces
From the previous examples, we learn that F V, # is a huge lattice it is complete!and that assaying subspaces may be complicated, such as Fréchet spaces, nonmetrizable spaces, and so forth.This situation suggests to choose an involutive sublattice I ⊂ F, indexed by I, such that i I is generating: ii every V r , r ∈ I, is a Hilbert space or a reflexive Banach space, iii there is a unique self-dual assaying subspace V o V o , which is a Hilbert space.
In that case, the structure V I : V, I, • | • is called, respectively, a lattice of Hilbert spaces LHS or a lattice of Banach spaces LBS .Both types are particular cases of the so-called indexed PIP-spaces 14 .Note that V # , V themselves usually do not belong to the family {V r , r ∈ I}, but they can be recovered as In the LBS case, the lattice structure takes the following forms: i V p∧q V p ∩ V q , with the projective norm These norms are usual in interpolation theory 19 .In the LHS case, one takes similar definitions with squared norms, in order to get Hilbert norms throughout.
In the rest of this section, we will list a series of concrete examples of LHS/LBSs.Some more examples, which are of particular interest in signal processing, will be given in Section 6.2.For simplicity, we will restrict ourselves to one dimension, although most spaces may be defined on R n , n > 1, as well.

Chains of Hilbert or Banach Spaces
Typical are the two examples described in Section 1.
The chain 1.2 is a totally ordered lattice.The corresponding lattice completion is obtained by adding "nonstandard" spaces such as The lattice completion is similar to the previous one, introducing analogous "nonstandard" spaces 14, Section 5.1 .

A LHS of Weighted 2 Spaces
In ω, with the compatibility # and the partial inner product defined in Section 2.2.1, we may take the lattice I { 2 r } of the weighted Hilbert spaces defined in 2.6 , with lattice operations: i infimum: 2 p ∧ q 2 p ∧ 2 q 2 r , r n min p n , q n , ii supremum: 2 p ∨ q 2 p ∨ 2 q 2 s , s n max p n , q n , iii duality: 2 p ∧ q ↔ 2 p ∨ q , 2 p ∨ q ↔ 2 p ∧ q .
As a matter of fact, the norms above are equivalent to the projective and inductive norms, respectively.Then, it is easy to show that the lattice I { 2 r } is generating in F ω, # .

Köthe Perfect Sequence Spaces
We have already noticed that the complete lattice F ω, # consists precisely of all K öthe perfect sequence spaces.Indeed, these are defined as the assaying subspaces corresponding to the compatibility #, which is called α-duality 15 .Among these, there is an interesting class, the so-called φ spaces associated to symmetric norming functions.From property n 5 ), it is clear that a symmetric norming function φ is entirely determined by its values on the set ϕ of finite, positive, nonincreasing sequences.Hence, from conditions n 2 ) and n 4 ), we deduce that To every symmetric norming function φ, one can associate a Banach space φ as follows.Given a sequence x ∈ ω, define its nth section as x n x 1 , x 2 , . . ., x n , 0, 0, . . . .Then the sequence φ x n is nondecreasing, so that one can define and then extend the norming function φ to the whole of φ by putting φ x lim n φ x n .This relation defines a norm φ on φ , for which it is complete, hence, a Banach space.In other words, we can also say that φ {x ∈ ω : φ x < ∞} is the natural domain of definition of the extended norming function φ.Clearly, one has φ ∞ ∞ and φ 1 1 .Similarly, p φ p , where φ p x n |x n | p 1/p .Thus every space φ contains 1 and is contained in ∞ .In addition, the set of Banach spaces φ constitutes a lattice.Given two symmetric norming functions φ and ψ, one defines their infimum and supremum, exactly as for the general case: i φ ∧ ψ : max{φ, ψ}, which defines on the space φ∧ψ : φ ∩ ψ a norm equivalent to φ x ψ x , ii φ ∨ ψ : min{φ, ψ}, which defines on the space φ∨ψ : φ ψ a norm equivalent to inf x y z {φ y ψ z }, x ∈ φ ψ , y ∈ φ , z ∈ ψ .
It remains to analyze the relationship of the spaces φ with the PIP-space structure of ω.Define, for any finite, positive, nonincreasing sequence y ∈ ϕ , The function φ thus defined is a symmetric norming function; hence, it can be extended to the corresponding Banach space φ .The function φ is said to be conjugate to φ and the space φ is the conjugate dual of φ with respect to the partial inner product, that is, φ φ # .Clearly one has φ φ; hence, In addition, it is easy to show that φ∧ψ φ∨ψ and φ∨ψ φ∧ψ .In other words, one gets the following result.

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Proposition 3.2.The family of Banach spaces φ , where φ is a symmetric norming function, is an involutive sublattice of the lattice F ω, # and a LBS.
Actually, since every φ satisfies the inclusions 1 ⊂ φ ⊂ ∞ , the family { φ } is also an involutive sublattice of the lattice F ∞ , # obtained by restricting to ∞ the PIP-space structure of ω.
These spaces { φ } may be generalized further to what is called the theory of Banach ideals of sequences.See 14, Section 4.3 for more details.

A LHS of Weighted L 2 Spaces
In L 1 loc R, dx , we may take the lattice I {L 2 r } of the weighted Hilbert spaces defined in 2.7 , with i infimum: iii duality: Here too, these norms are equivalent to the projective and inductive norms, respectively.

The Spaces
The spaces L p R, dx , 1 < p < ∞ do not constitute a scale, since one has only the inclusions L p ∩ L q ⊂ L s , p < s < q.Thus one has to consider the lattice they generate, with the following lattice operations: i L p ∧ L q L p ∩ L q , with projective norm, ii L p ∨ L q L p L q , with inductive norm.
For 1 < p, q < ∞, both spaces L p ∧ L q and L p ∨ L q are reflexive Banach spaces and their conjugate duals are, respectively, L p ∧ L q × L p ∨ L q and L p ∨ L q × L p ∧ L q .It is convenient to introduce the following unified notation: 3.9 Then, for 1 < p, q < ∞, L p,q is a reflexive Banach space, with conjugate dual L p,q .Next, if we represent p, q by the point of coordinates 1/p, 1/q , we may associate all the spaces L p,q 1 p, q ∞ in a one-to-one fashion with the points of a unit square J 0, 1 × 0, 1 see Figure 1 .Thus, in this picture, the spaces L p are on the main diagonal, intersections L p ∩ L q above it and sums L p L q below.The space L p,q is contained in L p ,q if p, q is on the left and/or above p , q .Thus the smallest space is and it corresponds to the upper-left corner, while the largest one is corresponding to the lower-right corner.Inside the square, duality corresponds to geometrical symmetry with respect to the center 1/2,1/2 of the square, which represents the space L 2 .The ordering of the spaces corresponds to the following rule: L p,q ⊂ L p ,q ⇐⇒ p, q p , q ⇐⇒ p p , q q .

3.12
With respect to this ordering, J is an involutive lattice with the operations p, q ∧ p , q p ∨ p , q ∧ q , p, q ∨ p , q p ∧ p , q ∨ q , p, q p, q ,

3.13
where p ∧ p min{p, p }, p ∨ p max{p, p }.It is remarkable that the lattice J generated by I {L p } is obtained at the first "generation".One has, for instance, L r,s ∧ L a,b L r∨a,s∧b , both as sets and as topological vector spaces.

