Eigenfunction expansions of ultradifferentiable functions and

EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS AND ULTRADISTRIBUTIONS. III. HILBERT SPACES AND UNIVERSALITY

arXiv:1812.01283v1 [math.FA] 4 Dec 2018

APARAJITA DASGUPTA AND MICHAEL RUZHANSKY 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 papers [4] and [5]. 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.

Contents 1. Introduction 2. Fourier Analysis on Hilbert Spaces 3. Sequence spaces and sequential linear mappings 4. Tensor representations and the adjointness 4.1. Duals and α-duals 4.2. Adjointness 5. Applications to universality 5.1. Extension to Komatsu classes References

1 3 4 4 5 10 18 21 21

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 Date: December 5, 2018. 1991 Mathematics Subject Classification. Primary 46F05; Secondary 22E30. Key words and phrases. Smooth functions; Hilbert spaces; Komatsu classes; sequence spaces; tensor representations; universality. 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

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III 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 λj denotes its eigenvalues in the ascending order, then smooth functions on X can be characterised in terms of their Fourier coefficients: f ∈ C ∞ (X) ⇔ ∀N ∃CN : |fb(j, l)| ≤ CN λ−N for all j ≥ 1, 1 ≤ l ≤ dj , j

(1.1)


where fb(j, l) = f, elj L2 with elj being the lth eigenfunction corresponding to the eigenvalue λj (of multiplicity dj ). 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., K¨othe [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 [), that is, any vector valued C ∞ -mapping f : V → E, from δ-mapping δ : V → E(V [) by a unique linear morphism V to a sequence space E, factors through δ : V → E(V [) → E as f = f˜ ◦ δ. The universality of the spaces of Gevrey functions on the f˜ : E(V 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 2

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III 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 (H, || · ||H ) be a separable Hilbert space and denote by U := {ejl : ejl ∈ H, 1 ≤ l ≤ dj , dj ∈ N}j∈N a collection of elements of H . We assume that U is a basis of the space H with the property (ejl , emn )H = δjm δln , j, m ∈ N and 1 ≤ l ≤ dj , 1 ≤ n ≤ dm , where δjm is the Kroneckar delta, equal to 1 for j = m, and to zero otherwise. Also let us fix a sequence of positive numbers Λ := {λj }j∈N such that 0 < λ1 ≤ λ2 ≤ λ3 ≤ ..., and the series ∞ X dj λj−s0 < ∞ (2.1) j=1

converges for some s0 > 0. For example, in a compact C ∞ manifold X of dimension n without boundary and with a fixed measure we have ∞ X n dj (1 + λj )−q < ∞ if and only if q > , ν j=1

where 0 < λ1 < λ2 < ... are eigenvalues of a positive elliptic pseudo-differential operator E of an integer order ν, with Hj ⊂ L2 (X) the corresponding eigenspace and dj := dim Hj , H0 := kerE, λ0 := 0, d0 := dim H0 . We associate to the pair {U, Λ} a linear self-adjoint operator E : H → H such that s

E f=

dj ∞ X X

λsj (f, ejl )ejl

(2.2)

j=1 l=1

for s ∈ R and those f ∈ H for which the series converges in H. Then E is densely defined since E s ejl = λsj ejl , 1 ≤ l ≤ dj , j ∈ N, and U is a basis of H. Also we write Hj = span{ejl }1≤l≤dj , and so dim Hj = dj . Then we have M H= Hj . j∈N

The Fourier transform for f ∈ H is defined as fb(j, l) := (f, ejl )H , j ∈ N, 1 ≤ l ≤ dj . We next define the following notions:

The spaces of smooth type functions are defined by \ HEs , HE∞ := s∈R

3

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III where HEs :=

(

φ∈H:

∞ X j=1

There exists a linear pairing

)

2 b λ2s j ||φ(j)||HS < ∞ .

(f, g)HEs = (E s f, E −s g)H , for f ∈ HEs and g ∈ HE−s . It is easy to see from this that every continuous linear functional on HEs is of the form f → (f, g)HEs for some g ∈ HE−s , that is (HEs )′ = HE−s . S −s HE . Then we denote the space of distributions as (HE∞ )′ = s∈R

3. Sequence spaces and sequential linear mappings We briefly recall that a sequence space V is a linear subspace of CZ = {a = (aj )|aj ∈ C, j ∈ Z}. The dual Vb (α-dual in the terminology of G. K¨othe [11]) is a sequence space defined by X Vb = {a ∈ CZ : |fj ||aj | < ∞ for all f ∈ V }. j∈Z

b A sequence space V is called perfect if Vb = V . A sequence space is called normal if f = (fj )j∈N ∈ V implies |f | = (|fj |)j∈N ∈ V. A dual space Vb is normal so that any perfect space is normal. A pairing h·, ·iV on V is a bilinear function on V × Vb defined by X hf, giV = fj gj ∈ C, j∈Z

which converges absolutely by the definition of Vb .

Definition 3.1. φ : V → C is called a sequential linear functional if there exists some a ∈ Vb such that φ(f ) = hf, aiV 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, c , the composed mapping g ◦ φ : V → C is in Vb . (2) for any g ∈ W

4. 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. 4

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III 4.1. Duals and α-duals. cs of the space H s , where In this section we first prove that the α-dual, H E E   dj ∞ X   X b l)||vjl | < ∞, for all φ ∈ H s , cs = v = (vj )j∈N , vj ∈ Cdj ; H | φ(j, E E   j=1 l=1

coincides with the space HE−s .

