Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.


Introduction
The present paper is a continuation of our papers [4] and [5].In [5], we analysed the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds.We also described the tensor structure of sequential mappings on spaces of Fourier coefficients and characterised their adjoint mappings.In particular, these classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, dual spaces of distributions and ultradistributions, in both Roumieu and Beurling settings.In another work, [4], we have characterised Komatsu spaces of ultradifferentiable functions and ultradistributions on compact manifolds in terms of the eigenfunction expansions related to positive elliptic operators.Here we note that, using properties of the elliptic operators and the Plancherel formula one can get such type of characterisation of smooth functions in terms of their Fourier coefficients.For example, if E is a positive elliptic pseudo-differential operator on a compact manifold X without boundary and λ j denotes its eigenvalues in the ascending order, then smooth functions on X can be characterised in terms of their Fourier coefficients: where f (j, l) = f, e l j L 2 with e l j being the l th eigenfunction corresponding to the eigenvalue λ j (of multiplicity d j ).Such characterisations for analytic functions were obtained by Seeley in [25], with a subsequent extension to Gevrey and, more generally, to Komatsu classes, in [4].The results obtained in [5] do not include the cases of smooth functions on compact manifolds.We will extend the results in [5] to the spaces of smooth functions.Moreover in this work, we aim at discussing an abstract analysis of the spaces of smooth type functions generated by basis elements of an arbitrary Hilbert space.Considering an abstract point of view has an advantage that the results will cover the analysis of smooth functions on different spaces like compact Lie groups and manifolds.In particular, we introduce a notion of smooth functions generated by elements of a Hilbert space H forming a basis.We will show that the appearing spaces of coefficients with respect to expansions in eigenfunctions of positive selfadjoint operators are perfect spaces in the sense of the theory of sequence spaces (see, e.g., Köthe [11]).Consequently, we obtain tensor representations for linear mappings between spaces of smooth type functions.Such discrete representations in a given basis are useful in different areas of time-frequency analysis, in partial differential equations, and in numerical investigations.
Using the obtained representations we establish the universality properties of the appearing spaces.In [29], L. Waelbroeck proved the so-called universality of the space of Schwartz distributions E(V ) with compact support on a C ∞ -manifold V, with the δ-mapping δ : V → E(V ), that is, any vector valued C ∞ -mapping f : V → E, from V to a sequence space E, factors through δ : V → E(V ) by a unique linear morphism f : E(V ) → E as f = f • δ.The universality of the spaces of Gevrey functions on the torus has been established in [27].As an application of our tensor representations, we prove the universality of the spaces of smooth functions on compact manifolds.
Our analysis is based on the global Fourier analysis on arbitrary Hilbert spaces using techniques similar to compact manifold which was consistently developed in [6], with a number of subsequent applications, for example to the spectral properties of operators [7], or to the wave equations for the Landau Hamiltonian [21].The corresponding version of the Fourier analysis is based on expansions with respect to orthogonal systems of eigenfunctions of a self-adjoint operator.The non self-adjoint version has been developed in [20], with a subsequent extension in [22].
The paper is organised as follows.In Section 2 we will briefly recall the constructions leading to the global Fourier analysis on arbitrary Hilbert spaces and define the smooth type function spaces.In Section 3 we very briefly recall the relevant definitions from the theory of sequence spaces.In Section 4 we present the main results of this paper and their proofs.In Section 5 we prove the universality results for the smooth functions on compact manifold.

Fourier Analysis on Hilbert Spaces
Let (H, || • || H ) be a separable Hilbert space and denote by a collection of elements of H .We assume that U is a basis of the space H with the property (e jl , e mn ) H = δ jm δ ln , j, m ∈ N and 1 where δ jm is the Kroneckar delta, equal to 1 for j = m, and to zero otherwise.Also let us fix a sequence of positive numbers Λ := {λ j } j∈N such that 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ ..., and the series converges for some s 0 > 0. For example, in a compact C ∞ manifold X of dimension n without boundary and with a fixed measure we have where 0 < λ 1 < λ 2 < ... are eigenvalues of a positive elliptic pseudo-differential operator E of an integer order ν, with H j ⊂ L 2 (X) the corresponding eigenspace and We associate to the pair {U, Λ} a linear self-adjoint operator E : H → H such that for s ∈ R and those f ∈ H for which the series converges in H. Then E is densely defined since E s e jl = λ s j e jl , 1 ≤ l ≤ d j , j ∈ N, and U is a basis of H. Also we write H j = span{e jl } 1≤l≤d j , and so dim H j = d j .Then we have The Fourier transform for f ∈ H is defined as We next define the following notions: The spaces of smooth type functions are defined by where There exists a linear pairing It is easy to see from this that every continuous linear functional on

