Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III. Hilbert spaces and Universality

TL;DR

This paper investigates the structure of smooth function spaces generated by Hilbert space elements, establishing their perfect sequence space nature, tensor structures, and demonstrating their universality on compact manifolds.

Contribution

It extends previous work by characterizing these function spaces as perfect sequence spaces and proving their universality on compact manifolds.

Findings

Spaces are perfect sequence spaces

Tensor structure of sequential mappings characterized

Universality of these spaces on compact manifolds proven

Abstract

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 previous papers in this series. 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.

Equations

$$f\in C^{\infty }(X)\iff \forall N\exists C_N:|\hat{f}(j,l)|\le C_N{\lambda }_j^{-N}\text{for all }j\ge 1,1\le l\le d_j,$$

$$\mathcal{U}:=\{e_{jl}:e_{jl}\in \mathcal{H},1\le l\le d_j,d_j\in \mathbb{N}{\}}_{j\in \mathbb{N}}$$

$$(e_{jl},e_{mn}{)}_{\mathcal{H}}={\delta }_{jm}{\delta }_{ln},j,m\in \mathbb{N}\text{and}1\le l\le d_j,1\le n\le d_m,$$

$$\sum _{j=1}^{\infty }{d}_j{\lambda }_j^{-s_0}<\infty$$

$$\sum _{j=1}^{\infty }{d}_j(1+{\lambda }_j{)}^{-q}<\infty \text{if and only if}q>\frac{n}{\nu },$$

$$d_j:=\dim H_j,H_0:=\text{ker}E,{\lambda }_0:=0,d_0:=\dim H_0.$$

$$E^{s}f=\sum _{j=1}^{\infty }\sum _{l=1}^{d_j}{\lambda }_j^s(f,e_{jl})e_{jl}$$

$$E^s{e}_{jl}={\lambda }_j^s{e}_{jl},1\le l\le d_j,j\in \mathbb{N},$$

$$\mathcal{H}=\underset{j\in \mathbb{N}}{⨁}{H}_j.$$

$$\hat{f}(j,l):=(f,e_{jl}{)}_{\mathcal{H}},j\in \mathbb{N},1\le l\le d_j.$$

$$H_E^{\infty }:=\bigcap _{s\in \mathbb{R}}{H}_E^s,$$

$$H_E^s:=\left\{\varphi \in \mathcal{H}:\sum _{j=1}^{\infty }{\lambda }_j^{2s}||\hat{\varphi }(j)|{|}_{\mathtt{HS}}^2<\infty \right\}.$$

$$(f,g{)}_{H_E^s}=(E^{s}f,E^{-s}g{)}_{\mathcal{H}},$$

$${\mathbb{C}}^{\mathbb{Z}}=\{a=(a_j)|a_j\in \mathbb{C},j\in \mathbb{Z}\}.$$

$$\hat{V}=\{a\in {\mathbb{C}}^{\mathbb{Z}}:\sum _{j\in \mathbb{Z}}|f_j||a_j|<\infty \text{ for all }f\in V\}.$$

$$⟨f,g{⟩}_V=\sum _{j\in \mathbb{Z}}{f}_j{g}_j\in \mathbb{C},$$

$$\widehat{H_E^s}=\left\{v=(v_j{)}_{j\in \mathbb{N}},v_j\in {\mathbb{C}}^{d_j};\sum _{j=1}^{\infty }\sum _{l=1}^{d_j}|\hat{\varphi }(j,l)||v_{jl}|<\infty ,\text{for all }\varphi \in H_E^s\right\},$$

$$w\in \widehat{H_E^s}⟹\sum _{j=1}^{\infty }{\lambda }_j^{-s-s_0/2}||w_j|{|}_{\mathtt{HS}}<\infty .$$

$$||w_j|{|}_{\mathtt{HS}}\le ||w_j|{|}_{l^1},$$

$$\sum _{j=1}^{\infty }{\lambda }_j^{-s-s_0/2}||w_j|{|}_{\mathtt{HS}}$$

$$\begin{gathered} \sum _{j=1}^{\infty }\sum _{l=1}^{d_j}|\hat{\varphi }(j,l)||u_{jl}| \\ \le {\left(\sum _{j=1}^{\infty }\sum _{l=1}^{d_j}{\lambda }_j^{2s}|\hat{\varphi }(j,l){|}^2\right)}^{1/2}{\left(\sum _{j=1}^{\infty }\sum _{l=1}^{d_j}{\lambda }_j^{-2s}|u_{jl}{|}^2\right)}^{1/2}<\infty ,\text{(by assumption)}. \end{gathered}$$

$$w(\varphi )=\sum _{j=1}^{\infty }{\lambda }_j^{-s_0/2}\hat{\varphi }(j)\cdot w_j=\sum _{j=1}^{\infty }{\lambda }_j^{-s_0/2}\sum _{l=1}^{d_j}\hat{\varphi }(j,l)w_{jl}.$$

$$|w(\varphi )|$$

$$||\widehat{{\varphi }_m}(j)-\hat{\varphi }(j)|{|}_{\mathtt{HS}}\le C_m{\lambda }_j^{-s},$$

$$|w({\varphi }_m-\varphi )|$$

$$v\in H_E^{\infty }⟺\sum _{j=1}^{\infty }{\lambda }_j^{2s}||v_j|{|}_{\mathtt{HS}}^2<\infty ,\forall s\in \mathbb{R},$$

$$\widehat{H_E^{\infty }}=\left\{v=(v_j),v_j\in {\mathbb{C}}^{d_j}:\sum _{j=1}^{\infty }\sum _{l=1}^{d_j}|v_{jl}||\hat{\varphi }(j,l)|<\infty ,\text{ for all}\varphi \in H_E^{\infty }\right\}.$$

$$v_{jl}=\hat{v}(j,l)={\lambda }_j^s,1\le l\le d_j.$$

$$\sum _{j=1}^{\infty }{\lambda }_j^{-2s}{\lambda }_j^{-s_0}||v_j|{|}_{\mathtt{HS}}^2=\sum _{j=1}^{\infty }{d}_j{\lambda }_j^{-2s}{\lambda }_j^{-s_0}{\lambda }_j^{2s}=\sum _{j=1}^{\infty }{d}_j{\lambda }_j^{-s_0}<\infty .$$

$$\sum _{j=1}^{\infty }{\lambda }_j^{2s}||w_j|{|}_{\mathtt{HS}}^2$$

Full text

The second author was partly supported by the FWO Odysseus Project, by the EPSRC Grant EP/K039407/1, and by the Leverhulme Research Grant RPG-2014-02.