Remark 4.1. Here we observe that, cs =⇒ w∈H E

∞ X

−s−s0 /2

λj

||wj ||HS < ∞.

j=1

b l) = λ−(s+s0 /2) for j ∈ N and 1 ≤ l ≤ dj . cs . Define φ(j, Indeed, let w ∈ H j E 1/2 −s−s0 /2 b Then ||φ(j)|| = d λ . It follows from (2.1) and the definition of HEs that HS j j φ ∈ HEs . From the definition of the α-dual and using the inequality, ||wj ||HS ≤ ||wj ||l1 , we can then conclude that, ∞ X

−s−s /2 λj 0 ||wj ||HS

≤

j=1

∞ X

−s−s0 /2

λj

||wj ||l1

j=1

=

dj ∞ X X j=1 l=1

b l)||wjl| < ∞. |φ(j,

(4.1)

Our first result is the identifiction of the topological dual with the α-dual. cs = H −s . Theorem 4.2. H E E

cs . Proof. First we will show HE−s ⊆ H E Let u ∈ HE−s . ∞ P λ−2s Then from the definition we have u(j)||2HS < ∞. We denote for j ∈ N, j ||b j=1

ujl = u b(j, l) ∈ Cdj , l = 1, 2, ..., dj . Using the Cauchy-Schwartz inequality, for any s φ ∈ HE we get dj ∞ X X j=1 l=1

b l)||ujl | |φ(j,

1/2 1/2   dj dj ∞ X ∞ X X X 2 b  λj−2s |ujl|2  < ∞, (by assumption). λ2s ≤ j |φ(j, l)| j=1 l=1

j=1 l=1

5

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III b s . We thus obtain H −s ⊆ H cs . This implies that u ∈ H E E E −s s c Next we will show that HE ⊆ HE . cs ⊆ By duality we know that (HEs )′ = HE−s . So it will be enough if we can prove H E s ′ (HE ) . cs . Then for φ ∈ H s we define Let w ∈ H E E w(φ) =

∞ X j=1

−s /2 b λj 0 φ(j)

· wj =

|w(φ)| ≤

−s /2 λj 0

j=1

Then using Remark 4.1 we have

∞ X

∞ X

−s0 /2

λj

j=1 ∞ X

≤ C

dj X l=1

b ||φ(j)|| HS ||wj ||HS

−s0 /2 −s λj ||wj ||HS

λj

b l)wjl . φ(j,

< ∞,

j=1

(4.2)

since φ ∈ HEs . So w(φ) is well defined. We next check that w is continuous. Let φm → φ as m → ∞ in HEs . This means, b ||φm − φ||HEs → 0 as m → ∞, which implies λsj ||φc m (j) − φ(j)||HS → 0 as m → ∞, for j ∈ N. So we have −s b ||φc m (j) − φ(j)||HS ≤ Cm λj , where Cm → 0 as m → ∞. Then |w(φm − φ)| ≤

∞ X

−s0 /2

λj

j=1

≤ Cm

∞ X

b ||φc m (j) − φ(j)||HS ||wj ||HS

−(s+s0 /2)

λj

||wj ||HS → 0,

j=1

as m → ∞. Hence w is continuous. This gives w ∈ (HEs )′ = HE−s . So we have cs ⊆ H −s , that implies H cs = H −s . H  E E E E From this we can have the following corollary.

∞ cs ⇐⇒ P λ−2s ||vj ||2 < ∞ and vj ∈ Cdj , we denote vjl = Corollary 4.3. v ∈ H HS j E j=1

vc jl = (v, ejl )H .

Remark 4.4. From the definition of HE∞ , v∈

HE∞

⇐⇒

∞ X

2 λ2s j ||vj ||HS < ∞, ∀s ∈ R,

j=1

where vj = b v (j) ∈ Cdj .

6

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III ∞ d We next define the α-dual of the space H E ,   dj ∞ X   X ∞ dj ∞ b d |vjl ||φ(j, l)| < ∞, for all φ ∈ HE . HE = v = (vj ), vj ∈ C :   j=1 l=1

T S s S −s ∞ \ d c = Also observe that H HEs = H HE . From this we can state the E = E s∈R

s∈R

s∈R

following lemma and the proof will follow from our above observation. ∞ d Lemma 4.5. v ∈ H E ⇐⇒ for some s ∈ R we have

∞ P

j=1

2 λ−2s j ||vj ||HS < ∞.

Next we proceed to prove that HE∞ is a perfect space. But before that let us prove the following lemma. ∞ ∧ d Lemma 4.6. We have w ∈ [H E ] if and only if ∞ ∧ d Proof. Let w ∈ [H E ] . Let s ∈ R, we define

∞ P

j=1

2 λ2s j ||wj ||HS < ∞ for all s ∈ R.

vjl = b v (j, l) = λsj , 1 ≤ l ≤ dj . −s−

Then ||vj ||2HS = dj λ2s j and so v ∈ HE ∞ X

−s0 2 λ−2s j λj ||vj ||HS

=

j=1

∞ d Then v ∈ H E =

s0 2

∞ X

because in view of (2.1) we have

−s0 2s dj λ−2s j λj λj

=

j=1

S

s∈R

HE−s

which gives

∞ X

dj λj−s0 < ∞.

(4.3)

j=1

dj ∞ P P

|wjl||vjl | < ∞. Now we observe first that

j=1 l=1

∞ X

2 λ2s j ||wj ||HS

∞ X

≤

j=1

j=1 ∞ X

=

2 dj λ2s j ||wj ||HS

||vj ||2HS ||wj ||2HS ,

(4.4)

j=1

since dj ≥ 1. We want to show that

∞ P

||vj ||2HS ||wj ||2HS < ∞. To prove this we will use the following

j=1

identity: n X i=1

From this we get dj X i=1

|wji |2

dj X l=1

a2i

n X

b2l =

a2i b2i +

dj X

n X

a2i

i=1

i=1

l=1

|vjl |2 =

n X

|vji |2 |wji |2 +

i=1

dj X i=1

7

n X

b2l − b2i

l=1

!

