FUNCTIONAL CALCULUS IN THE ALGEBRA OF GENERALIZED HYPERFUNCTIONS ON THE CIRCLE AND APPLICATIONS

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FUNCTIONAL CALCULUS IN THE ALGEBRA OF GENERALIZED HYPERFUNCTIONS ON THE

CIRCLE AND APPLICATIONS

V. Valmorin

To cite this version:

V. Valmorin. FUNCTIONAL CALCULUS IN THE ALGEBRA OF GENERALIZED HYPERFUNC-

TIONS ON THE CIRCLE AND APPLICATIONS. 2019. �hal-02099190�

GENERALIZED HYPERFUNCTIONS ON THE CIRCLE AND APPLICATIONS

V. VALMORIN

Abstract. This paper deals with a functional calculus in the al- gebra H pTq of generalized hyperfunctions on the circle. This is done introducing an inductive family of complete ultrametric sub- algebras. Power series expansions of classical functions such as the exponential, logarithm or power ones are considered. As an appli- cation, a nonlinear Cauchy problem involving fractional powers of generalized hyperfunctions is studied.

1. Introduction

This paper aims to provide the algebra H pTq of generalized periodic hy- perfunctions with a functional calculus based on elementary functions but with high nonlinearities. This becomes essential when dealing with nonlinear differential or functional equations. The algebra H pTq was introduced in [18] and its ultrametric topology in [17]. Earlier a first version was given in [16] involving real 2π-periodic smooth functions.

Later on, using the framework of sequence spaces, see [5, 6, 7], the au- thor and his collaborators have given a general topological description of various algebras of generalized functions including H pTq . This des- crition involves projective and inductive limits of locally convex spaces.

It is well-known that contrary to projective limits inductive limits have a bad inheritance of completeness. Moreover it has never been proved that H pTq was a complete space or not. Then to overcome such a situ- ation, we introduce an inductive family p H

pTqq

of complete ultra- metric differential algebras in such a way that H pTq ind lim

H

pTq in a set theoritical sense. Therefore it is shown that the induced induc- tive limit topology on H pTq is finer that its original one. Recall that the initial ultrametric topology of H pTq is given by ω p f, g q ν p f g q where ν is the so-called indicator introduced in [17]. We point out that ν p λ q 1 for all nonzero complex number λ. It follows that p H pTq , ω q is

1

2010 Mathematics Subject Classification : 30H50, 46F15, 46F30, 54E50.

Keywords: Generalized periodic hyperfunctions, Complete algebras, Functional cal- culus, Cauchy problem.

1

not a classical topological algebra over the field C of complex numbers since the multiplication by a nonzero complex number is not contin- uous. Nevertheless ν induces a complete ultrametric structure on the associated algebra C of generalized complex numbers over which H pTq is a classical topological algebra but it should be noticed that C is not a field nor a domain. In the same way the topology of each algebra H

pTq is defined by an indicator ν

. Endowed with the ultrametric ω

such that ω p f, g q ν

p f g q , H

pTq is a complete algebras.

For the basic theory of Colombeau generalized functions, we refer to [3, 4, 9, 10, 13, 14]. Topological results on generalized functions can be found in [7, 13]. For the theory of periodic hyperfunctions we refer to [1, 2, 11, 12]. We notice that a product of hyperfunctions on the circle is defined in [8] in a more classical setting. This is done using conditions on Fourier coefficients. In the setting of Colombeau algebras, the first work on product of hyperfunctions has been done in [15].

The paper is organized as follows. Section 2 presents some preliminaries on the algebra H pTq which are useful for the sequel. References for this section are mainly [12, 17, 18]. In Section 3 we define and study the algebras H

pTq , r ¡ 1. They are proved to be complete and the same is done for the algebra C of generalized numbers endowed with the ultrametric ω. In Section 4 we give necessary of sufficient conditions for the existence of log p h q , exp p h q or h

, s P R where h P H pTq . Section 4 is concerned with the resolution of a nonlinear Cauchy problem in H pTq where the introduced functional calculus is used.

2. Preliminaries

2.1. The algebra of generalized hyperfunctions on the circle.

For this section we refer mainly to [12, 17, 18]. For r ¡ 1 let C

t z P C , 1 { r   | z |   r u and } f }

sup

zPCr

| f p z q|

for every bounded continuous function f defined in C

. We denote by O

the Banach space of bounded holomorphic functions in C

en- dowed with the norm } }

. Then, the topological space of real analytic functions on the unit circle T is

A pTq ind lim

O

.

If X pTq is the set of sequences of functions p f

q

with f

P A pTq , we denote by X

pTq the subset of X pTq whose elements p f

q

satisfy:

D a ¡ 0, D η P N , D r ¡ 1, f

P O

, } f

}

¤ a

, n ¡ η.