1. 1 Introduction

2. 2 Fourier Analysis on Hilbert Spaces

3. 3 Sequence spaces and sequential linear mappings

4. 4 Tensor representations and the adjointness

5. 4.1 Duals and $\alpha$-duals

6. 4.2 Adjointness

7. 5 Applications to universality

8. 5.1 Extension to Komatsu classes

1. 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 $\lambda_{j}$ denotes its eigenvalues in the ascending order, then smooth functions on $X$ can be characterised in terms of their Fourier coefficients:

[TABLE]

where $\widehat{f}(j,l)=\left(f,e^{l}_{j}\right)_{L^{2}}$ with $e^{l}_{j}$ being the $l^{th}$ eigenfunction corresponding to the eigenvalue $\lambda_{j}$ (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 $\mathcal{H}$ forming a basis. We will show that the appearing spaces of coefficients with respect to expansions in eigenfunctions of positive self-adjoint 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 $\mathcal{E}(V)$ with compact support on a $C^{\infty}$-manifold $V,$ with the $\delta$-mapping $\delta:V\rightarrow\widehat{\mathcal{E}(V)},$ that is, any vector valued $C^{\infty}$-mapping $f:V\rightarrow E,$ from $V$ to a sequence space $E$, factors through $\delta:V\rightarrow\widehat{\mathcal{E}(V)}$ by a unique linear morphism $\tilde{f}:\widehat{\mathcal{E}(V)}\rightarrow E$ as $f=\tilde{f}\circ\delta.$ 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.

2. Fourier Analysis on Hilbert Spaces

Let $(\mathcal{H},||\cdot||_{\mathcal{H}})$ be a separable Hilbert space and denote by

[TABLE]

a collection of elements of $\mathcal{H}$ . We assume that $\mathcal{U}$ is a basis of the space $\mathcal{H}$ with the property

[TABLE]

where $\delta_{jm}$ is the Kroneckar delta, equal to 1 for $j=m$, and to zero otherwise. Also let us fix a sequence of positive numbers $\Lambda:=\{\lambda_{j}\}_{j\in\mathbb{N}}$ such that $0<\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq...,$ and the series

[TABLE]

converges for some $s_{0}>0.$ For example, in a compact $C^{\infty}$ manifold $X$ of dimension $n$ without boundary and with a fixed measure we have

[TABLE]

where $0<\lambda_{1}<\lambda_{2}<...$ are eigenvalues of a positive elliptic pseudo-differential operator $E$ of an integer order $\nu$, with $H_{j}\subset L^{2}(X)$ the corresponding eigenspace and

[TABLE]

We associate to the pair $\{\mathcal{U},\Lambda\}$ a linear self-adjoint operator $E:\mathcal{H}\rightarrow\mathcal{H}$ such that

[TABLE]

for $s\in\mathbb{R}$ and those $f\in\mathcal{H}$ for which the series converges in $\mathcal{H}.$ Then $E$ is densely defined since

[TABLE]

and $\mathcal{U}$ is a basis of $\mathcal{H}$. Also we write $H_{j}=\text{span}\{e_{jl}\}_{1\leq l\leq d_{j}},$ and so $\dim H_{j}=d_{j}.$ Then we have

[TABLE]

The Fourier transform for $f\in\mathcal{H}$ is defined as

[TABLE]

We next define the following notions:

 

The spaces of smooth type functions are defined by

[TABLE]

where

[TABLE]

There exists a linear pairing

[TABLE]

for $f\in H^{s}_{E}$ and $g\in H^{-s}_{E}.$ It is easy to see from this that every continuous linear functional on $H^{s}_{E}$ is of the form $f\rightarrow(f,g)_{H^{s}_{E}}$ for some $g\in H^{-s}_{E},$ that is $(H^{s}_{E})^{\prime}=H^{-s}_{E}.$ Then we denote the space of distributions as $\left(H^{\infty}_{E}\right)^{\prime}=\bigcup\limits_{s\in\mathbb{R}}H^{-s}_{E}$.

3. Sequence spaces and sequential linear mappings

We briefly recall that a sequence space $V$ is a linear subspace of

[TABLE]

The dual $\widehat{V}$ ($\alpha$-dual in the terminology of G. Köthe [11]) is a sequence space defined by

[TABLE]

A sequence space $V$ is called perfect if $\widehat{\widehat{V}}=V$. A sequence space is called normal if $f=(f_{j})_{j\in\mathbb{N}}\in V$ implies $|f|=(|f_{j}|)_{j\in\mathbb{N}}\in V.$ A dual space $\widehat{V}$ is normal so that any perfect space is normal.

A pairing ${\langle\cdot,\cdot\rangle}_{V}$ on $V$ is a bilinear function on $V\times\widehat{V}$ defined by

[TABLE]

which converges absolutely by the definition of $\widehat{V}.$

Definition 3.1**.**

$\phi:V\rightarrow\mathbb{C}$* is called a sequential linear functional if there exists some $a\in\widehat{V}$ such that $\phi(f)=\langle f,a\rangle_{V}$ for all $f\in V.$ We abuse the notation by also writing $a:V\rightarrow\mathbb{C}$ for this mapping.*

Definition 3.2**.**

A mapping $\phi:V\rightarrow W$ between two sequence spaces is called a sequential linear mapping if

1. (1)

$\phi$* is algebraically linear,* 2. (2)

for any $g\in\widehat{W},$ the composed mapping $g\circ\phi:V\rightarrow\mathbb{C}$ is in $\widehat{V}.$

4. Tensor representations and the adjointness

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

4.1. Duals and $\alpha$-duals

In this section we first prove that the $\alpha$-dual, $\widehat{H^{s}_{E}}$ of the space $H^{s}_{E},$ where

[TABLE]

coincides with the space $H^{-s}_{E}.$

Remark 4.1**.**

Here we observe that,

[TABLE]

*Indeed, let $w\in\widehat{H^{s}_{E}}.$ Define $\widehat{\phi}(j,l)=\lambda_{j}^{-(s+s_{0}/2)}$ for $j\in\mathbb{N}$ and $1\leq l\leq d_{j}.$

Then $||\widehat{\phi}(j)||_{\mathtt{HS}}=d_{j}^{1/2}\lambda_{j}^{-s-s_{0}/2}.$ It follows from (2.1) and the definition of $H^{s}_{E}$ that $\phi\in H^{s}_{E}.$

From the definition of the $\alpha$-dual and using the inequality,*

[TABLE]

we can then conclude that,

[TABLE]

Our first result is the identifiction of the topological dual with the $\alpha$-dual.