.

  dj X |vjl |2 − |vji |2  . |wji |2  l=1

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III We consider the second term of the above inequality, that is,   dj dj dj dj dj X X X X X
2 2 2 2 2 0 |vjl | ≤ |wji |2 Cλ2s+s , |vjl | − |vji | ≤ |wji | |wji | j i=1

i=1

l=1

−s−s0 /2

since v ∈ HE

i=1

l=1

. Then we get

∞ X

||vj ||2HS ||wj ||2HS

dj ∞ X X

≤

0 |wji |2 |vji |2 + Cλ2s+s j

j=1 i=1

j=1

dj ∞ X X

≤

|wji |2 |uji |2

j=1 i=1



≤ C

dj ∞ X X j=1 i=1

0 where |uji|2 = |vji |2 + Cλ2s+s . Now j

∞ X

0 λ−2s−2s ||uj ||2HS = j

j=1

∞ X

2

|wji ||uji| ,

0 λ−2s−2s ||vj ||2HS + C j

j=1

≤ C′

∞ X




∞ X

(4.5)

0 2s+s0 dj λ−2s−2s λj j

j=1

λj−s0 + C

j=1

∞ X

dj λj−s0 < ∞,

(4.6)

j=1

∞ d which implies that u ∈ HE−s−s0 , that is u ∈ H E . This gives that dj ∞ X X

|wji ||uji | < ∞,

(4.7)

j=1 i=1

∞ ∧ ∞ d d as w ∈ [H E ] and u ∈ HE . So we have from (4.4), (4.5) and (4.7) that ∞ X

2 λ2s j ||wj ||HS < ∞.

j=1

Next we proceed to prove the opposite direction. Let  w ∈ Σ = v = (vj )j∈N , vj ∈ Cdj

be such that

∞ P

j=1

2 ∞ d λ2s j ||wj ||HS < ∞ for all s ∈ R. Let v ∈ HE =

we have v ∈ HE−s for some s ∈ R. We have to show dj ∞ X X

|wjl||vjl | < ∞.

j=1 l=1

8

S

s∈R

HE−s . In particular,

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III By the Cauchy-Schwartz inequality we have dj ∞ X X j=1 l=1

dj ∞ X X

|wjl ||vjl | =

λsj |wjl|λ−s j |vjl |

j=1 l=1

1/2   1/2 dj dj ∞ X ∞ X X X 2 2  ≤  λ2s λ−2s < ∞. (4.8) j |wjl | j |vjl | j=1 l=1

j=1 l=1

∞ ∧ d It follows that w ∈ [H E ] , completing the proof.



Now using Remark 4.4 and Lemma 4.6 we can prove that the spaces HE∞ are perfect spaces. Theorem 4.7. HE∞ is a perfect space. ∞ ∧ d Proof. From the definition we always have HE∞ ⊆ [H E ] . We will prove the other dj ∞ ∧ d direction. Let w ∈ [H E ] , w = (wj )j∈N and wj ∈ C . Define dj ∞ X X wjl ejl . φ= j=1 l=1

The series is convergent and φ ∈ H since ∞ dj X X wjl ejl ||φ||H = j=1 l=1 H  1/2  1/2 dj dj ∞ X ∞ X X X ≤  λsj 0 |wjl |2   λj−s0 ||ejl ||2H  j=1 l=1

j=1 l=1

1/2  dj ∞ X X =  λsj 0 |wjl |2  j=1 l=1

∞ X

dj λj−s0

j=1

!1/2

< ∞,

(4.9)

∞ ∧ d since w ∈ [H E ] and using (2.1) and Lemma 4.6. Also from the property (ejl , emn )H = δjm δln , for j, m ∈ N and 1 ≤ l ≤ dj , 1 ≤ n ≤ b l) = (φ, ejl)H = wjl . This gives ||φ(j)|| b dm , it is obvious that φ(j, HS = ||wj ||HS . So by Lemma 4.6, ∞ ∧ d w ∈ [H E ]

=⇒ =⇒

∞ X

j=1 ∞ X j=1

2 λ2s j ||wj ||HS < ∞, for all s ∈ R

2 b λ2s j ||φ(j)||HS < ∞,

∞ ∞ ∧ d and so from Remark 4.4 we have w ∈ HE∞ which implies that [H E ] ⊆ HE holds. 

9

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III 4.2. Adjointness. Before proving the adjointness theorem we first prove the following lemma, ∞ d Lemma 4.8. Let v ∈ HE∞ and w ∈ H E . Then we have ∞ X

||vj ||HS ||wj ||HS < ∞,

(4.10)

j=1

where vj = (vjl )1≤l≤dj = vb(j, l), and j ∈ N, and the same for w. Moreover, suppose ∞ d that for all v ∈ HE∞ , (4.10) holds. Then we must have w ∈ H E . Also, if for all ∞ ∞ d w∈H E , (4.10) holds, we have v ∈ HE . Proof. Let us assume first that ∞ X

||vj ||HS ||wj ||HS < ∞,

j=1

∞ d for v ∈ HE∞ or w ∈ H E . We observe that dj X

|vjl ||wjl | ≤ ||vj ||HS ||wj ||HS .

l=1

And so we have

dj ∞ X X

|vjl ||wjl| ≤

∞ X

||vj ||HS ||wj ||HS < ∞.

j=1

j=1 l=1

Now we prove the other direction. Here we will use the following inequality, ! n n n n n X X X X X |ai | |bl | = |ai ||bi | + |ai | |bl | − |bi | , i=1

i=1

l=1

i=1

l=1

for any ai , bi ∈ R, yielding

||vj ||HS ||wj ||HS ≤

dj X

|wji |

i=1

=

dj X

dj X

|vjl |

l=1

|vji ||wji | +

i=1

dj X i=1

  dj X |wji |  |vjl | − |vji | .

(4.11)

l=1

We consider the second term of the above inequality, that is,   dj dj dj X X X
|wji |  |vjl | − |vji | ≤ |wji | Cdj λ−s , ∀s ∈ R, j i=1

i=1

l=1

since v ∈ HE∞ and so v ∈ HEs , ∀s ∈ R. Then we get ∞ X j=1

||vj ||HS ||wj ||HS ≤

dj ∞ X X j=1 i=1

10


|wji | |vji | + Cdj λ−s . j

(4.12)

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III Let |uji| = |vji | + Cdj λ−s j , for i = 1, 2, ..., dj . Then dj X

|uji |

2

=

i=1

dj X

2

|vji | +

i=1

=

||vj ||2HS

+

dj X

C 2 d2j λ−2s j

i=1 2 3 −2s C d j λj

+

dj X

2Cdj λ−s j |vji |

i=1 2 −2s 2Cdj λj .