Mixed-Norm Lebesgue Spaces L p,q m
An interesting class of function spaces, close relatives to the Lebesgue L p spaces, consists of the so-called L P spaces with mixed norm.Let X, μ and Y, ν be two σ-finite measure spaces and 1 p, q ∞ in the general case, one considers n such spaces and n-tuples

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P : p 1 , p 2 , . . ., p n .Then, a function f x, y measurable on the product space X × Y is said to belong to L p,q X × Y if the number obtained by taking successively the p-norm in x and the q-norm in y, in that order, is finite exchanging the order of the two norms leads in general to a different space .If p, q < ∞, the norm reads The analogous norm for p or q ∞ is obvious.For p q, one gets the usual space L p X × Y .These spaces enjoy a number of properties similar to those of the L p spaces: i each space L p,q is a Banach space and it is reflexive if and only if 1 < p, q < ∞; ii the conjugate dual of L p,q is L p,q , where, as usual, p −1 p −1 1, q −1 q −1 1; thus the topological conjugate dual coincides with the K öthe dual; iii the mixed-norm spaces satisfy a generalized H ölder inequality and have nice interpolation properties.
The case X Y R d with Lebesgue measure is the important one for signal processing 20, Section 11.1 .More generally, one can add a weight function m and obtain the spaces L p,q m R d we switch to a notation more suitable for the applications : Here the weight function m is a nonnegative locally integrable function on R 2d , assumed to be v-moderate, that is, The typical weights are of polynomial growth: v s z 1 |z| s , s 0. The space L p,q m R 2d is a Banach space for the norm • p,q m .The duality property is, as expected, L p,q m × L p,q 1/m .Of course, things simplify when Concerning lattice properties of the family of L p,q m spaces, we cannot expect more than for the L p spaces.Two L p,q m spaces are never comparable, even for the same weight m, so one has to take the lattice generated by intersection and duality.
A different type of mixed-norm spaces is obtained if one takes X Y Z d , with the counting measure.Thus one gets the space p,q m Z 2d , which consists of all sequences a a kn , k, n ∈ Z d , for which the following norm is finite: Contrary to the continuous case, here we do have inclusion relations: if p 1 p 2 , q 1 q 2 and m 2 Cm 1 , then m 2 .Discrete mixed-norm spaces have been used extensively in functional analysis and signal processing.For instance, they are key to the proof that certain operators are bounded between two given function spaces, such as modulation spaces see below or p spaces.In general, a mixed-norm space will prove useful whenever one has a signal consisting Advances in Mathematical Physics 13 of sequences labeled by two indices that play different roles.An obvious example is timefrequency or time-scale analysis: a Gabor or wavelet basis or frame is written as {ψ j,k , j, k ∈ Z}, where j indexes the scale or frequency and k the time.More generally, this applies whenever signals are expanded with respect to a dictionary with two indices.

Köthe Function Spaces
The mixed-norm Lebesgue spaces L p,q m are special cases of a very general class, the so-called Köthe function spaces.These have been introduced and given that name by Dieudonné 16 and further studied by Luxemburg-Zaanen 21 .The procedure here is entirely parallel to that used in Section 3.2.2above for introducing the sequence spaces φ .
Let X, μ be a σ-finite measure space and M the set of all measurable, non-negative functions on X, where two functions are identified if they differ at most on a μ-null set.A function norm is a mapping ρ : A function norm ρ is said to have the Fatou property if and only if 0 Given a function norm ρ, it can be extended to all complex measurable functions on X by defining ρ f ρ |f| .Denote by L ρ the set of all measurable f such that ρ f < ∞.With the norm f ρ f , L ρ is a normed space and a subspace of the vector space V of all measurable μ-a.e.finite, functions on X.Furthermore, if ρ has the Fatou property, L ρ is complete, that is, a Banach space.
A function norm ρ is said to be saturated if, for any measurable set E ⊂ X of positive measure, there exists a measurable subset F ⊂ E such that μ F > 0 and ρ χ F < ∞ χ F is the characteristic function of F .
Let ρ be a saturated function norm with the Fatou property.Define ρ f sup X fg dμ : ρ g 1 .

3.17
Then ρ is a saturated function norm with the Fatou property and ρ ≡ ρ ρ.Hence, L ρ is a Banach space.Moreover, one has also ρ f sup X fg dμ : ρ g 1 .

3.18
For each ρ as above, L ρ is a Banach space and L ρ L ρ # , that is, each L ρ is assaying.The pair L ρ , L ρ is actually a dual pair, although V # , V is not.The space L ρ is called the Köthe dual or α-dual of L ρ and denoted by L ρ α .
However, L ρ is in general only a closed subspace of the Banach conjugate dual L ρ × ; thus, the Mackey topology τ L ρ , L ρ is coarser than the ρ-norm topology, which is Advances in Mathematical Physics τ L ρ , L ρ × .This defect can be remedied by further restricting ρ.A function norm ρ is called . .0 pointwise a.e. on X.For instance, the Lebesgue L p -norm is absolutely continuous for 1 p < ∞, but the L ∞ -norm is not!Also, even if ρ is absolutely continuous, ρ need not be.Yet, this is the appropriate concept, in view of the following results: Let ρ be a saturated, absolutely continuous function norm on X, with the Fatou property and such that ρ is also absolutely continuous.Then L ρ , L ρ is a reflexive dual pair of Banach spaces.In addition, the set J of all function norms with these properties is an involutive lattice with respect to the following partial order: ρ 1 ρ 2 if and only if ρ 1 f ρ 2 f , for every measurable f.The lattice operations are the following: For the corresponding Banach spaces, we have the relations

3.19
Consider now the usual space V L 1 loc X, dμ , with the compatibility and partial inner product defined in Section 2.2.2, so that V # L ∞ c X, dμ .Then the construction outlined above provides L 1 loc X, dμ with the structure of a LBS.Indeed, one has the following result.
Proposition 3.3.Let J be the set of saturated, absolutely continuous function norms ρ on X, with the Fatou property and such that ρ is also absolutely continuous.Let I denote the set I : {L ρ : ρ ∈ J and L ρ ⊂ L 1 loc }.Then I is a LBS, with the lattice operations defined above.
More general situations may be considered, for which we refer to 14, Section 4.4 .