Sequence spaces and sequential linear mappings
We briefly recall that a sequence space V is a linear subspace of The dual V (α-dual in the terminology of G. Köthe [11]) is a sequence space defined by A dual space V is normal so that any perfect space is normal.
A pairing •, • V on V is a bilinear function on V × V defined by which converges absolutely by the definition of V .
Definition 3.1.φ : V → C is called a sequential linear functional if there exists some a ∈ V such that φ(f ) = f, a V for all f ∈ V. We abuse the notation by also writing a : V → C for this mapping.
Definition 3.2.A mapping φ : V → W between two sequence spaces is called a sequential linear mapping if (1) φ is algebraically linear, (2) for any g ∈ W , the composed mapping g • φ : V → C is in V .

Tensor representations and the adjointness
In this section we discuss α-duals of the spaces, tensor representations for mappings between these spaces and their α-duals, and obtain the corresponding adjointness theorem.

Duals and α-duals.
In this section we first prove that the α-dual, H s E of the space H s E , where coincides with the space H −s E .Remark 4.1.Here we observe that, . It follows from (2.1) and the definition of From the definition of the α-dual and using the inequality, Our first result is the identifiction of the topological dual with the α-dual.
This implies that u ∈ H s E .We thus obtain φ(j, l)w jl .
Then using Remark 4.1 we have From this we can have the following corollary.
We next define the α-dual of the space From this we can state the following lemma and the proof will follow from our above observation.
Next we proceed to prove that H ∞ E is a perfect space.But before that let us prove the following lemma. Then since d j ≥ 1.
We want to show that To prove this we will use the following identity: From this we get We consider the second term of the above inequality, that is, where ) So we have from (4.4), (4.5) and (4.7) that Next we proceed to prove the opposite direction.Let In particular, we have v ∈ H −s E for some s ∈ R. We have to show By the Cauchy-Schwartz inequality we have Proof.From the definition we always have w jl e jl .
The series is convergent and φ ∈ H since since w ∈ [ H ∞ E ] ∧ and using (2.1) and Lemma 4.6.Also from the property (e jl , e mn ) H = δ jm δ ln , for j, m ∈ N and 1 We observe that And so we have We consider the second term of the above inequality, that is, Then for any t > 0 we have, using (4.13), that Now since v ∈ H ∞ E , in particular we can have 2s = 2t + 3s 0 , for any t > 0, which gives, using (2.1) completing the proof.
We next prove the adjointness theorem, also recalling Definition 3.1.Let H, G be two Hilbert spaces and E and F be the operators defined by (2.2) corresponding to the bases {e j } j∈N , {h k } k∈N , respectively, where d j = dim X j and g k = dim Y k , and We denote the corresponding spaces to the operators E and F in the Hilbert space H and G respectively by H ∞ E and G ∞ F .Theorem 4.9.A linear mapping f : and Furthermore, the adjoint mapping f : G ∞ F → H ∞ E defined by the formula f (v) = v • f is also sequential, and the transposed matrix (f kj ) t represents f , with f and f related by Let us summarise the ranges for indices in the used notation as well as give more explanation to (4.16).For f : so that ) and where we view f kj as a matrix, f kj ∈ C g k ×d j , and the product of the matrices has been explained in (4.17).
Remark 4.10.Let us now describe how the tensor Then v ki ∈ G ∞ F , and since f is sequential we have v ki • f ∈ H ∞ E , and we can write the l th component of the vector The formula (4.21) will be shown in the proof of Theorem 4.9.In particular, since for φ ∈ H ∞ E we have f (φ) ∈ G ∞ F , it will be a consequence of (4.32) and (4.33) later on that so that the tensor (f kjli ) is describing the transformation of the Fourier coefficients of φ into those of f (φ).
To prove Theorem 4.9 we first establish the following lemma.
Proof of Lemma 4.11.Let u ∈ H ∞ E and u ≈ ( u(j)) j∈N .Define u n := u (n) (j) j∈N by setting Then for any w ∈ H ∞ E , we get u − u n , w H ∞ E → 0 as n → ∞.This is true since F and from (4.15) and (4.16) we have where Now we have the mapping f : Then from (4.15) and (4.16) we have u, The proof is complete.Remark 4.12.This proof does not require sequentiality and it can be used to improve the argument in [5,Theorem 4.7].
Proof of Theorem 4.9.Let us assume first that the mapping f : and ) p∈N be such that for some j, l where j ∈ N, 1 ≤ l ≤ d j , we have We now first show that where The way in which f has been defined we have We can then write In particular using the definition of u 1 and (4.26) we get for any j ∈ N and 1 Then we consider the series so that we have Let ǫ = (ǫ i ) 1≤i≤d k , k ∈ N, be such that ǫ i ∈ C and |ǫ i | ≤ C, for all i and such that Then It follows from Lemma 4.11 that So we proved that if (f kjli ) satisfies we have from (4.27) and (4.29), respectively, that f kjl u(j, l), (4.30) So by (ii) we get that Now to prove the converse part we assume that f : We have to show that f can be represented as f ≈ (f kjli ) k,j∈N,1≤l≤d j ,1≤i≤g k and satisfies (4.15) and (4.16).
Define for k, i where k ∈ N and 1 ≤ i ≤ g k , the sequence u ki = u ki j j∈N such that u ki j ∈ C d j and u ki j (l) = u ki (j, l), given by We can also write