+

(4.13)

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

2 λ2t j ||uj ||HS

∞ X

=

j=1

j=1 ∞ X

≤

2 λ2t j ||vj ||HS

+C

2

2 2 λ2t j ||vj ||HS + C

j=1

∞ X

3 −2s λ2t j d j λj

j=1 ∞ X

+ 2C

∞ X

2 −2s λ2t j d j λj

j=1

dj λj2t+2s0 λ−2s j

j=1

+2C

∞ X

λj2t+2s0 λj−2s .

(4.14)

j=1

Now since v ∈ HE∞ , in particular we can have 2s = 2t + 3s0 , for any t > 0, which gives, using (2.1) ∞ X

2 λ2t j ||uj ||HS

≤

∞ X

2 λ2t j ||vj ||HS

+C

2

∞ d So u ∈ HE∞ . Then we have for w ∈ H E , ∞ X

||vj ||HS ||wj ||HS ≤

j=1

completing the proof.

dj λj−s0

+ 2C

j=1

j=1

j=1

∞ X

dj ∞ X X

∞ X

λj−s0 < ∞.

j=1

|wji ||uji | < ∞,

j=1 i=1



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 {ej }j∈N , {hk }k∈N , respectively, where dj = dim Xj and gk = dim Yk , and L L dj k Yk . We denote Xj and G = , Yk = span{hki }gi=1 . Also H = Xj = span{ejl }l=1 j∈N

k∈N

the corresponding spaces to the operators E and F in the Hilbert space H and G respectively by HE∞ and G∞ F .

Theorem 4.9. A linear mapping f : HE∞ → G∞ F is sequential if and only if f is represented by an infinite tensor (fkjli ), k, j ∈ N, 1 ≤ l ≤ dj and 1 ≤ i ≤ gk such that ∞ d for any u ∈ HE∞ and v ∈ G F we have dj ∞ X X

|fkjli||b u(j, l)| < ∞, for all k ∈ N, i = 1, 2, ..., gk ,

(4.15)

j=1 l=1

and

gk ∞ X X k=1 i=1

! ∞ X |(vk )i | fkj u b(j) < ∞. j=1 i 11

(4.16)

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III ∞ ∞ b d d Furthermore, the adjoint mapping fb : G F → HE defined by the formula f (v) = v ◦ f t is also sequential, and the transposed matrix (fkj ) represents fb, with f and fb related = hu, fb(v)iHE∞ . by hf (u), viG∞ F

Let us summarise the ranges for indices in the used notation as well as give more ∞ explanation to (4.16). For f : HE∞ → G∞ F and u ∈ HE we write C

gk

∋ f (u)k =

∞ X j=1

so that

k ∈ N,

(4.17)

k, j ∈ N, 1 ≤ l ≤ dj , 1 ≤ i ≤ gk ,

(4.18)

fkj u b(j) =

fkjl ∈ Cgk , fkjli ∈ C,

dj ∞ X X j=1 l=1

fkjlu b(j, l),

and C ∋ (f (u)k )i = f (u)ki =

dj ∞ X X j=1 l=1

fkjli u b(j, l),

k ∈ N, 1 ≤ i ≤ gk ,

(4.19)

where we view fkj as a matrix, fkj ∈ Cgk ×dj , and the product of the matrices has been explained in (4.17). Remark 4.10. Let us now describe how the tensor (fkjli ), k, j ∈ N, 1 ≤ l ≤ dj , 1 ≤ i ≤ gk , is constructed given a sequential mapping f : HE∞ → G∞ F . For every
ki ki k ∈ N and 1 ≤ i ≤ gk , define the family v = vj j∈N such that each vjki ∈ Cdj is defined by ( 1, j = k, l = i, vjki (l) = (4.20) 0, otherwise. ki ∞ ∞ d d Then v ki ∈ G F ,
and since f is sequential we have v ◦ f ∈ HE , and we can write v ki ◦ f = v ki ◦ f j∈N , where (v ki ◦ f )j ∈ Cdj . Then for each 1 ≤ l ≤ dj we set

fkjli := (v ki ◦ f )j (l),

th

ki

(4.21)

dj

the l component of the vector (v ◦ f )j ∈ C . The formula (4.21) will be shown in the proof of Theorem 4.9. In particular, since for φ ∈ HE∞ we have f (φ) ∈ G∞ F , it will be a consequence of (4.32) and (4.33) later on that v ◦ f (φ) = (fd (φ))(k, i) = ki

dj ∞ X X j=1 l=1

b l), fkjli φ(j,

(4.22)

so that the tensor (fkjli ) 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. Lemma 4.11. Let f : HE∞ → G∞ F be a linear mapping represented by an infinite tensor (fkjli )k,j∈N,1≤l≤dj ,1≤i≤gk satisfying (4.15) and (4.16). Then for all u ∈ HE∞ and ∞ d v∈G F , we have ! ! ∞ gk gk ∞ X ∞ X X X X X |(vk )i | fkj u b(j) = lim |(vk )i | fkj u b(j) . n→∞ 1≤j≤n j=1 k=1 i=1 k=1 i=1 i i 12

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III Proof of Lemma 4.11. Let u ∈ HE∞ and u ≈ (b u(j))j∈N . Define un := u b(n) (j) setting ( u b(j), j ≤ n, u b(n) (j) = 0, j > n.




j∈N

by

n ∞ d ∞ Then P∞ for any w ∈ HE , we get hu − u , wiHE → 0 as n → ∞. This is true since u(j) · wj | < ∞ so that j=1 |b X hu − un , wiH ∞ ≤ |ubj · wj | → 0 E j≥n