Theorem 4.2**.**

$\widehat{H^{s}_{E}}=H^{-s}_{E}.$**

Proof.

First we will show $H^{-s}_{E}\subseteq\widehat{H^{s}_{E}}.$

Let $u\in H^{-s}_{E}.$

Then from the definition we have $\sum\limits_{j=1}^{\infty}\lambda_{j}^{-2s}||\widehat{u}(j)||^{2}_{{\mathtt{HS}}}<\infty.$ We denote for $j\in\mathbb{N},$ $u_{jl}=\widehat{u}(j,l)\in\mathbb{C}^{d_{j}},$ $l=1,2,...,d_{j}.$ Using the Cauchy-Schwartz inequality, for any $\phi\in H^{s}_{E}$ we get

[TABLE]

This implies that $u\in\widehat{H}^{s}_{E}.$ We thus obtain $H^{-s}_{E}\subseteq\widehat{H^{s}_{E}}.$

Next we will show that $\widehat{H^{s}_{E}}\subseteq H^{-s}_{E}.$

By duality we know that $(H^{s}_{E})^{\prime}=H^{-s}_{E}.$ So it will be enough if we can prove $\widehat{H^{s}_{E}}\subseteq(H^{s}_{E})^{\prime}.$

Let $w\in\widehat{H^{s}_{E}}.$ Then for $\phi\in{H^{s}_{E}}$ we define

[TABLE]

Then using Remark 4.1 we have

[TABLE]

since $\phi\in H^{s}_{E}$. So $w(\phi)$ is well defined.

We next check that $w$ is continuous. Let $\phi_{m}\rightarrow\phi$ as $m\rightarrow\infty$ in $H^{s}_{E}.$ This means, $||\phi_{m}-\phi||_{H^{s}_{E}}\rightarrow 0$ as $m\rightarrow\infty,$ which implies $\lambda_{j}^{s}||\widehat{\phi_{m}}(j)-\widehat{\phi}(j)||_{\mathtt{HS}}\rightarrow 0$ as $m\rightarrow\infty,$ for $j\in\mathbb{N}$. So we have

[TABLE]

where $C_{m}\rightarrow 0$ as $m\rightarrow\infty.$ Then

[TABLE]

as $m\rightarrow\infty.$ Hence $w$ is continuous. This gives $w\in(H^{s}_{E})^{\prime}=H^{-s}_{E}.$ So we have $\widehat{H^{s}_{E}}\subseteq H^{-s}_{E},$ that implies $\widehat{H^{s}_{E}}=H^{-s}_{E}.$ ∎

From this we can have the following corollary.

Corollary 4.3**.**

$v\in\widehat{H^{s}_{E}}$* $\iff$ $\sum\limits_{j=1}^{\infty}\lambda_{j}^{-2s}||v_{j}||^{2}_{\mathtt{HS}}<\infty$ and $v_{j}\in\mathbb{C}^{d_{j}},$ we denote $v_{jl}=\widehat{v_{jl}}=(v,e_{jl})_{\mathcal{H}}.$*

Remark 4.4**.**

From the definition of $H^{\infty}_{E}$,

[TABLE]

where $v_{j}=\widehat{v}(j)\in\mathbb{C}^{d_{j}}.$

We next define the $\alpha$-dual of the space $\widehat{H^{\infty}_{E}},$

[TABLE]

Also observe that $\widehat{H^{\infty}_{E}}=\widehat{\bigcap\limits_{s\in\mathbb{R}}H^{s}_{E}}=\bigcup\limits_{s\in\mathbb{R}}\widehat{H^{s}_{E}}=\bigcup\limits_{s\in\mathbb{R}}H^{-s}_{E}.$ From this we can state the following lemma and the proof will follow from our above observation.

Lemma 4.5**.**

$v\in\widehat{H^{\infty}_{E}}\iff$* for some $s\in\mathbb{R}$ we have $\sum\limits_{j=1}^{\infty}\lambda_{j}^{-2s}||v_{j}||^{2}_{\mathtt{HS}}<\infty$.*

Next we proceed to prove that $H^{\infty}_{E}$ is a perfect space. But before that let us prove the following lemma.

Lemma 4.6**.**

We have $w\in[\widehat{H^{\infty}_{E}}]^{\wedge}$ if and only if $\sum\limits_{j=1}^{\infty}\lambda_{j}^{2s}||w_{j}||^{2}_{\mathtt{HS}}<\infty$ for all $s\in\mathbb{R}.$

Proof.