.32)
We have If we denote f ki jl by f ki jl = f kjli , we can say that f is represented by the tensor This completes the proof of Theorem 4.9.

Applications to universality
In this section we give an application of the developed analysis to the universality problem.We start with the spaces of smooth functions, and then make some remarks how the same arguments can be extended to the Komatsu classes setting from [4].

First we recall the notations:
Let H, G be two Hilbert spaces and let E and F be the operators corresponding to the bases {e j } j∈N , {h k } k∈N , as in (2.2), where d j = dim X j and g k = dim Y k , and We denote the spaces of smooth type functions corresponding to the operators E and F in the Hilbert space H and G, respectively, by H ∞ E and G ∞ F .
The main application of Theorem 4.9 will be in the setting when X, Y are compact manifold without boundary, where H = L 2 (X) and Using Theorem 4.9 we prove the universality of the spaces of the smooth type functions, C ∞ (X), where we can write Further details of such spaces can be found in [6].In particular, if E is an elliptic pseudo-differential operator of positive order, then this is just the usual space of smooth functions on X. Definition 5.1.Let E be a self-adjoint, positive operator.A mapping f : X → W from the compact manifold X to a sequence space W , is said to be a H ∞ E -mapping if for any u ∈ W , the composed mapping u • f : X → C belongs to H ∞ E .Next we prove the universality of the spaces of smooth type functions.Theorem 5.2.Let X be a compact manifold.
φ(j, l)(δ x ) jl = φ(x), for all φ ∈ H ∞ E , x ∈ X, is a H ∞ E -mapping.(ii) If g : H ∞ E → G ∞ F is a sequential linear mapping, then the composed mapping g • δ : X → G ∞ F is a H ∞ E -mapping.(iii) For any H ∞ E -mapping f : X → G ∞ F , there exists a unique sequential linear mapping f : We define the composed mapping This is well-defined since v ∈ H ∞ E and δ Since the above is true for any s ∈ R, we have [ Using same argument as in the proof of (i) and from the definition of the mapping δ we have u • g • δ x = u • g(x) and u • g • δ belongs to H ∞ E .So from Definition 5.1, g • δ is a H ∞ E -mapping.
(iii) Existence of f : Hence by Theorem 4.9 there is an adjoint mapping, we denote it by f , where f : ∧ is a sequential mapping.By the definition of the adjoint mapping f we have where •, • is the bilinear function on H ∞ E × H ∞ E defined in Section 3. The above can be written as for any v ∈ G ∞ F .This proves f = f • δ.
Suppose f • δ = f = 0. We have to show that f = 0 on H ∞ E .Since f is sequential, there exists g :

Lemma 4 . 11 .
Let f : H ∞ E → G ∞F be a linear mapping represented by an infinite tensor (f kjli ) k,j∈N,1≤l≤d j ,1≤i≤g k satisfying (4.15) and (4.16).Then for all completing the proof.Now using Remark 4.4 and Lemma 4.6 we can prove that the spaces H ∞ E are perfect spaces.
Before proving the adjointness theorem we first prove the following lemma, || HS ||w j || HS < ∞, (4.10)where v j = (v jl ) 1≤l≤d j = v(j,l), and j ∈ N, and the same for w.Moreover, suppose that for all v ∈ H ∞ E , (4.10) holds.Then we must have w ∈ H ∞ E .Also, if for all w ∈ H ∞ E , (4.10) holds, we have v ∈ H ∞ E .Proof.Let us assume first that 2 HS < ∞, and so from Remark 4.4 we have w ∈ H ∞ E which implies that [ H ∞ E ] ∧ ⊆ H ∞ E holds.4.2.Adjointness.
From the definition of H ∞ E -mapping we have to show that u• g • δ ∈ H ∞ E .Now by given condition g : [H ∞ E ] ′ → G ∞ F , so we have u • g ∈ [H ∞ E ] ′′ .Here we claim that [H ∞ E ] ′′ = H ∞ E .Note that H ∞ E ⊆ [H ∞ E ] ′′ .Recall that, [H ∞ E ] ′ = s∈RH −s E and so for any s ∈ R,