∞ d as n → ∞. Now for any u ∈ HE∞ and v ∈ G F and from (4.15) and (4.16) we have ! ∞ ∞ ∞ X X X fkj u b(j) · vk = (f (u))k · vk = hf (u), viG∞ F k=1

k=1

j=1

dj

=

gk ∞ XX ∞ X X k=1 j=1 ℓ=1 i=1 dj

=

∞ X X j=1 ℓ=1

where

fkjℓi u b(j, ℓ)(vk )i =

u b(j, ℓ)

∞ X

fkjℓ · vk =

∞ X

(∞ X

u b(j, ℓ)

gk ∞ X X

fkjℓi (vk )i

k=1 i=1

u b(j) · (v ◦ f )j = hu, v ◦ f iHE∞ , (4.23)

fkjℓ · vk

k=1

and

j=1 ℓ=1

j=1

k=1

Cdj ∋ (v ◦ f )j =

dj ∞ X X

)dj

,

j ∈ N,

ℓ=1

v ◦ f = {(v ◦ f )j }∞ j=1 . ∞ ∞ d Now we have the mapping f : HE∞ → G∞ F and v ∈ GF , so we have v ◦ f : HE → Σ,  ∞ d where Σ = v = (vj )j∈N , vj ∈ Cdj . For any u ∈ HE∞ and v ∈ G F we have

= hf (u), viG∞ F

dj ∞ gk ∞ X X XX j=1 ℓ=1 k=1 i=1

fkjℓi u b(j, ℓ)(vk )i = hu, v ◦ f iHE∞ .

Then from (4.15) and (4.16) we have hu, (v ◦ f )i < ∞, and since HE∞ is perfect, we ∞ P |(v ◦ f )j · u b(j)| is convergent. So have for any u ≈ (b u(j))j∈N ∈ HE∞ that the series j=1

∞ d then v ◦ f ∈ H E . Then we have

= hu − un , v ◦ f iHE∞ → 0 hf (u) − f (un ), viG∞ F

as n → ∞. Therefore, , = lim hf (un ), viG∞ hf (u), viG∞ F F n→∞

13

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III ∞ ∞ ∞ d d for all u ∈ HE∞ and v ∈ G F . Hence for any u ∈ HE and v ∈ GF we have ! ! ∞ ∞ ∞ X X X X vk · lim fkj u b(j) = vk · fkj u b(j) , n→∞

1≤j≤n

k=1

k=1

that is,

gk ∞ X X lim (vk )i

n→∞

k=1 i=1

X

1≤j≤n

fkj u b(j)

!

= i

gk ∞ X X

j=1

(vk )i

k=1 i=1

∞ X j=1

fkj u b(j)

!

Now we will use the fact that if u ∈ HE∞ then |u| ∈ HE∞ where |u| = c ∈ Rdj , with |u| j





. i

  b j |u|

j∈N

,

u(j, 1)|   |b    |b  u (j, 2)| c :=  , |u| j ..     .   |b u(j, dj )|

∧ in view of Theorem 4.7. The same is true for the dual space [G∞ F ] . So then this argument gives ! ! ∞ gk gk ∞ X ∞ X X X X X fkj u b(j) = |(vk )i | lim |(vk )i | fkj u b(j) . n→∞ 1≤j≤n j=1 k=1 i=1 k=1 i=1 i i

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 : HE∞ → G∞ F can be represented by f = (fkjli )k,j∈N,1≤l≤dj ,1≤i≤gk , an infinite tensor such that dj ∞ X X

|fkjli||b u(j, l)| < ∞, for all k ∈ N, i = 1, 2, . . . , gk ,

(4.24)

j=1 l=1

and gk ∞ X X k=1 i=1

! X ∞ fkj u b(j) < ∞ |(vk )i | j=1 i

(4.25)

∞ d hold for all u ∈ HE∞ and v ∈ G F . Let u b1 = (ub1 (p))p∈N be such that for some j, l where j ∈ N, 1 ≤ l ≤ dj , we have ( 1, p = j, q = l, ub1 (p, q) = 0, otherwise.

14

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III Then u1 ∈ HE∞ so f u1 = f (u1 ) ∈ G∞ F and ∞ X

(f u1 )k =

p=1

fkp u b1 (p)

dp ∞ X X

=

p=1 q=1

dj X

=

q=1

fkpq ub1 (p, q)

fkjq ub1 (j, q)

= fkjl ∈ Cgk .

(4.26)

We now first show that d (f u)(k) =

dj ∞ X X j=1 l=1

fkjl u b(j, l),

where fkjli ∈ C for each k, j ∈ N, 1 ≤ l ≤ dj and 1 ≤ i ≤ gk . The way in which f has been defined we have (f u)k =

dj ∞ X X j=1 l=1

fkjl u b(j, l),

fkjl ∈ Cgk .

  d Also since u ∈ HE∞ , from our assumption we have f u ∈ G∞ and f u ≈ (f u)(j) F d so that (f u)k ≈ (f u)(k).

j∈N

,

dj ∞ P P ∧ d We can then write (f u)(k) = fkjl u b(j, l). Since we know that v ∈ [G∞ F ] and j=1 l=1

f u ∈ G∞ F , we have gk ∞ X X k=1 i=1

d |(vk )i ||((f u)(k))i | =

gk ∞ X X

|(vk )i ||

k=1 i=1

dj ∞ X X j=1 l=1

fkjli u b(j, l)| < ∞.

In particular using the definition of u1 and (4.26) we get dp gk gk ∞ X ∞ X ∞ X X X X |(vk )i ||fkjli | < ∞, fkpqi ub1 (p, q) = |(vk )i | k=1 i=1

p=1 q=1

k=1 i=1

for any j ∈ N and 1 ≤ l ≤ dj . Now for any u ∈ HE∞ consider

dj gk ∞ X ∞ X X X (vk )i fkjli ||b u(j, l)|. J= | j=1 l=1

k=1 i=1

Then we consider the series

dj gk ∞ X X X X (vk )i fkjli ||b u(j, l)|, | In := 1≤j≤n l=1

k=1 i=1

15

(4.27)

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III so that we have In =

dj gk ∞ X X X X | (vk )i fkjli ||b u(j, l)|

1≤j≤n l=1

=

k=1 i=1

dj gk ∞ X X X X (vk )i fkjli u b(j, l)|. |

1≤j≤n l=1

k=1 i=1

Let ǫ = (ǫi )1≤i≤dk , k ∈ N, be such that ǫi ∈ C and |ǫi | ≤ C, for all i and such that |

gk ∞ X X k=1 i=1

Then

In =

(vk )i fkjli u b(j, l)| = dj ∞ gk X X XX

1≤j≤n l=1 k=1 i=1

≤ C

gk ∞ X X k=1 i=1

gk ∞ X X k=1 i=1

(vk )i fkjli u b(j, l)ǫi .