Comparing PIP-Spaces
The definition of LBS/LHS given in Section 3 leads to the proper notion of comparison between two linear compatibilities on the same vector space.Namely, we shall say that a compatibility # 1 is Now, suppose that a linear compatiblity # is given on V .Then, every involutive cofinal sublattice of F V, # defines a coarser PIP-space, and vice versa.Thus coarsening is always possible, and will ultimately lead to a minimal PIP-space, consisting of V # and V only, that is, the situation of distribution spaces.However, the operation of refining is not always possible; in particular there is no canonical solution, a fortiori no unique maximal solution.There are exceptions, however, for instance, when one is given explicitly a larger set of assaying subspaces that also form, or generate, a larger involutive sublattice.To give an example, the weighted L 2 spaces of Section 3.3.1 form an involutive sublattice of the involutive lattice I of K öthe function spaces of Section 3.3.4;thus, I is a genuine refinement of the original LHS.
In the case of a LHS, refining is possible, with infinitely many solutions, by use of interpolation methods or the spectral theorem for self-adjoint operators, which are essentially equivalent in this case.In particular, one may always refine a discrete scale of Hilbert spaces into a nonunique continuous one.Indeed, for the scale described in Section 1, Example ii , one has, by definition, H n D A n , the domain of A n , equipped with the graph norm f n A n f , f ∈ D A n , for n ∈ N.Then, for each 0 α 1, one may define where {E s , 1 ≤ s < ∞} is the spectral family of A. With the inner product H n α is a Hilbert space and one has the continuous embeddings One may go further, as follows.Let ϕ be any continuous, positive function on 1, ∞ such that ϕ t is unbounded for t → ∞, but increases slower than any power t α 0 < α 1 .An example is ϕ t log t t 1 .Then ϕ A is a well-defined self-adjoint operator, with domain With the corresponding inner product D ϕ A becomes a Hilbert space H ϕ .For every α, 0 < α 1, one has, with proper inclusions and continuous embeddings, This can be continued as far as one wants, with the result that every scale of Hilbert spaces possesses infinitely many proper refinements which are themselves chains of Hilbert spaces 14, Chapter 5 .
Another type of refinement consists in refining a RHS Φ ⊂ H ⊂ Φ × , by inserting a number of intermediate spaces, called interspaces, namely, spaces E such that Φ → E → Φ × which implies that the conjugate dual E × is also an interspace .Upon some additional conditions, the most important of which being that Φ be dense in E ∩ F with its projective topology, for any pair E, F of interspaces, one obtains in that way a proper refining of the original RHS.With this construction, which goes under the name of multiplication framework, one succeeds, for instance, in defining a valid partial multiplication between distributions.A thorough analysis may be found in 14, Section 6.3 .

General Definitions
As already mentioned, the basic idea of indexed PIP-spaces is that vectors should not be considered individually, but only in terms of the subspaces V r r ∈ F or r ∈ I , the building blocks of the structure; see 3.1 .Correspondingly, an operator on a PIP-space should be defined in terms of assaying subspaces only, with the proviso that only bounded operators between Hilbert or Banach spaces are allowed.Thus an operator is a coherent collection of bounded operators.More precisely, one has the following.
, where d A is a nonempty subset of I, ii for every q ∈ d A , there exists a p ∈ I such that the restriction of A to V q is linear and continuous into V p we denote this restriction by A pq , iii A has no proper extension satisfying i and ii .
The linear bounded operator A pq : V q → V p is called a representative of A. In terms of the latter, the operator A may be characterized by the set j A { q, p ∈ I × I : A pq exists}.Thus the operator A may be identified with the collection of its representatives:

A
A pq : V q −→ V p : q, p ∈ j A .

5.1
By condition ii , the set d A is obtained by projecting j A on the "first coordinate" axis.The projection i A on the "second coordinate" axis plays, in a sense, the role of the range of A. More precisely, d A q ∈ I : there is a p such that A pq exists , i A p ∈ I : there is a q such that A pq exists .

5.2
The following properties are immediate see the see Figure 2 : i d A is an initial subset of I: if q ∈ d A and q < q, then q ∈ d A , and A pq A pq E qq , where E qq is a representative of the unit operator this is what we mean by a 'coherent' collection , ii i A is a final subset of I: if p ∈i A and p > p, then p ∈i A and A p q E p p A pq .
iii j A ⊂d A ×i A , with strict inclusion in general.
We denote by Op V I the set of all operators on V I .Of course, a similar definition may be given for operators A : V I → Y K between two LHSs or LBSs.Since V # is dense in V r , for every r ∈ I, an operator may be identified with a separately continuous sesquilinear form on V # × V # .Indeed, the restriction of any representative to V # × V # is such a form, and all these restrictions coincide.Equivalently, an operator may be identified with a continuous linear map from V # into V continuity with respect to the respective Mackey topologies .But the idea behind the notion of operator is to keep also the algebraic operations on operators; namely, we define the following operations: i Adjoint: Every A ∈ Op V I has a unique adjoint A × ∈ Op V I , defined by the relation that is, A × rs A sr * usual Hilbert/Banach space adjoint .¡list-item¿¡label/¿ It follows that A ×× A, for every A ∈ Op V I : no extension is allowed, by the maximality condition iii of Definition 5.1.
ii Partial Multiplication: The product AB is defined if and only if there is a q ∈ i B ∩ d A , that is, if and only if there is a continuous factorization through some V q : It is worth noting that, for a LHS/LBS, the domain D A is always a vector subspace of V this is not true for a general PIP-space .Therefore, Op V I is a vector space and a partial * -algebra 22 .
The concept of PIP-space operator is very simple, yet it is a far-reaching generalization of bounded operators.It allows indeed to treat on the same footing all kinds of operators, from bounded ones to very singular ones.By this, we mean the following, loosely speaking.Take

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Three cases may arise: ii if A oo does not exist, but only A or : V r → V o , with r < o, then A corresponds to an unbounded operator, with domain iii if no A or exists, but only A sr : V r → V s , with r < o < s, then A corresponds to a singular operator, with Hilbert space domain possibly reduced to {0}.

Special Classes of Operators on PIP-Spaces
Exactly as for Hilbert or Banach spaces, one may define various types of operators between PIP-spaces, in particular LBS/LHSs.We discuss briefly the most important classes, namely, regular operators, homomorphisms and isomorphisms, unitary operators, symmetric operators, and orthogonal projections.Further details may be found in the monograph 14 .

Regular and Totally Regular Operators
An operator A on a nondegenerate PIP-space V I , with positive-definite partial inner product, in particular, a LBS/LHS, is called regular if d A i A I or, equivalently, if A : V # → V # and A : V → V continuously for the respective Mackey topologies.This notion depends only on the pair V # , V , not on the particular compatibility #.The set of all regular operators V I → V I is denoted by Reg V I .Thus a regular operator may be multiplied both on the left and on the right by an arbitrary operator.Clearly, the set Reg V I is a * -algebra and can often be identified with an O * -algebra 22, 23 .
We give two examples.
1 If V ω, V # ϕ, then Op ω consists of arbitrary infinite matrices and Reg ω of infinite matrices with finite rows and finite columns.
2 If V S × , V # S, then Op S × consists of arbitrary tempered kernels, while Reg S × contains those kernels that can be extended to S × and map S into itself.A nice example is the Fourier transform.
An operator A is called totally regular if j A contains the diagonal of I × I, that is, A rr exists for every r ∈ I or A maps every V r into itself continuously.