Let $w\in[\widehat{H^{\infty}_{E}}]^{\wedge}.$ Let $s\in\mathbb{R},$ we define

[TABLE]

Then $||v_{j}||^{2}_{\mathtt{HS}}=d_{j}\lambda_{j}^{2s}$ and so $v\in H^{-s-\frac{s_{0}}{2}}_{E}$ because in view of (2.1) we have

[TABLE]

Then $v\in\widehat{H^{\infty}_{E}}=\bigcup\limits_{s\in\mathbb{R}}H^{-s}_{E}$ which gives $\sum\limits_{j=1}^{\infty}\sum\limits_{l=1}^{d_{j}}|w_{jl}||v_{jl}|<\infty.$ Now we observe first that

[TABLE]

since $d_{j}\geq 1.$

We want to show that $\sum\limits_{j=1}^{\infty}||v_{j}||_{\mathtt{HS}}^{2}||w_{j}||^{2}_{\mathtt{HS}}<\infty.$ To prove this we will use the following identity:

[TABLE]

From this we get

[TABLE]

We consider the second term of the above inequality, that is,

[TABLE]

since $v\in H^{-s-s_{0}/2}_{E}.$ Then we get

[TABLE]

where $|u_{ji}|^{2}=|{v}_{ji}|^{2}+C\lambda_{j}^{2s+s_{0}}.$ Now

[TABLE]

which implies that $u\in H^{-s-s_{0}}_{E},$ that is $u\in\widehat{H^{\infty}_{E}}.$ This gives that

[TABLE]

as $w\in[\widehat{H^{\infty}_{E}}]^{\wedge}$ and $u\in\widehat{H^{\infty}_{E}}.$ So we have from (4.4), (4.5) and (4.7) that

[TABLE]

Next we proceed to prove the opposite direction. Let

[TABLE]

be such that $\sum\limits_{j=1}^{\infty}\lambda_{j}^{2s}||w_{j}||^{2}_{\mathtt{HS}}<\infty$ for all $s\in\mathbb{R}.$ Let $v\in\widehat{H^{\infty}_{E}}=\bigcup\limits_{s\in\mathbb{R}}H^{-s}_{E}.$ In particular, we have $v\in H^{-s}_{E}$ for some $s\in\mathbb{R}.$ We have to show

[TABLE]

By the Cauchy-Schwartz inequality we have

[TABLE]

It follows that $w\in[\widehat{H^{\infty}_{E}}]^{\wedge},$ completing the proof.∎

Now using Remark 4.4 and Lemma 4.6 we can prove that the spaces $H^{\infty}_{E}$ are perfect spaces.

Theorem 4.7**.**

$H^{\infty}_{E}$* is a perfect space.*

Proof.

From the definition we always have $H^{\infty}_{E}\subseteq[\widehat{H^{\infty}_{E}}]^{\wedge}.$ We will prove the other direction. Let $w\in[\widehat{H^{\infty}_{E}}]^{\wedge},$ $w=(w_{j})_{j\in\mathbb{N}}$ and $w_{j}\in\mathbb{C}^{d_{j}}.$

Define

[TABLE]

The series is convergent and $\phi\in\mathcal{H}$ since

[TABLE]

since $w\in[\widehat{H^{\infty}_{E}}]^{\wedge}$ and using (2.1) and Lemma 4.6.

Also from the property $(e_{jl},e_{mn})_{\mathcal{H}}=\delta_{jm}\delta_{ln},$ for $j,m\in\mathbb{N}$ and $1\leq l\leq d_{j},$ $1\leq n\leq d_{m},$ it is obvious that $\widehat{\phi}(j,l)=(\phi,e_{jl})_{\mathcal{H}}=w_{jl}.$ This gives $||\widehat{\phi}(j)||_{\mathtt{HS}}=||w_{j}||_{\mathtt{HS}}.$ So by Lemma 4.6,

[TABLE]

and so from Remark 4.4 we have $w\in H^{\infty}_{E}$ which implies that $[\widehat{H^{\infty}_{E}}]^{\wedge}\subseteq H^{\infty}_{E}$ holds. ∎

4.2. Adjointness

Before proving the adjointness theorem we first prove the following lemma,

Lemma 4.8**.**

Let $v\in H^{\infty}_{E}$ and $w\in\widehat{H^{\infty}_{E}}$. Then we have

[TABLE]

where $v_{j}=(v_{jl})_{1\leq l\leq d_{j}}=\widehat{v}(j,l),$ and $j\in\mathbb{N},$ and the same for $w.$ Moreover, suppose that for all $v\in H^{\infty}_{E},$ (4.10) holds. Then we must have $w\in\widehat{H^{\infty}_{E}}.$ Also, if for all $w\in\widehat{H^{\infty}_{E}},$ (4.10) holds, we have $v\in H^{\infty}_{E}.$

Proof.

Let us assume first that

[TABLE]

for $v\in H^{\infty}_{E}$ or $w\in\widehat{H^{\infty}_{E}}$. We observe that

[TABLE]

And so we have

[TABLE]

Now we prove the other direction. Here we will use the following inequality,

[TABLE]

for any $a_{i},b_{i}\in\mathbb{R},$ yielding

[TABLE]

We consider the second term of the above inequality, that is,

[TABLE]

since $v\in H^{\infty}_{E}$ and so $v\in H^{s}_{E},\forall s\in\mathbb{R}.$ Then we get

[TABLE]

Let $|u_{ji}|=|v_{ji}|+Cd_{j}\lambda_{j}^{-s},$ for $i=1,2,...,d_{j}.$ Then

[TABLE]

Then for any $t>0$ we have, using (4.13), that

[TABLE]

Now since $v\in H^{\infty}_{E},$ in particular we can have $2s=2t+3s_{0},$ for any $t>0,$ which gives, using (2.1)

[TABLE]

So $u\in H^{\infty}_{E}.$ Then we have for $w\in\widehat{H^{\infty}_{E}},$

[TABLE]

completing the proof. ∎

We next prove the adjointness theorem, also recalling Definition 3.1. Let $\mathcal{H},\mathcal{G}$ be two Hilbert spaces and $E$ and $F$ be the operators defined by (2.2) corresponding to the bases $\{e_{j}\}_{j\in\mathbb{N}},$ $\{h_{k}\}_{k\in\mathbb{N}},$ respectively, where $d_{j}=\dim X_{j}$ and $g_{k}=\dim Y_{k},$ and $X_{j}=\text{span}\{e_{jl}\}_{l=1}^{d_{j}},$ $Y_{k}=\text{span}\{h_{ki}\}_{i=1}^{g_{k}}.$ Also $\mathcal{H}=\bigoplus\limits_{j\in\mathbb{N}}X_{j}$ and $\mathcal{G}=\bigoplus\limits_{k\in\mathbb{N}}Y_{k}.$ We denote the corresponding spaces to the operators $E$ and $F$ in the Hilbert space $\mathcal{H}$ and $\mathcal{G}$ respectively by $H^{\infty}_{E}$ and $G^{\infty}_{F}.$