(vk )i fkjli u b(j, l)ǫi

X X dj fkjli )b u(j, l)ǫi . |(vk )i | 1≤j≤n l=1

(4.28)

It follows from Lemma 4.11 that ∞ dj dj gk gk ∞ X ∞ X X X X X X X fkjli u b(j, l)ǫi < ∞. |(vk )i | fkjli u b(j, l)ǫi = |(vk )i | lim n→∞ j=1 l=1 k=1 i=1 1≤j≤n l=1 k=1 i=1 Then

dj gk ∞ X ∞ X X X J= | (vk )i fkjli ||b u(j, l)| < ∞. j=1 l=1

(4.29)

k=1 i=1

So we proved that if (fkjli ) satisfies •

dj ∞ P P

|fkjli ||b u(j, l)| < ∞, ! P gk ∞ ∞ P P |(vk )i | fkj u b(j) < ∞, • j=1 k=1 i=1 j=1 l=1

HE∞

then for any u ∈ and v ∈ that gk ∞ P P |(vk )i ||fkjli| < ∞, (i) (ii)

k=1 i=1 dj ∞ P P

|

gk ∞ P P

i ∞ ∧ [GF ]

we have from (4.27) and (4.29), respectively,

(vk )i fkjli )||b u(j, l)| < ∞.

j=1 l=1 k=1 i=1

Now recall that for f : HE∞ → G∞ F we have (f (u))k =

dj ∞ X X j=1 l=1

16

fkjl u b(j, l),

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III ∧ ∞ for any u ∈ HE∞ , then for any v ∈ [G∞ F ] , the composed mapping v ◦ f : HE → C is given by ∞ X

(v ◦ f )(u) =

vk · (f (u))k =

k=1 i=1

k=1

dj gk ∞ X ∞ X X X

=

gk ∞ X X

j=1 l=1

k=1 i=1

So by (ii) we get that

  dj ∞ XX (vk )i  fkjli u b(j, l) j=1 l=1

!

(vk )i fkjli u b(j, l).

(4.30)

dj gk ∞ X ∞ X X X | |(v ◦ f )(u)| ≤ (vk )i fkjli ||b u(j, l)| < ∞. j=1 l=1

k=1 i=1

gk ∞ P P ∞ d (vk )i fkjli ∈ H So fb(v) = (fb(v)jl )j∈N,1≤l≤dj , with fb(v)jl = E (from the definition k=1 i=1

∞ b d ∞ ∞ of H E ), that is f is sequential. And then hf (u), viGF = hu, f (v)iHE is also true.

Now to prove the converse part we assume that f : HE∞ → G∞ F is sequential. We have to show that f can be represented as f ≈ (fkjli )k,j∈N,1≤l≤dj ,1≤i≤gk and satisfies (4.15) and (4.16).
Define for k, i where k ∈ N and 1 ≤ i ≤ gk , the sequence uki = uki j j∈N such that ki (j, l), given by uki ∈ Cdj and uki (l) = uc j

j

( 1, j = k, l = i, c ki uki j (l) = u (j, l) = 0, otherwise.

ki ∞ ∧ ki ki ∞ d Then  u ∈
[G  F ] . Now since f is sequential we have u ◦ f ∈ HE
and u ◦ f = uki ◦ f j , where (uki ◦ f )j ∈ Cdj . We denote uki ◦ f = fjki j∈N , where j∈N

ki dj ∞ d = (u ◦ f )j . Then (fjki )j∈N ∈ H E and fj ∈ C .   b Then for any φ ≈ φ(j) ∈ HE∞ we have

fjki

ki

j∈N

dj ∞ X X j=1 l=1

b l)| < ∞. |fjlki ||φ(j,

For φ ∈ HE∞ we can write f (φ) ∈ G∞ F . We can also write   f (φ) = fd (φ)(p) 17

p∈N

.

(4.31)

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III So ki

u ◦ f (φ) =

∞ X

\ uki j (f (φ))j

j=1

=

dj ∞ X X

\ uki jl (f (φ))(j, l)

j=1 l=1

= (fd (φ))(k, i) (from the definition of uki ).

∞ d We have uki ◦ f = (f ki ) ∈ H E , so

∞ X

(uki ◦ f )(φ) =

j=1

b fjki φ(j)

dj ∞ X X

=

j=1 l=1

From (4.32) and (4.33) we have (fd (φ))(k, i) = dj ∞ P P

(4.32)

b l). fjlki φ(j,

dj ∞ P P

j=1 l=1

(4.33)

b l). fjlki φ(j,

b l), k ∈ N, and 1 ≤ i ≤ gk , that is f is represented by fjlki φ(j, Hence (f (φ))ki = j=1 l=1  the tensor (fjlki ) k,j∈N,1≤i≤g ,1≤l≤d . j

k

If we denote fjlki by fjlki = fkjli , we can say that f is represented by the tensor ∞ ∞ ∞ d (fkjli )k,j∈N,1≤l≤dj ,1≤i≤gk . Also let v ∈ G F . Since f (φ) ∈ GF for φ ∈ HE , then from ∞ d the definition of G F we have gk ∞ X X

|(vk )i |

k=1 i=1

dj ∞ X X j=1 l=1

This completes the proof of Theorem 4.9.