Homomorphisms
Let V I , Y K be two LHSs or LBSs.An operator We denote by Hom V I , Y K the set of all homomorphisms from V I into Y K .The following properties are easy to prove: B, for any two elements of A, B ∈ Hom V I , W L , where V I is any PIP-space.Typical examples of monomorphisms are the inclusion maps resulting from the restriction of a support.Take for instance, L 1 loc X, dμ , the space of locally integrable functions on a measure space X, μ .Let Ω be a measurable subset of X and Ω its complement, both of nonzero measure, and construct the space L 1 loc Ω, dμ , which is a PIP-subspace of L 1 loc X, dμ see Section 5.2.5 .Given f ∈ L 1 loc X, dμ , define f Ω fχ Ω , where χ Ω is the characteristic function of χ Ω .Then we obtain an injection monomorphism M Ω : L 1 loc Ω, dμ → L 1 loc X, dμ as follows: If we consider the lattice of weighted Hilbert spaces {L 2 r } in this PIP-space, then the correspondence r ↔ r Ω rχ Ω is a bijection between the corresponding involutive lattices.The homomorphism A ∈ Hom V I , Y K is an isomorphism if there exists a homomorphism B ∈ Hom Y K , V I such that BA 1 V , AB 1 Y , where 1 V , 1 Y denote the identity operators on V I , Y K , respectively.
For instance, the Fourier transform is an isomorphism of the Schwartz RHS S ⊂ L 2 ⊂ S × onto itself and, similarly, of the Feichtinger triplet 6.16 onto itself.In both cases, the property extends to several scales of lattices interpolating between the two extreme spaces, for instance, the Hilbert chain of the Hermite representation of tempered distributions.

Unitary Operators and Group Representations
The operator U ∈ Op V I , Y K is unitary if U × U and UU × are defined and U × U 1 V , UU × 1 Y .We emphasize that unitary operators need not be homomorphisms !In fact, unitarity is a rather weak property and it is insufficient for group representations.
Thus a unitary representation of a group G into a PIP-space V I is defined as a homomorphism of G into the unitary isomorphisms of V I .Given such a unitary representation U of G into V I , where the latter has the central Hilbert space H 0 , consider the representative U 00 g of U g in H 0 .Then g → U 00 g is a unitary representation of G into H 0 , in the usual sense.
To give an example, let V I be the scale built on the powers of the operator Hamiltonian H −Δ v r on L 2 R 3 , dx , where Δ is the Laplacian on R 3 and v is a nice rotation invariant potential.The system admits as symmetry group G SO 3 the full symmetry group might be larger; for instance, the Coulomb potential admits SO 4 as Advances in Mathematical Physics symmetry group for its bound states.and the representation U 00 is the natural representation of SO 3 in L 2 R 3 : Then U 00 extends to a unitary representation U by totally regular isomorphisms of V I .Angular momentum decompositions, corresponding to irreducible representations of SO 3 , extend to V I as well.In addition, this is a good setting also for representations of the Lie algebra so 3 .Notice that the representation U is totally regular, but this need not be the case.
For instance, if the potential v is not rotation invariant, U will no longer be totally regular, although it is still an isomorphism.

Symmetric Operators
Since one has A ×× A, no extension is allowed, by the maximality condition.Thus symmetric operators behave essentially like bounded self-adjoint operators in a Hilbert space.Yet, they can be very singular, as indicated above, for a chain 5.8 In this case, the question is whether a symmetric operator A ∈ Op V I has a self-adjoint restriction to the central Hilbert space V o .In a Hilbert space context, an answer is given by the celebrated KLMN theorem KLMN stands for Kato, Lax, Lions, Milgram, Nelson .Actually, this classical result already has a distinct PIP-space flavor.Thus is not surprising that the KLMN theorem has a natural generalization to a LHS or a PIP-space with positive-definite partial inner product and central Hilbert space V o V o , and a quadratic form version as well 14, Section 3.3.5 .An interesting application is a correct description of very singular Hamiltonians in quantum mechanics, typically with δ or δ interactions.For instance, one can treat in this way the Kronig-Penney crystal model, which consists of a δ potential at each node of a lattice, in one, two, or three dimensions 24, 25 .

Orthogonal Projections, Bases, Frames
An operator P on a nondegenerate PIP-space V , respectively, a LBS/LHS V I , is an orthogonal projection if P ∈ Hom V I and P 2 P P × .It follows immediately from the definition that an orthogonal projection is totally regular, that is, j P contains the diagonal I × I, or still that P leaves every assaying subspace invariant.Equivalently, P is an orthogonal projection if P is an idempotent operator that is, P 2 P such that {Pf} # ⊇ {f} # for every f ∈ V and g | Pf Pg | f whenever f#g.We denote by Proj V the set of all orthogonal projections in V and similarly for Proj V I .
These projection operators enjoy several properties similar to those of Hilbert space projectors.Two of them are of special interest in the present context.
i Given a nondegenerate PIP-space V , there is a natural notion of subspace, called orthocomplemented, which guarantees that such a subspace W of V is again a nondegenerate PIP-space with the induced compatibility relation and the restriction of the partial inner product.There are equivalent topological conditions, so that orthocomplemented subspaces are the proper PIP-subspaces 26 .Then the basic theorem about projections states that a subspace W of V is orthocomplemented if and only if W is the range of an orthogonal projection P ∈ Proj V , that is, W PV .Then V W ⊕ Z, where Z is another orthocomplemented subspace.
ii An orthogonal projection P is of finite rank if and only if W Ran P ⊂ V # and W ∩ W ⊥ {0} this condition is, of course, superfluous when the partial inner product is positive definite .
There is a natural partial order on the set of projections: but the lattice properties of Proj V are unknown.Thus we expect that quantum logic may be reformulated in a PIP-space language only under additional restrictions on V .Property ii has important consequences for the structure of bases.First we recall that a sequence {e n , n 1, 2, . ..} of vectors in a Banach space E is a Schauder basis if, for every f ∈ E, there exists a unique sequence of scalar coefficients Then one may write c k e k .

5.10
The basis is unconditional if the series 5.10 converges unconditionally in E i.e., it keeps converging after an arbitrary permutation of its terms .
A standard problem is to find, for instance, a sequence of functions that is an unconditional basis for all the spaces L p R , 1 < p < ∞.In the PIP-space language, this statement means that the basis vectors must belong to V # 1<p<∞ L p R .Also, since 5.10 means that f may be approximated by finite sums, the property ii of orthogonal projections implies that all the basis vectors must belong to V # .Some examples are given in Section 6.2.5.
Actually, in the context of signal processing, orthogonal in the Hilbert sense bases are not enough; one needs also biorthogonal bases and, more generally, frames.We recall that a countable family of vectors {ψ n } in a Hilbert space H is called a frame if there are two positive constants m, M, with 0

5.11
The two constants m, M are called frame bounds.If m M, the frame is said to be tight.Consider the analysis operator C : H → 2 defined by C : f → {| ψ n | f |} and the frame operator S C * C. Then the vectors ψ n S −1 ψ n also constitute a frame, called the canonical dual frame, and one has the strongly converging expansions Then the considerations made above for bases should apply to frame vectors as well, that is, the vectors ψ n , ψ n should also belong to V # .

Applications in Mathematical Physics
PIP-spaces have found many applications in mathematical physics, mostly in quantum physics.We will sketch a few of them in this section.Most of what follows is described in detail in 14 .