Theorem 4.9**.**

A linear mapping $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ is sequential if and only if $f$ is represented by an infinite tensor $(f_{kjli}),$   $k,j\in\mathbb{N},$ $1\leq l\leq d_{j}$ and $1\leq i\leq g_{k}$ such that for any $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}}$ we have

[TABLE]

and

[TABLE]

Furthermore, the adjoint mapping $\widehat{f}:\widehat{G^{\infty}_{F}}\rightarrow\widehat{H^{\infty}_{E}}$ defined by the formula $\widehat{f}(v)=v\circ f$ is also sequential, and the transposed matrix ${(f_{kj})}^{t}$ represents $\widehat{f}$, with $f$ and $\widehat{f}$ related by $\langle f(u),v\rangle_{G^{\infty}_{F}}=\langle u,\widehat{f}(v)\rangle_{H^{\infty}_{E}}.$

Let us summarise the ranges for indices in the used notation as well as give more explanation to (4.16). For $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ and $u\in H^{\infty}_{E}$ we write

[TABLE]

so that

[TABLE]

and

[TABLE]

where we view $f_{kj}$ as a matrix, $f_{kj}\in\mathbb{C}^{g_{k}\times d_{j}}$, and the product of the matrices has been explained in (4.17).

Remark 4.10**.**

Let us now describe how the tensor $(f_{kjli})$, $k,j\in\mathbb{N},$ $1\leq l\leq d_{j}$, $1\leq i\leq g_{k}$, is constructed given a sequential mapping $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$. For every $k\in\mathbb{N}$ and $1\leq i\leq g_{k}$, define the family $v^{ki}=\left(v^{ki}_{j}\right)_{j\in\mathbb{N}}$ such that each $v^{ki}_{j}\in{\mathbb{C}}^{d_{j}}$ is defined by

[TABLE]

Then $v^{ki}\in\widehat{G^{\infty}_{F}}$, and since $f$ is sequential we have $v^{ki}\circ f\in\widehat{H^{\infty}_{E}}$, and we can write $v^{ki}\circ f=\left(v^{ki}\circ f\right)_{j\in{\mathbb{N}}},$ where $(v^{ki}\circ f)_{j}\in{\mathbb{C}}^{d_{j}}.$ Then for each $1\leq l\leq d_{j}$ we set

[TABLE]

the $l^{th}$ component of the vector $(v^{ki}\circ f)_{j}\in{\mathbb{C}}^{d_{j}}.$ The formula (4.21) will be shown in the proof of Theorem 4.9. In particular, since for $\phi\in H^{\infty}_{E}$ we have $f(\phi)\in G^{\infty}_{F},$ it will be a consequence of (4.32) and (4.33) later on that

[TABLE]

so that the tensor $(f_{kjli})$ is describing the transformation of the Fourier coefficients of $\phi$ into those of $f(\phi)$.

To prove Theorem 4.9 we first establish the following lemma.

Lemma 4.11**.**

Let $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ be a linear mapping represented by an infinite tensor $(f_{kjli})_{k,j\in{{\mathbb{N}}},1\leq l\leq d_{j},1\leq i\leq g_{k}}$ satisfying (4.15) and (4.16). Then for all $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}},$ we have

[TABLE]

Proof of Lemma 4.11.

Let $u\in H^{\infty}_{E}$ and $u\approx\left(\widehat{u}(j)\right)_{j\in{\mathbb{N}}}.$ Define $u^{n}:=\left(\widehat{u}^{(n)}(j)\right)_{j\in{\mathbb{N}}}$ by setting

[TABLE]

Then for any $w\in\widehat{H^{\infty}_{E}},$ we get $\langle u-u^{n},w\rangle_{H^{\infty}_{E}}\rightarrow 0$ as $n\rightarrow\infty.$ This is true since $\sum_{j=1}^{\infty}\left|\widehat{u}(j)\cdot w_{j}\right|<\infty$ so that

[TABLE]

as $n\rightarrow\infty.$ Now for any $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}}$ and from (4.15) and (4.16) we have

[TABLE]

where

[TABLE]

and

[TABLE]

Now we have the mapping $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ and $v\in\widehat{G^{\infty}_{F}},$ so we have $v\circ f:{H^{\infty}_{E}}\rightarrow\Sigma,$ where $\Sigma=\left\{v=(v_{j})_{j\in\mathbb{N}},v_{j}\in\mathbb{C}^{d_{j}}\right\}.$ For any $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}}$ we have

[TABLE]

Then from (4.15) and (4.16) we have $\left\langle u,(v\circ f)\right\rangle<\infty,$ and since $H^{\infty}_{E}$ is perfect, we have for any $u\approx(\widehat{u}(j))_{j\in{\mathbb{N}}}\in H^{\infty}_{E}$ that the series $\sum\limits_{j=1}^{\infty}\left|(v\circ f)_{j}\cdot\widehat{u}(j)\right|$ is convergent. So then $v\circ f\in\widehat{H^{\infty}_{E}}.$ Then we have

[TABLE]

as $n\rightarrow\infty.$ Therefore,

[TABLE]

for all $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}}.$ Hence for any $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}}$ we have

[TABLE]

that is,

[TABLE]

Now we will use the fact that if $u\in H^{\infty}_{E}$ then $|u|\in H^{\infty}_{E}$ where $|u|=\left(\widehat{|u}|_{j}\right)_{j\in\mathbb{N}},$ $\widehat{|u|}_{j}\in\mathbb{R}^{d_{j}},$ with

[TABLE]

in view of Theorem 4.7. The same is true for the dual space $\left[G^{\infty}_{F}\right]^{\wedge}.$ So then this argument gives

[TABLE]

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 $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ can be represented by $f=(f_{kjli})_{k,j\in\mathbb{N},1\leq l\leq d_{j},1\leq i\leq g_{k}},$ an infinite tensor such that