b l)| < ∞. fkjli φ(j,



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 H, G be two Hilbert spaces and let E and F be the operators corresponding to the bases {ej }j∈N , {hk }k∈N , as in (2.2), where dj = dim Xj and gk = dim Yk , and L L dj k . Also H = Xj and G = Yk . Xj = span{ejl }l=1 , Yk = span{hki }gi=1 j∈N

k∈N

We denote the spaces of smooth type functions corresponding to the operators E and F in the Hilbert space H and G, respectively, by HE∞ and G∞ F . 18

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III The main application of Theorem 4.9 will be in the setting when X, Y are compact manifold without boundary, where H = L2 (X) and HE∞ = C ∞ (X), and G = L2 (Y ), ∞ G∞ F = C (Y ). Using Theorem 4.9 we prove the universality of the spaces of the smooth type functions, C ∞ (X), where we can write f ∈ C ∞ (X) ⇐⇒ ∀N∃CN : |fb(j, k)| ≤ CN (1 + λj )−N for all j, k.

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 HE∞ -mapping if c, the composed mapping u ◦ f : X → C belongs to H ∞ . for any u ∈ W E Next we prove the universality of the spaces of smooth type functions.

Theorem 5.2. Let X be a compact manifold. ∞ d (i) The delta mapping δ : X → H E defined by δ(x) = δx ,

and δx (φ) = hδx , φiHE∞ =

dj ∞ X X j=1 l=1

b l)(δx )jl = φ(x), for all φ ∈ H ∞ , x ∈ X, φ(j, E

is a HE∞ -mapping. ∞ ∞ d (ii) If g˜ : H E → GF is a sequential linear mapping, then the composed mapping ∞ g˜ ◦ δ : X → GF is a HE∞ -mapping. (iii) For any HE∞ -mapping f : X → G∞ F , there exists a unique sequential linear ∞ ∞ b d mapping f : HE → GF such that f = fb ◦ δ.

S −s ∞ d HE = [HE∞ ]′ . Proof. (i) Recall that H E = s∈R

∞ ∞ ∧ d Let v ∈ [H E ] = HE . We define the composed mapping

v ◦ δx = hδx , viHE∞ =

dj ∞ X X

vjl (δx )jl = v(x) (by definition of δ).

j=1 l=1

∞ ∞ d This is well-defined since v ∈ HE∞ and δx ∈ [HE∞ ]′ = H E . Also since v ∈ HE , we see that v ◦ δ ∈ HE∞ and that implies δ is a HE∞ -mapping.

∞ ′ ∞ d (ii) Let u ∈ G F = [GF ] . From the definition of HE∞ -mapping we have to show that u ◦ g˜ ◦ δ ∈ HE∞ . Now by given condition g˜ : [HE∞ ]′ → G∞ ˜ ∈ [HE∞ ]′′ . F , so we have u ◦ g Here we claim that [HE∞ ]′′ = HE∞ .

19

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III Note that HE∞ ⊆ [HE∞S ]′′ . ∞ ′ Recall that, [HE ] = HE−s and so for any s ∈ R, s∈R

HE−s ⊆ [HE∞ ]′ ⇒ [HE∞ ]′′ ⊆ [HE−s ]′ ⇒ [HE∞ ]′′ ⊆ HEs . T s Since the above is true for any s ∈ R, we have [HE∞ ]′′ ⊆ HE = HE∞ and this gives s∈R

[HE∞ ]′′ = HE∞ . Then u◦˜ g ∈ HE∞ . Using same argument as in the proof of (i) and from the definition of the mapping δ we have u ◦ g˜ ◦ δx = u ◦ g˜(x) and u ◦ g˜ ◦ δ belongs to HE∞ . So from Definition 5.1, g˜ ◦ δ is a HE∞ -mapping. ∞ ∞ d (iii) Existence of fb : H E → GF .

∞ ∞ ∞ d By hypothesis, f : X → G∞ F is a HE -mapping so that for any v ∈ GF , v ◦ f ∈ HE . ∞ ∞ ˜ d A sequential linear mapping f˜ : G F → HE can be defined by f (v) = v ◦ f, and . v(f (x)) = hf (x), viG∞ F Hence by Theorem 4.9 there is an adjoint mapping, we denote it by fb, where fb : ∞ ∞ ∧ b d d H E → [GF ] is a sequential mapping. By the definition of the adjoint mapping f we have ∞ ∧ ˜ b hu, f(v)i H ∞ = hf ◦ u, viG∞ , u ∈ [H ] , E

E

F

∞ d where h·, ·i is the bilinear function on HE∞ × H E defined in Section 3. The above can be written as ∞ d . hu, v ◦ f iH ∞ = hfb ◦ u, viG∞ , u ∈ H F

E

E

For u = δx , this gives

, = hδx , v ◦ f iHE∞ = (v ◦ f )(x) = v(f (x)) = hf (x), viG∞ hfb ◦ δx , viG∞ F F

∞ b d for any v ∈ G F . This proves f = f ◦ δ. ∞ ∞ d Uniqueness of fb : H E → GF .

∞ b d Suppose fb ◦ δ = f = 0. We have to show that fb = 0 on H E . Since f is sequential, ∞ ∞ d there exists g : G F → HE such that

∞ d Take u = δx ∈ H E , then

= hu, g ◦ viHE∞ . hfb ◦ u, viG∞ F

= hδx , g ◦ viHE∞ = g(v(x)) = 0 hfb ◦ δx , viG∞ F

∞ ∞ ∞ b d d d ∞ for any v ∈ G F , that is, g = 0 on GF . From hf ◦ u, viGF = 0 for any v ∈ GF , we get ∞ ∞ b d d fb ◦ u = 0 for any u ∈ H  E , that is f = 0 on HE .