Dirac Formalism in Quantum Mechanics
The mathematical description of a quantum system rests on three basic principles: i The superposition principle, which implies that the set of states of the system has a linear structure; ii The notion of transition amplitude, given by an inner product: and iii The probabilistic interpretation, which requires that ψ | ψ ψ 2 > 0, whenever ψ / 0. Combining these basic principles implies that the set of states of the system is a positive-definite inner product space Φ, that is, a pre-Hilbert space.On this basis, Dirac developed a formalism for quantum physics with great computational capacity and broad predictive power.The essential features of Dirac's formalism are the following.
i Physical observables are represented by linear operators in the space Φ and these operators form an algebra.Therefore, it makes sense to arbitrarily add and multiply operators to form new operators.
ii For a given quantum physical system, there exist complete systems of commuting observables CSCO in the algebra of observables.The system of eigenvectors for a chosen CSCO provides a basis for the space Φ, that is, every vector φ ∈ Φ can be expanded into the eigenvectors of the CSCO.
In the simplest case, a CSCO consists of only one observable A, with a mixed spectrum consisting of discrete eigenvalues {λ n } σ p A and a continuous part {λ} σ c A .The corresponding eigenvectors, written as |λ n , |λ , respectively, obey "orthogonality" relations Then every φ ∈ Φ can be expanded as Clearly the eigenvectors |λ cannot belong to the pre-Hilbert space Φ, nor to its completion H. Thus Dirac's formalism, while extremely practical and used by physicists on a daily basis, is not mathematically well defined.
For that reason, von Neumann formulated a rigorous version of quantum mechanics, in a pure Hilbert space language.His formulation consists in the following two axioms: i Pure states are represented by rays i.e., one-dimensional subspaces in a Hilbert space H; and ii Observables are represented by self-adjoint operators in H.This formulation is well defined mathematically, but too restrictive.Nonnormalizable eigenvectors, corresponding to points of a continuous spectrum, cannot belong to H, yet they are extremely useful and often have a clear physical meaning plane waves, for instance .Observables may be unbounded, so that domain considerations must be taken into account.In particular, unbounded operators may not always be multiplied.Thus it is understandable that the large majority of physicists stay with Dirac's formalism.
This had prompted several authors 27-31 to propose a rigorous version in terms of a RHS Φ ⊂ H ⊂ Φ × .In this scheme, the space Φ is constructed from the basic observables labeled observables of the system at hand and is interpreted as the space of all physically preparable states.The conjugate dual Φ × contains idealized states probes , identified with measurement devices.In that context, let A be an observable, represented by a self-adjoint operator in H such that A : Φ → Φ continuously.Then A may be transposed by duality to a linear operator A × : Φ × → Φ × , which is an extension of A † : A * Φ, where A * is the usual Hilbert space adjoint operator, namely, For such an operator, the vector ξ λ ∈ Φ × is called a generalized eigenvector of A, with eigenvalue λ ∈ C, if it satisfies Then the spectral theorem of Gel'fand-Maurin 5, 6 states that A possesses in Φ × a complete orthonormal set of generalized eigenvectors ξ λ ∈ Φ × , λ ∈ R.This means that, for any two φ, ψ ∈ Φ, one has we split again into the discrete and the continuous part of the spectrum of where ρ is a non-negative integrable function.In that way one recovers essentially Dirac's bra-and-ket formalism.This approach is based on a RHS, but the construction is such that a PIP-space version is easily obtained-and is in fact closer to Dirac's spirit.For more details, see 14, Section 7.1.1 .

Symmetries, Singular Interactions in Quantum Mechanics
Several other topics in quantum mechanics can be advantageously formulated in a RHS or PIP-space language, for instance, the implementation of symmetries, with the two dual points of view, the active one and the passive one 32 .A symmetry group is represented by a unitary representation in H that extends to a unitary representation U in the enveloping PIP-space, in the sense defined in Section 5.2.3.Then, in accordance with the physical interpretation given above, the active point of view corresponds to the action of U in V # ≡ Φ, the passive one to the action on V ≡ Φ × .Another problem is a correct definition of a Hamiltonian with a singular interaction, already mentioned in Section 5.2.4.In the simplest case, the standard definition is H − Δ/2m V , where the interaction V is given by some reasonable function potential .However, there are cases where a singular interaction is needed, for instance when V is replaced formally by a δ function or a δ function, with support in a point or several or a manifold of lower dimension.Then the usual formulation is based on von Neumann's theory of self-adjoint extensions of symmetric operators, sometimes coupled with Krein's formula 33 .But here the PIP-space approach is a convenient substitute to that approach, as shown in 24, 25 and 14, Section 7.1.3 .

Quantum Scattering Theory
In scattering theory, it is common to use scales of Hilbert spaces built on the powers of A 1 : 1 |x| 2 or A 2 : 1 |p| 2 , and the LHS obtained by combinations of both.This example contains the Sobolev spaces the scale built on A 2 , the weighted spaces L 2 s the scale built on A 1 , and spaces of mixed type.In particular, operators of the form f x g p , for suitable functions f, g, play an essential role in the so-called phase-space approach to scattering theory and they may be controlled by this LHS.For instance, their trace ideal properties may be derived in this way and they are used for proving the absence of singular continuous spectrum by the limiting absorption principle.
On the other hand, the Weinberg-van Winter WVW formulation of scattering theory 34-36 has a very natural interpretation in terms of a LHS, whose components, including the extreme ones, are Hilbert spaces consisting of functions analytic in a sector; thus the indexing parameter is the opening angle of that sector.This technique has allowed to show that the WVW formalism is a particular case of the Complex Scaling Method 14, Section 7.2.3 , a result hitherto unknown.

Quantum Field Theory
Mathematically rigorous formulations of QFT rely heavily on a RHS or a PIP-space approach, primarily Wightman's axiomatic formulation.There, indeed, a quantum field is defined as an operator-valued distribution, which is customarily written in terms of an unsmeared field field at a point A x , as Under quite reasonable assumptions, the unsmeared field can be defined as a map from S R 4 into Op V , where V is a conveniently chosen PIP-space.This allows to give the previous formula a rigorous mathematical meaning 14, Section 7.3.1 .
Another PIP-space version of QFT is the Fock construction tensor algebra over the RHS where V m denotes the forward mass shell V m {p ∈ R 4 : p 2 m 2 , p 0 > 0} and dμ the Lorentz invariant measure d 3 p/p 0 on it.Write Φ 1 S V m , the space of "good" one-particle states.
Then define where the right-hand side denotes the symmetrized tensor product of n copies of Φ 1 , corresponding to n-boson states.Again Φ n is reflexive, complete, and nuclear with respect to its natural topology, and it can be described as the end space of a scale of Hilbert spaces.Finally, define that is, the topological direct sum of the component spaces.Elements of Φ are finite sequences f {f 0 , f 1 , . . ., f n , . ..}, f 0 ∈ C, f n ∈ Φ n , that is, totally symmetric functions of Schwartz type.The space Φ is reflexive, complete, and nuclear with respect to the direct sum topology.Its dual is the topological product Thus we get a suitable RHS, in which the central Hilbert space H is Fock space, that is, the tensor algebra Other examples are the construction of QFT via the Borchers algebra, Nelson's Euclidean field theory, or the precise treatment of unsmeared fields fields at a point .See 14, Section 7.3 for a detailed presentation.