[TABLE]

and

[TABLE]

hold for all $u\in H^{\infty}_{E}$ and $v\in\widehat{G^{\infty}_{F}}.$

Let $\widehat{u}_{1}=\left(\widehat{u_{1}}(p)\right)_{p\in{\mathbb{N}}}$ be such that for some $j,l$ where $j\in{\mathbb{N}},$ $1\leq l\leq d_{j}$, we have

[TABLE]

Then $u_{1}\in H^{\infty}_{E}$ so $fu_{1}=f(u_{1})\in G^{\infty}_{F}$ and

[TABLE]

We now first show that

[TABLE]

where $f_{kjli}\in\mathbb{C}$ for each $k,j\in\mathbb{N},$ $1\leq l\leq d_{j}$ and $1\leq i\leq g_{k}.$ The way in which $f$ has been defined we have

[TABLE]

Also since $u\in H^{\infty}_{E}$, from our assumption we have $fu\in G^{\infty}_{F}$ and $fu\approx\left(\widehat{(fu)}(j)\right)_{j\in\mathbb{N}}$, so that $(fu)_{k}\approx\widehat{(fu)}(k).$

We can then write $\widehat{(fu)}(k)=\sum\limits_{j=1}^{\infty}\sum\limits_{l=1}^{d_{j}}f_{kjl}\widehat{u}(j,l).$ Since we know that $v\in{\left[G^{\infty}_{F}\right]^{\wedge}}$ and $fu\in G^{\infty}_{F},$ we have

[TABLE]

In particular using the definition of $u_{1}$ and (4.26) we get

[TABLE]

for any $j\in\mathbb{N}$ and $1\leq l\leq d_{j}.$

Now for any $u\in H^{\infty}_{E}$ consider

[TABLE]

Then we consider the series

[TABLE]

so that we have

[TABLE]

Let $\epsilon=(\epsilon_{i})_{1\leq i\leq d_{k}},$ $k\in{\mathbb{N}}$, be such that $\epsilon_{i}\in\mathbb{C}$ and $|\epsilon_{i}|\leq C,$ for all $i$ and such that

[TABLE]

Then

[TABLE]

It follows from Lemma 4.11 that

[TABLE]

Then

[TABLE]

So we proved that if $(f_{kjli})$ satisfies

• •

$\sum\limits_{j=1}^{\infty}\sum\limits_{l=1}^{d_{j}}|f_{kjli}||\widehat{u}(j,l)|<\infty$,

• •

$\sum\limits_{k=1}^{\infty}\sum\limits_{i=1}^{g_{k}}\left|(v_{k})_{i}\right|\left|{\left(\sum\limits_{j=1}^{\infty}f_{kj}\widehat{u}(j)\right)_{i}}\right|<\infty$,

then for any $u\in{H^{\infty}_{E}}$ and $v\in\left[G^{\infty}_{F}\right]^{\wedge}$ we have from (4.27) and (4.29), respectively, that

1. (i)

$\sum\limits_{k=1}^{\infty}\sum\limits_{i=1}^{g_{k}}|(v_{k})_{i}||f_{kjli}|<\infty$, 2. (ii)

$\sum\limits_{j=1}^{\infty}\sum\limits_{l=1}^{d_{j}}|\sum\limits_{k=1}^{\infty}\sum\limits_{i=1}^{g_{k}}(v_{k})_{i}f_{kjli})||\widehat{u}(j,l)|<\infty.$

Now recall that for $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ we have

[TABLE]

for any $u\in{H^{\infty}_{E}},$ then for any $v\in\left[G^{\infty}_{F}\right]^{\wedge}$, the composed mapping $v\circ f:H^{\infty}_{E}\rightarrow\mathbb{C}$ is given by

[TABLE]

So by (ii) we get that

[TABLE]

So $\widehat{f}(v)=(\widehat{f}(v)_{jl})_{j\in\mathbb{N},1\leq l\leq d_{j}},$ with $\widehat{f}(v)_{jl}=\sum\limits_{k=1}^{\infty}\sum\limits_{i=1}^{g_{k}}(v_{k})_{i}f_{kjli}\in\widehat{H^{\infty}_{E}}$ (from the definition of $\widehat{H^{\infty}_{E}}$), that is $f$ is sequential. And then $\langle f(u),v\rangle_{G^{\infty}_{F}}=\langle u,\widehat{f}(v)\rangle_{H^{\infty}_{E}}$ is also true.

Now to prove the converse part we assume that $f:H^{\infty}_{E}\rightarrow G^{\infty}_{F}$ is sequential. We have to show that $f$ can be represented as $f\approx(f_{kjli})_{k,j\in\mathbb{N},1\leq l\leq d_{j},1\leq i\leq g_{k}}$ and satisfies (4.15) and (4.16).

Define for $k,i$ where $k\in\mathbb{N}$ and $1\leq i\leq g_{k},$ the sequence $u^{ki}=\left(u^{ki}_{j}\right)_{j\in\mathbb{N}}$ such that $u^{ki}_{j}\in{\mathbb{C}}^{d_{j}}$ and $u^{ki}_{j}(l)=\widehat{u^{ki}}({j,l})$, given by

[TABLE]

Then $u^{ki}\in\left[G^{\infty}_{F}\right]^{\wedge}.$ Now since $f$ is sequential we have $u^{ki}\circ f\in\widehat{H^{\infty}_{E}}$ and $u^{ki}\circ f=\left(\left(u^{ki}\circ f\right)_{j}\right)_{j\in\mathbb{N}},$ where $(u^{ki}\circ f)_{j}\in{\mathbb{C}}^{d_{j}}.$ We denote $u^{ki}\circ f=\left(f^{ki}_{j}\right)_{j\in{{\mathbb{N}}}},$ where $f^{ki}_{j}=(u^{ki}\circ f)_{j}.$ Then $(f^{ki}_{j})_{j\in{\mathbb{N}}}\in\widehat{H^{\infty}_{E}}$ and $f^{ki}_{j}\in{\mathbb{C}}^{d_{j}}.$

Then for any $\phi\approx\left(\widehat{\phi}(j)\right)_{j\in\mathbb{N}}\in H^{\infty}_{E}$ we have

[TABLE]

For $\phi\in H^{\infty}_{E}$ we can write $f(\phi)\in G^{\infty}_{F}.$ We can also write