20

Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III 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 Γ{Mk } (X) on a compact manifold X, where M{k} be a sequence of positive numbers such that (1) M0 = 1, (2) Mk+1 ≤ AH k Mk , k = 0, 1, 2, ..., (3) M2k ≤ AH k min Mq Mk−q , k = 0, 1, 2, ..., for some A, H > 0. 0≤q≤k

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 {Mk }and {Nk } satisfy conditions (M.0)− (M.3). A linear mapping f : Γ{Mk } (X) → Γ{Nk } (X) is sequential if and only if f is represented by an infinite tensor (fkjli ), k, j ∈ N0 , 1 ≤ l ≤ dj and 1 ≤ l ≤ dk such \ that for any u ∈ Γ{Mk } (X) and v ∈ Γ{N (X) we have k} dj ∞ X X

|fkjli | |b u(j, l)| < ∞, f or all k ∈ N0 , i = 1, 2, ..., dk ,

j=0 l=1

and

dk ∞ X X k=0 i=1

! ∞ X |(vk )i | fkj uˆ(j) < ∞. j=0 i

\ \ Furthermore, the adjoint mapping fb : Γ{N (X) → Γ{M (X) defined by the formula k} k} fb(v) = v ◦ f is sequential, and the transposed matrix (fkj )t represents fb, with f and fb related by hf (u), vi = hu, fb(v)i.

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. References [1] R. D. Carmichael, A. Kami´ nski, and S. Pilipovi´c. Boundary values and convolution in ultradistribution spaces, volume 1 of Series on Analysis, Applications and Computation. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.

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Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III [2] A. Delcroix, M. F. Hasler, S. Pilipovi´c, and V. Valmorin. Embeddings of ultradistributions and periodic hyperfunctions in Colombeau type algebras through sequence spaces. Math. Proc. Cambridge Philos. Soc., 137(3):697–708, 2004. [3] A. Dasgupta and M. Ruzhansky. Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces. Bull. Sci. Math., 138(6):756–782, 2014. [4] A. Dasgupta and M. Ruzhansky. Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc., 368 (12):8481– 8498, 2016. [5] A. Dasgupta and M. Ruzhansky, Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations, Trans. Amer. Math. Soc. Ser. B., 5 (2018), 81–101. [6] J. Delgado and M. Ruzhansky. Fourier multipliers, symbols and nuclearity on compact manifolds. J. Anal. Math, 135 (2018), 757–800. [7] J. Delgado and M. Ruzhansky. Schatten classes on compact manifolds: kernel conditions. J. Funct. Anal., 267(3):772–798, 2014. [8] L. H¨ ormander. The spectral function of an elliptic operator. Acta Math., 121:193–218, 1968. [9] C. Garetto and M. Ruzhansky. Wave equation for sums of squares on compact Lie groups. J. Differential Equations, 258(12):4324–4347, 2015. [10] T. Gramchev, S. Pilipovic and L. Rodino. Eigenfunction expansions in Rn . Proc. Amer. Math. Soc., 139(12):4361–4368, 2011. [11] G. K¨othe. Topological vector spaces. I. Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159. Springer-Verlag New York Inc., New York, 1969. [12] H. Komatsu. Ultradistributions. I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20:25–105, 1973. [13] H. Komatsu. Ultradistributions. II. The kernel theorem and ultradistributions with support in a submanifold. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24(3):607–628, 1977. [14] H. Komatsu. Ultradistributions. III. Vector-valued ultradistributions and the theory of kernels. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29(3):653–717, 1982. [15] S. Pilipovi´c and B. Prangoski. On the convolution of Roumieu ultradistributions through the ǫ tensor product. Monatsh. Math., 173(1):83–105, 2014. [16] S. Pilipovi´c and D. Scarpalezos. Regularity properties of distributions and ultradistributions. Proc. Amer. Math. Soc., 129(12):3531–3537 (electronic), 2001. [17] L. Rodino. Linear partial differential operators in Gevrey spaces. World Scientific Publishing Co., Inc., River Edge, NJ, 1993. [18] C. Roumieu. Ultra-distributions d´efinies sur Rn et sur certaines classes de vari´et´es diff´erentiables. J. Anal. Math., 10:153–192, 1962/1963. [19] W. Rudin. Real and complex analysis. McGraw-Hill Book Co., New York-D¨ usseldorfJohannesburg, second edition, 1974. McGraw-Hill Series in Higher Mathematics. [20] M. Ruzhansky and N. Tokmagambetov. Nonharmonic analysis of boundary value problems. Int. Math. Res. Not. IMRN, 12:3548–3615, 2016. [21] M. Ruzhansky and N. Tokmagambetov. Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field. Lett. Math. Phys., 107(4):591–618, 2017. [22] M. Ruzhansky and N. Tokmagambetov. Nonharmonic analysis of boundary value problems without WZ condition. Math. Model. Nat. Phenom., 12(1):115–140, 2017. [23] M. Ruzhansky and N. Tokmagambetov, Convolution, Fourier analysis, and distributions generated by Riesz bases. Monatsh. Math., 187:147–170, 2018. [24] R. T. Seeley. Integro-differential operators on vector bundles. Trans. Amer. Math. Soc., 117:167–204, 1965. [25] R. T. Seeley. Eigenfunction expansions of analytic functions. Proc. Amer. Math. Soc., 21:734– 738, 1969. [26] Y. Taguchi. Fourier coefficients of periodic functions of Gevrey classes and ultradistributions. Yokohama Math. J., 35:51–60, 1987.

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Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III [27] Y. Taguchi. The universality of the spaces of ultradistributions Cs (T)∧ , C(s) (T)∧ (0 < s ≤ ∞), C0 (T)∧ and Exp(C× )∧ . Tokyo J. Math., 10(2):391–401, 1987. [28] D. Vuckovi´c and J. Vindas. Eigenfunction expansions of ultradifferentiable functions and ultradistributions in Rn . J. Pseudo-Differ. Oper. Appl., 7(4):519–531, 2016. [29] L. Waelbroeck. Differentiable mappings into b-spaces, J. Funct. Anal., 1, 409–418, 1967. Aparajita Dasgupta: Department of Mathematics Indian Institute of Technology, Delhi, Hauz Khas New Delhi-110016 India E-mail address [email protected] Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium and School of Mathematical Sciences Queen Mary University of London United Kingdom E-mail address [email protected]

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