Representations of Lie Groups and Lie Algebras
Let us return to the situation described in Section 5.2.3.We start with a strongly continuous unitary representation U 00 of a Lie group G in a Hilbert space H 0 and seek to build a PIP-space V I , with H 0 being its central Hilbert space, such that U 00 extends to a unitary representation U into V I .
The solution of this problem is well known from Nelson's theory of analytic vectors.Let Δ be the closure of the Nelson operator Δ : n j 1 X 2 j , where {X j , j 1, . . ., n} are the representatives under U 00 of the elements of a basis of the Lie algebra g of G. Δ is essentially Advances in Mathematical Physics self-adjoint on the so-called Gårding domain H G 0 , Δ is self-adjoint, and Δ 0. Define V I : {H n , n ∈ Z} as the canonical scale of Hilbert spaces generated by the operator Δ 1 : Next, for every g ∈ G, U 00 g leaves each H n , n ∈ N, invariant and its restriction U nn g : H n → H n is continuous; thus it can be transposed to a continuous map U nn g −1 : H n → H n .It follows that U 00 extends to a unitary representation U in the LHS V I .Corresponding to the triplet 6.11 , we have three representations of G, namely, U 00 , its restriction U ∞∞ , and the dual U ∞∞ of the latter, which is an extension of the first two.All three are continuous.Moreover, if one of the three is topologically irreducible i.e., there is no proper invariant closed subspace , so are the other two.
In addition to the representations of the group G, the scale V I is the natural tool for studying the properties of the operators representing elements of the Lie algebra g or the universal enveloping algebra U g of G.For every element x ∈ g or L ∈ U g , the representative U x , respectively U L , originally defined on H G 0 , extends to a regular operator on V I .These regular operators have in general no {0, 0}-representative, since x and L are represented in H 0 by unbounded operators.As in the case of the group G, one gets three * -representations of the enveloping algebra U g , and in particular of the Lie algebra g, in the three spaces of the triplet 6.11 .Namely, one has, for every where L ↔ L is the involution on U g .These representations have the same irreducibility properties as the corresponding ones of the group.See 14, Section 7 for further details.

Applications in Analysis and Signal Processing
Many families of function spaces of interest in analysis or signal processing are, or contain, lattices of Banach spaces.To quote a few: amalgam spaces, modulation spaces, Besov spaces, α-modulation spaces, coorbit spaces, which contain many of the previous cases.We shall describe them briefly in succession.For further information about those spaces, we refer to our monograph 14, Chapters 4 and 8 .

Amalgam Spaces
A situation intermediate between the mixed-norm spaces L p,q R 2d for m ≡ 1 and the spaces p,q Z 2d is that of the so-called amalgam spaces.They were introduced specifically to overcome the inability of the L p norms to distinguish between the local and the global behavior of functions.The simplest ones are the spaces W L p , q of Wiener 37 , which consist of functions on R which are locally in L p with q behavior at infinity, in the sense that the L p norms over the intervals n, n 1 form an q sequence.It is a Banach space for the norm , 1 p, q < ∞.

6.13
The following inclusion relations, with all embeddings continuous, derive immediately from those of the L p and the q spaces.
i If q 1 q 2 , then W L p , q 1 ⊂ W L p , q 2 .
ii If p 1 p 2 , then W L p 2 , q ⊂ W L p 1 , q .
Thus the smallest space is W L ∞ , 1 and the largest space is W L 1 , ∞ .As for duality, one has W L p , q × W L p , q , for 1 p, q < ∞.
The interesting fact is that, for 1 p, q ∞, the set J of all amalgam spaces {W L p , q } may be represented by the points p, q of the same unit square J as in the example of the L p spaces, with the same order structure.However, J is not a lattice with respect to the order 3.12 .One has indeed W L p , q ∧ W L p , q ⊃ W L p∨p , q∧q , W L p , q ∨ W L p , q ⊂ W L p∧p , q∨q , 6.14 where again ∧ means intersection with projective norm and ∨ means vector sum with inductive norm, but equality is not obtained.Thus, as in the previous case, one gets chains by varying either p or q, but not both.
A very useful class of amalgam spaces is the family W FL p , q , 1 p, q ∞, where FL p denotes the set of Fourier transforms of L p functions one may even add weights on both spaces .These spaces have, for instance, nice inclusion and convolution properties.
Among these, the most interesting one is S 0 W FL 1 , 1 , called the Feichtinger algebra.The space S 0 has many interesting properties; for instance, one has the following.i S 0 is a Banach space for the norm f S 0 V g 0 f 1 , and S → S 0 → L 2 , with all embeddings continuous with dense range.Here g 0 is the Gaussian and V g f denotes the Short-Time Fourier or Gabor Transform of f ∈ L 2 R d , given in 6.17 below.
ii S 0 is a Banach algebra with respect to pointwise multiplication and convolution.
iii Time-frequency shifts T x M ω are isometric on S 0 : T x M ω f S 0 f S 0 , where T x g y g y − x translation and M ω h y e 2πiyω h y modulation .S 0 is continuously embedded in any Banach space with the same property and containing g 0 ; thus it is the smallest Banach space with this property.iv The Fourier transform is an isometry on S 0 : Ff S 0 f S 0 .
As for the conjugate dual S × 0 of S 0 , it is a Banach space with norm f S × 0 V g f ∞ .The space S × 0 contains both the δ function and the pure frequency χ ω x e −2πixω .In virtue of i above, we have where all embeddings are continuous and have dense range.It turns out that the triplet is the prototype of a Banach Gel'fand triple, that is, a RHS or LBS in which the extreme spaces are nonreflexive Banach spaces.This is often a very convenient substitute for Schwartz' RHS and it is widely used in signal processing.

Modulation Spaces and Gabor Analysis (Time-Frequency Analysis)
Modulation spaces are closely linked to, and in fact defined in terms of, the Short-Time Fourier or Gabor Transform.Given a C ∞ window function g / 0, the Short-Time Fourier Transform STFT of f ∈ L 2 R d is defined by where, as usual, T x g y g y − x translation and M ω h y e 2πiyω h y modulation .Then, given a v-moderate weight function m x, ω , see Section 3.3.3 the modulation space M p,q m is defined in terms of a mixed norm of an STFT: For p q, one writes M p m ≡ M p,p m .The space M p,q m is a Banach space for the norm f M p,q m : V g f L p,q m .