[TABLE]

So

[TABLE]

We have $u^{ki}\circ f=(f^{ki})\in\widehat{H^{\infty}_{E}},$ so

[TABLE]

From (4.32) and (4.33) we have $(\widehat{f(\phi)})(k,i)=\sum\limits_{j=1}^{\infty}\sum\limits_{l=1}^{d_{j}}f^{ki}_{jl}\widehat{\phi}(j,l).$

Hence $(f(\phi))_{ki}=\sum\limits_{j=1}^{\infty}\sum\limits_{l=1}^{d_{j}}f^{ki}_{jl}\widehat{\phi}(j,l),~{}~{}k\in\mathbb{N},$ and $1\leq i\leq g_{k},$ that is $f$ is represented by the tensor $\left\{(f^{ki}_{jl})\right\}_{k,j\in\mathbb{N},1\leq i\leq g_{k},1\leq l\leq d_{j}}$.

If we denote $f^{ki}_{jl}$ by $f^{ki}_{jl}=f_{kjli},$ we can say that $f$ is represented by the tensor $(f_{kjli})_{k,j\in\mathbb{N},1\leq l\leq d_{j},1\leq i\leq g_{k}}.$ Also let $v\in\widehat{G^{\infty}_{F}}.$ Since $f(\phi)\in G^{\infty}_{F}$ for $\phi\in H^{\infty}_{E},$ then from the definition of $\widehat{G^{\infty}_{F}}$ we have

[TABLE]

This completes the proof of Theorem 4.9. ∎

5. 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 $\mathcal{H},\mathcal{G}$ be two Hilbert spaces and let $E$ and $F$ be the operators corresponding to the bases $\{e_{j}\}_{j\in\mathbb{N}},$ $\{h_{k}\}_{k\in\mathbb{N}},$ as in (2.2), where $d_{j}=\dim X_{j}$ and $g_{k}=\dim Y_{k},$ and $X_{j}=\text{span}\{e_{jl}\}_{l=1}^{d_{j}},$ $Y_{k}=\text{span}\{h_{ki}\}_{i=1}^{g_{k}}.$ Also $\mathcal{H}=\bigoplus\limits_{j\in\mathbb{N}}X_{j}$ and $\mathcal{G}=\bigoplus\limits_{k\in\mathbb{N}}Y_{k}.$

We denote the spaces of smooth type functions corresponding to the operators $E$ and $F$ in the Hilbert space $\mathcal{H}$ and $\mathcal{G},$ respectively, by $H^{\infty}_{E}$ and $G^{\infty}_{F}.$

The main application of Theorem 4.9 will be in the setting when $X,Y$ are compact manifold without boundary, where $\mathcal{H}=L^{2}(X)$ and $H^{\infty}_{E}=C^{\infty}(X)$, and $\mathcal{G}=L^{2}(Y),$ $G^{\infty}_{F}=C^{\infty}(Y).$

Using Theorem 4.9 we prove the universality of the spaces of the smooth type functions, $C^{\infty}(X),$ where we can write

[TABLE]

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 $f:X\rightarrow W$ from the compact manifold $X$ to a sequence space $W$, is said to be a $H^{\infty}_{E}$-mapping if for any $u\in\widehat{W},$ the composed mapping $u\circ f:X\rightarrow\mathbb{C}$ belongs to $H^{\infty}_{E}$.

Next we prove the universality of the spaces of smooth type functions.

Theorem 5.2**.**

Let $X$ be a compact manifold.

• (i)

The delta mapping $\delta:X\rightarrow\widehat{H^{\infty}_{E}}$ defined by

[TABLE]

and

[TABLE]

is a $H^{\infty}_{E}$-mapping.

• (ii)

If $\tilde{g}:\widehat{H^{\infty}_{E}}\rightarrow G^{\infty}_{F}$ is a sequential linear mapping, then the composed mapping $\tilde{g}\circ\delta:X\rightarrow G^{\infty}_{F}$ is a $H^{\infty}_{E}$-mapping.

• (iii)

For any $H^{\infty}_{E}$-mapping $f:X\rightarrow G^{\infty}_{F},$ there exists a unique sequential linear mapping $\widehat{f}:\widehat{H^{\infty}_{E}}\rightarrow G^{\infty}_{F}$ such that $f=\widehat{f}\circ\delta.$

Proof.

(i)   Recall that $\widehat{H^{\infty}_{E}}=\bigcup\limits_{s\in\mathbb{R}}H^{-s}_{E}=[H^{\infty}_{E}]^{\prime}.$

Let $v\in[\widehat{H^{\infty}_{E}}]^{\wedge}=H^{\infty}_{E}.$ We define the composed mapping

[TABLE]

This is well-defined since $v\in H^{\infty}_{E}$ and $\delta_{x}\in[H^{\infty}_{E}]^{\prime}=\widehat{H^{\infty}_{E}}.$ Also since $v\in H^{\infty}_{E}$, we see that $v\circ\delta\in H^{\infty}_{E}$ and that implies $\delta$ is a $H^{\infty}_{E}$-mapping.

(ii)  Let $u\in\widehat{G^{\infty}_{F}}=[G^{\infty}_{F}]^{\prime}.$

From the definition of $H^{\infty}_{E}$-mapping we have to show that $u\circ\tilde{g}\circ\delta\in H^{\infty}_{E}.$

Now by given condition $\tilde{g}:[H^{\infty}_{E}]^{\prime}\rightarrow G^{\infty}_{F},$ so we have $u\circ\tilde{g}\in\left[{H^{\infty}_{E}}\right]^{\prime\prime}.$

Here we claim that $\left[{H^{\infty}_{E}}\right]^{\prime\prime}=H^{\infty}_{E}.$

Note that $H^{\infty}_{E}\subseteq\left[{H^{\infty}_{E}}\right]^{\prime\prime}.$

Recall that, $[H^{\infty}_{E}]^{\prime}=\bigcup\limits_{s\in\mathbb{R}}H^{-s}_{E}$ and so for any $s\in\mathbb{R},$

[TABLE]

Since the above is true for any $s\in\mathbb{R},$ we have $[H^{\infty}_{E}]^{\prime\prime}\subseteq\bigcap\limits_{s\in\mathbb{R}}H^{s}_{E}=H^{\infty}_{E}$ and this gives $[H^{\infty}_{E}]^{\prime\prime}=H^{\infty}_{E}.$

Then $u\circ\tilde{g}\in H^{\infty}_{E}.$ Using same argument as in the proof of $(i)$ and from the definition of the mapping $\delta$ we have $u\circ\tilde{g}\circ\delta_{x}=u\circ\tilde{g}(x)$ and $u\circ\tilde{g}\circ\delta$ belongs to $H^{\infty}_{E}.$

So from Definition 5.1, $\tilde{g}\circ\delta$ is a $H^{\infty}_{E}$-mapping.

(iii) Existence of $\widehat{f}:\widehat{H^{\infty}_{E}}\rightarrow{G^{\infty}_{F}}.$

By hypothesis, $f:X\rightarrow{G^{\infty}_{F}}$ is a $H^{\infty}_{E}$-mapping so that for any $v\in\widehat{G^{\infty}_{F}},$ $v\circ f\in H^{\infty}_{E}.$ A sequential linear mapping $\tilde{f}:\widehat{G^{\infty}_{F}}\rightarrow H^{\infty}_{E}$ can be defined by $\tilde{f}(v)=v\circ f,$ and $v(f(x))=\langle f(x),v\rangle_{G^{\infty}_{F}}.$

Hence by Theorem 4.9 there is an adjoint mapping, we denote it by $\widehat{f},$ where $\widehat{f}:\widehat{H^{\infty}_{E}}\rightarrow[\widehat{G^{\infty}_{F}}]^{\wedge}$ is a sequential mapping. By the definition of the adjoint mapping $\widehat{f}$ we have

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the bilinear function on $H^{\infty}_{E}\times\widehat{H^{\infty}_{E}}$ defined in Section 3. The above can be written as

[TABLE]

For $u=\delta_{x},$ this gives

[TABLE]

for any $v\in\widehat{G^{\infty}_{F}}$. This proves $f=\widehat{f}\circ\delta.$

 

Uniqueness of $\widehat{f}:\widehat{H^{\infty}_{E}}\rightarrow G^{\infty}_{F}.$

Suppose $\widehat{f}\circ\delta=f=0.$ We have to show that $\widehat{f}=0$ on $\widehat{H^{\infty}_{E}}.$ Since $\widehat{f}$ is sequential, there exists $g:\widehat{G^{\infty}_{F}}\rightarrow H^{\infty}_{E}$ such that

[TABLE]

Take $u=\delta_{x}\in\widehat{H^{\infty}_{E}},$ then

[TABLE]

for any $v\in\widehat{G^{\infty}_{F}},$ that is, $g=0$ on $\widehat{G^{\infty}_{F}}.$ From $\langle\widehat{f}\circ u,v\rangle_{G^{\infty}_{F}}=0$ for any $v\in\widehat{G^{\infty}_{F}},$ we get $\widehat{f}\circ u=0$ for any $u\in\widehat{H^{\infty}_{E}},$ that is $\widehat{f}=0$ on $\widehat{H^{\infty}_{E}}.$ ∎

5.1. Extension to Komatsu classes

Here we briefly outline how the analysis above can be extended to the setting of Komatsu classes from [4, 5].

Remark 5.3**.**

In another work ([5]) we studied the Komatsu classes of ultra-differentiable functions $\Gamma_{\{M_{k}\}}(X)$ on a compact manifold $X,$ where $M_{\{k\}}$ be a sequence of positive numbers such that

1. (1)

$M_{0}=1,$** 2. (2)

$M_{k+1}\leq AH^{k}M_{k},~{}k=0,1,2,...,$** 3. (3)

$M_{2k}\leq AH^{k}\min\limits_{0\leq q\leq k}M_{q}M_{k-q},k=0,1,2,...,$* for some $A,H>0.$*

In [5] we have characterised the dual spaces of these Komatsu classes and have shown that these spaces are perfect spaces, i.e, these spaces coincide with their second dual spaces. Furthermore, in [5, Theorem 4.7] we proved the following theorem for Komatsu classes of functions on a compact manifold $X$:

Theorem 5.4** (Adjointness Theorem).**

Let $\{M_{k}\}$and $\{N_{k}\}$ satisfy conditions $(M.0)-(M.3)$. A linear mapping $f:\Gamma_{\{M_{k}\}}(X)\rightarrow\Gamma_{\{N_{k}\}}(X)$ is sequential if and only if $f$ is represented by an infinite tensor $(f_{kjli}),$ $k,j\in\mathbb{N}_{0},$ $1\leq l\leq d_{j}$ and $1\leq l\leq d_{k}$ such that for any $u\in\Gamma_{\{M_{k}\}}(X)$ and $v\in\widehat{\Gamma_{\{N_{k}\}}(X)}$ we have

[TABLE]

and

[TABLE]

Furthermore, the adjoint mapping $\widehat{f}:\widehat{\Gamma_{\{N_{k}\}}(X)}\rightarrow\widehat{\Gamma_{\{M_{k}\}}(X)}$ defined by the formula $\widehat{f}(v)=v\circ f$ is sequential, and the transposed matrix $(f_{kj})^{t}$ represents $\widehat{f},$ with $f$ and $\widehat{f}$ related by $\langle f(u),v\rangle=\langle u,\widehat{f}(v)\rangle.$

The above theorem described the tensor structure of sequential mappings on spaces of Fourier coefficients and characterised their adjoint mappings. Now in particular the considered classes include spaces of analytic and Gevrey functions (which are perfect spaces too), as well as spaces of ultradistributions, yielding tensor representations for linear mappings between these spaces on compact manifolds. Now using [5, Theorem 4.7] and the same techniques used in this paper to prove the universality of smooth functions in Theorem 5.2, on compact manifolds in Section 5, one also obtains the universality of the Gevrey classes of ultradifferentiable functions on compact groups (from [3]) and Komatsu classes of functions in compact manifolds. As the proof would be a repetition of those arguments, we omit it here.