6.19
Actually, there is a slightly more restrictive definition, which uses the weight function m s x, ω ≡ w s ω 1 |ω| s , s 0, or, equivalently, m s x, ω 1 |ω| 2 s/2 , so that the norm reads Equivalently, one may define a modulation space as the inverse Fourier transform of a Wiener amalgam space: This space is independent of the choice of window g, in the sense that different window functions define equivalent norms.The lattice properties of the family {M p,q m , 1 p, q ∞} are, of course, the same as those of the mixed-norm spaces L p,q m .As for duality, one has M p,q m × M p,q 1/m .Inclusion relations hold, leading again to a lattice structure: if p 1 p 2 , q 1 q 2 , and m 2 Cm 1 , for some constant C > 0, then M p 1 ,q 1 m 1 ⊆ M p 2 ,q 2 m 2 .In particular, one has The class of modulation spaces M p,q w s contains several well-known spaces, such as the following: i The Bessel potential spaces or fractional Sobolev spaces H s M 2 m s : iii The Feichtinger algebra S 0 M 1 and its dual S × 0 M ∞ .
By construction, modulation spaces are function spaces well adapted to Gabor analysis, although they can often be replaced by amalgam spaces.A wealth of information about the spaces and their application in Gabor analysis may be found in the monograph of Gr öchenig 20 .Here we just indicate a few relevant points, especially those that are of a PIP-space nature.We consider in particular the action of several types of operators on such spaces.
i Translation and Modulation Operators a Every amalgam space W L p , q and every mixed-norm space L p,q m are invariant under translation, that is, T y is a totally regular operator in the corresponding PIP-space.b Every modulation space M p,q m is invariant under time-frequency shifts translation and modulation , that is, T y and M ξ are totally regular operators.
ii Fourier Transform a For 1 p, q 2, F maps W L p , q into WL q , p continuously, that is, J F contains every pair p, q , q, p .b If m ξ, −x Cm x, ξ , then every space M p m is invariant under Fourier transform.
iii Gabor Frame Operators Given a nonzero window function g ∈ L 2 R d and lattice parameters α, β > 0, the set of vectors is called a Gabor system.The system G g, α, β is a Gabor frame if there exist two constants m> 0 and M< ∞ such that

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The associated Gabor frame operator S g,g is given by S g,g f : The main results of the Gabor time-frequency analysis stem from the following proposition.
Proposition 6.1.If G g, α, β is a Gabor frame, there exists a dual window g S −1 gg g such that G g, α, β is a frame, called the dual frame.Then one has, for every 27 The outcome of the theory is that the modulation spaces M p,q m turn out to be the natural class of function spaces for Gabor analysis 20 .Define indeed the following operator, generalizing 6.26 slightly: 6.28 Then one has the following results they are highly nontrivial and their proof requires deep analysis i If g, g ∈ W L ∞ , 1 , then the Gabor frame operator S g,g is bounded on every ii If g, g ∈ M 1 v , then S g,g is bounded on M p,q m for all 1 p, q ∞, all v-moderate weights m, and all α, β.
iii If g is a dual window of g, that is, S g, g 1 on L 2 , then the two expansions in 6. 27 converge unconditionally in M p,q m if p, q < ∞.
Clearly statements i and ii can be translated into PIP-space language, by saying that S g,g is a totally regular operator in the chain {L p , 1 p ∞}, respectively, any PIP-space built from modulation spaces.
These results should suffice to convince the reader that the modulation spaces M p,q m are the "natural" spaces for Gabor analysis.Actually, most of this remains true if one replaces modulation spaces by amalgam spaces W L p , q m .Second, it is obvious that most of the statements have a distinctly PIP-space flavor: it is not some individual space M p,q m or W L p , q m that counts, but the whole family, with many operators being regular in the sense of PIP-spaces.

Besov Spaces and Wavelet Analysis (Time-Scale Analysis)
Besov spaces were introduced around 1960 for providing a precise control on the smoothness of solutions of certain partial differential equations.Later on, it was discovered that they are closely linked to wavelet analysis, exactly as the much more recent modulation spaces are structurally adapted to Gabor analysis.In fact, there are many equivalent definitions of Besov spaces.We restrict ourselves to a "discrete" formulation, based on a dyadic partition of unity.Other definitions may be found in the literature quoted in 14 , in particular 19, 38-40 .Let us consider a weight function ϕ ∈ S R with the following properties: Then one defines the following functions by their Fourier transform: i ϕ j ξ ϕ 2 −j ξ , j ∈ Z : high "frequency" for j > 0, low "frequency" for j < 0, ii ψ ξ 1 − ∞ j 1 ϕ 2 −j ξ : low "frequency" part.
Given the weight function ϕ, the inhomogeneous Besov space B s pq is defined as where • s pq denotes the norm The space B s pq is a Banach space and it does not depend on the choice of the weight function ϕ, since a different choice defines an equivalent norm.Note that B s 22 H s , the fractional Sobolev space or Bessel potential space.
For f ∈ B s pq , one may write the following weakly converging expansion, known as a dyadic Littlewood-Paley decomposition: Clearly the first term represents the relatively uninteresting low-"frequency" part of the function, whereas the second term analyzes in detail the high-"frequency" component.Besov spaces enjoy many familiar properties for more details, we refer to the literature, e.g., 19, Section 6.2 or 39, Chapter 2 .

(i) Inclusion Relations
The following relations hold, where all embeddings are continuous: In the terminology of Section 4, the first statement means that the spaces B s pq are interspaces for the RHS S → L 2 → S × .The inclusion relations above mean that the family of spaces B s pq contains again many chains of Banach spaces.

(ii) Interpolation
Besov spaces enjoy nice interpolation properties, in all three parameters s, p, q.
(iii) Duality One has B s pq × B −s p q s ∈ R .

(iv) Translation and Dilation Invariance
Every space B s pq is invariant under translation and dilation the unitary dilation operator in L 2 reads D a f x a −1/2 f x/a .

(v) Regularity Shift
Let J σ : S × → S × denote the operator J s f Then J σ is an isomorphism from B s pq onto B s−σ pq .Thus J σ is totally regular for σ 0, but not for σ > 0. It is also useful to consider the homogeneous Besov space Ḃs pq , defined as the set of all f ∈ S × for which f ˙ s pq < ∞, where the quasinorm • ˙ s pq is defined by pq for s > 0, 1 p, q ∞.In particular, every space Ḃs pq is invariant under translation and dilation, which is not surprising, since these spaces are in fact based on the ax b group, consisting precisely of dilations and translations of the real line, via the coorbit space construction see Section 6.2.4 below .
Besov spaces are well adapted to wavelet analysis, because the definition 6.29 essentially relies on a dyadic partition powers of 2 .Historically, the connection was made with the discrete wavelet analysis, for that reason.Indeed, there exists an equivalent definition given in terms of decay of wavelet coefficients.More precisely, if a function f is expanded in a wavelet basis, the decay properties of the wavelet coefficients allow to characterize precisely to which Besov space the function f belongs, as we shall see below.In addition, the Besov spaces may also be characterized in terms of the continuous wavelet transform see 14, Section 8.4 .
In order to go into details, we have to recall some basic facts about the wavelet transform for simplicity, we restrict ourselves to one dimension .Whereas the STFT is defined in terms of translation and modulation, the continuous wavelet transform is based on translations and dilations Note that we use here the so-called L 1 normalization.It is more frequent to use the L 2 normalization, in which the prefactor is a −1/2 instead of a −1 .In this relation, the wavelet ψ is assumed to satisfy the admissibility condition which implies that ∞ −∞ ψ x dx 0. This condition is only necessary, but becomes sufficient under some mild restrictions, so that it is commonly used as admissibility condition in practice.
However, discretizing the two parameters a and b in 6.33 leads in general only to frames.In order to get orthogonal wavelet bases, one relies on the so-called multiresolution analysis of L 2 R .This is defined as an increasing sequence of closed subspaces of L 2 R : with j ∈ Z V j {0} and j ∈ Z V j dense in L 2 R and such that 2 there exists a function φ ∈ V 0 , called a scaling function, such that the family {φ • − k , k ∈ Z} is an orthonormal basis of V 0 .
Combining conditions 1 and 2 , one sees that {φ jk ≡ 2 j/2 φ 2 j •−k , k ∈ Z} is an orthonormal basis of V j .The space V j can be interpreted as an approximation space at resolution 2 j .Defining W j as the orthogonal complement of V j in V j 1 , that is, V j ⊕ W j V j 1 , we see that W j contains the additional details needed to improve the resolution from 2 j to 2 j 1 .Thus one gets the decomposition L 2 R ⊕ j∈ Z W j .The crucial theorem then asserts the existence of a function ψ, called the mother wavelet, explicitly computable from φ, such that {ψ jk ≡ 2 j/2 ψ 2 j • −k , k ∈ Z} constitutes an orthonormal basis of W j and thus {ψ jk ≡ 2 j/2 ψ 2 j • −k , j, k ∈ Z} is an orthonormal basis of L 2 R : these are the orthonormal wavelets.Thus the expansion of an arbitrary function f ∈ L 2 into an orthogonal wavelet basis {ψ jk , j, k ∈ Z} reads f j,k∈Z c jk ψ jk , with c jk ψ jk | f .

6.36
Additional regularity conditions can be imposed to the scaling function φ.Given r ∈ N, the multiresolution analysis corresponding to φ is called r-regular if d n φ dx n c m 1 |x| m , for all n r and all integers m ∈ N.

6.37
Well-known examples include the Haar wavelets, the B-splines, and the various Daubechies wavelets.As a result of the "dyadic" definition 6.29 -6.30 , it is natural that Besov spaces can be characterized in terms of an r-regular multiresolution analysis {V j }.Let E j : L 2 → V j be the orthogonal projection on V j and D j E j 1 − E j that on W j .Let 0 < s < r and f ∈ L p R .Then, f ∈ B s pq R if and only if D j f p 2 −js δ j , where δ j ∈ q N , and one has means equivalence of norms  6.38 Specializing to p q 2, one gets a similar result for Sobolev spaces: given f ∈ H −r R and |s| < r, f ∈ H s R if, and only if, E 0 f ∈ L 2 R and D j f 2 2 −js j , j ∈ N, where j ∈ 2 N .But there is more.Indeed, modulation spaces and Besov spaces admit decomposition of elements into wavelet bases and each space can be uniquely characterized by the decay properties of the wavelet coefficients.To be precise, let {ψ jk , j, k ∈ Z} be an orthogonal wavelet basis coming from an r-regular multiresolution analysis based on the scaling function φ.Then the following results are typical 41, Chapters II.9 and VI.10 .where β k ∈ p and k∈Z |c jk | p 1/p 2 −j s 1/2−1/p γ j , with γ j ∈ q Z .
ii Homogeneous Besov Spaces: Let |s| < r.Then, if f ∈ Ḃs pq R , its wavelet coefficients c jk verify k∈Z |c jk | p 1/p 2 −j s 1/2−1/p γ j , where γ j ∈ q Z .Conversely, if this condition is satisfied, then f g P , where g ∈ Ḃs pq and P is a polynomial.

α-Modulation Spaces, Coorbit Spaces
The α-modulation spaces α ∈ 0, 1 are spaces intermediate between modulation and Besov spaces, to which they reduce for α 0 and α → 1, respectively.As for coorbit spaces, they are a far-reaching generalization, based on integrable group representations 42 .They contain most of the previous spaces, but we will refrain from describing them in detail, for lack of space.

Unconditional Bases
We conclude this section with some examples of unconditional wavelet bases, as announced in Section 5.2.5.Actually, the concept of wavelet basis can be further generalized to biorthogonal bases, obtained by considering two scales of the type 6.35 and imposing crossorthogonality relations 43, Section 8.3 .For precise definitions, we refer to the literature.
i The Haar wavelet basis is defined by the scaling function φ H χ 0,1 and the mother wavelet ψ H χ 0,1/2 − χ 1/2,1 .It is a standard result that the Haar system is an unconditional basis for every L p R , 1 < p < ∞.
ii The Lemarié-Meyer wavelet basis is an unconditional basis for all L p spaces, Sobolev spaces, and homogeneous Besov spaces Ḃs pq 1 p, q < ∞ 44 .iii There is a class of wavelet bases Wilson bases of exponential decay that are unconditional bases for every modulation space M p,q m , 1 p, q < ∞, but not for L p , 1 < p < ∞, p / 2 45 .

Conclusion
Most families of function spaces used in analysis and in signal processing come in scales or lattices and in fact are, or contain, PIP-spaces.The lattice indices defining the partial order characterize the properties of the corresponding functions or distributions: smoothness, local integrability, decay at infinity.Thus it seems natural to formulate the properties of various operators globally, using the theory of PIP-space operators; in particular the set j A of an operator encodes its properties in a very convenient and visual fashion.In addition, it is often possible to determine uniquely whether a function belongs to one of those spaces simply by estimating the asymptotic behavior of its Gabor or wavelet coefficients, a real breakthrough in functional analysis 41 .
A legitimate question is whether there are instances where a PIP-space is really needed, or a RHS could suffice.The answer is that there are plenty of examples, among the applications we have enumerated in Section 6 without details, for lack of space .We may therefore expect that the PIP-space formalism will play a significant role in Gabor/wavelet analysis, as well as in mathematical physics.
Concerning the applications in mathematical physics, in almost all cases, the relevant structure is a scale or a chain of Hilbert spaces, which allows a finer control on the behavior of operators.For instance all details may be found in our monograph 14 the following are considered.
i For the singular interactions in quantum mechanics δ or δ potentials , the approach of Grossmann et al. 18, 24, 25 is definitely the most appropriate; a RHS would be irrelevant.
ii The very formulation of the WVW approach to quantum scattering theory 34-36 requires a LHS, whose end spaces are in fact Hilbert spaces.
iii For quantum field theory, the energy bounds of Fredenhagen and Hertel 46 rely in an essential way on the scale generated by the Hamiltonian, and so does Nelson's approach to Euclidean field theory 47 .
As for the applications in signal processing, all families of spaces routinely used are, or contain, chains of Banach spaces, which are needed for a fine tuning of elements Advances in Mathematical Physics usually, distributions and operators on them.Such are, for instance, L p spaces, amalgam spaces, modulation spaces, Besov spaces, or coorbit spaces, mentioned above.Here again, a RHS, even with Banach end spaces like the Feichtinger algebra and its conjugate dual, is clearly not sufficient.The whole Section 6.2 illustrates the statement.

3 . 5 2
The scale 1.4 of Hilbert spaces {H n , n ∈ Z} built on powers of A A * 1.

Figure 1 :
Figure 1: The unit square describing the lattice J.

Figure 2 :
Figure 2: Characterization of the operator A, in the case of a scale.

i
Inhomogeneous Besov Spaces: f ∈ B s pq R if it can be written as and only if ρ is absolutely continuous;ii L ρ is reflexive if and only if ρ and ρ are absolutely continuous and ρ has the Fatou property.