We denote by N

pTq the subset of X

pTq constituted of elements p f

q

satisfying:

@ b P p 0, 1 q , D η P N , D r ¡ 1, f

P O

, } f

}

¤ b

, n ¡ η.

Clearly X

pTq is an algebra for usual termwise operations and N

pTq is an ideal of X

pTq .

Proposition 2.1. [18, Proposition 3.1] If p f

q

P X pTq , then:

(i) p f

q

P X

pTq if and only if

D a ¡ 0, D η P N , D r ¡ 1, | p f

p k q| ¤ a

r

, n ¡ η, k P Z . (ii) p f

q

P N

pTq if and only if

@ b P p 0, 1 q , D η P N , D r ¡ 1, | p f

p k q| ¤ b

r

, n ¡ η, k P Z . The algebra of generalized hyperfunctions on T is the factor algebra

H pTq X

pTq{ N

pTq

The class of p f

q

in H pTq will be denoted by cl p f

q .

Embedding of B pTq and A pTq in H pTq . The space B pTq of periodic hyperfunctions is the topological dual of A pTq . For n P N we set

ϕ

p z q ¸

|k|¤n

z

.

Then we have ϕ

ϕ

ϕ

and lim

ϕ

δ in B pTq where δ is the periodic Dirac distribution. If H P B pTq , then p H ϕ

qp z q

°

|k|¤n

H p p k q z

and lim

H ϕ

H in B pTq . Moreover, the maps i : B pTq Ñ X

pTq defined by i p H q p H ϕ

q

and i

: A pTq Ñ X

pTq defined by i

p f q p f

q

with f

f , satisfy the following:

(i) i and i

are linear embeddings;

(ii) i

is a morphism of algebras.

We denote by B

be the differential operator defined for f P O

, by B

f iz df

dz where z P C

. It follows that for every k P Z ,

pB z

f qp k q ik f ˆ p k q .

Henceforth, H pTq is endowed with two structures of differential algebra defined by

df dz cl

df

dz

and B

f cl pB

f

q

where f P H pTq and p f

q

is any representative of f. Passing to the quotient spaces we get a linear embedding ¯i and an injective morphism of algebras ¯i

such that ¯i |

¯i

. For any H P B pTq one has

¯i p dH dz q d

dz ¯i p H q

and ¯i pB

H q B

¯i p H q .

2.2. The algebra of generalized numbers of exponential type.

Let C

be the algebra of complex valued sequences p z

q

such that:

D a ¡ 0, D η P N

, @ n P E

, | z

| ¤ a

.

Elements of C

are said to be of exponential growth. In the same way, we define I

as the set of elements p z

q

P C

for which

@ b P p 0, 1 q , D η P N

, @ n P E

, | z

| ¤ b

.

The elements of I

are said to be of exponential decrease. It may be seen that C

is a subalgebra of C and that I

is an ideal of C

.

Definition 2.1. The algebra of complex generalized numbers of expo- nential type, is the quotient algebra C C

{ I

.

The complex number z is identified with a generalized number cl p z

q where z

z for all n. We denote by ˜ T the subalgebra of C constituted of elements z with a representative in T

.

Definition 2.2. [18, Definition 3.3] Let f P H pTq and z P T ˜ . The value f p z q of f at z is the generalized number f p z q cl p f

p z

qq where f cl p f

q and z cl p z

q with p z

q

P T

.

2.2.1. Fourier coefficients of a generalized hyperfunction.

Definition 2.3. The Fourier coefficent of rank k P Z of the generalized hyperfunction f is the generalized number

f ˆ p k q cl 1

2iπ

»

|z|1

f

p z q z

dz

where p f

q

is an arbitrary representative of f.

The Fourier coefficients do not depend on the chosen representative and we have the following:

Proposition 2.2. [18, Proposition 3.8 ] If f P H pTq , then:

(i) There exists F P H pTq such that B

F f if and only if f ˆ p 0 q 0.

(ii) There exists F P H pTq such that dF

dz f if and only if f ˆ p 1 q 0.

2.3. Invertibility. We denote by C

the subset of invertible elements in C . It follows from [18, Threorem 3.9], that z P C

if and only if z admits a representative p z

q

such that

D b P p 0, 1 q , D η P N

, @ n ¡ η, | z

| ¥ b

.

Let H

pTq denote the subset of invertible elements of H pTq . From [18, Theorem 3.10], we know that f P H

pTq if and only if it admits a representative p f

q

for which there is r ¡ 1 such that f

P O

and:

D b P p 0, 1 q , D η P N

, @ n ¡ η, inf

zPCr

| f

p z q| ¥ b

.

This means that the generalized number cl p inf

| f

p z q|q is invertible.

Moreover this condition does not depend on the chosen representative.

2.4. The topological structure of H pTq .

Definition 2.4. [17, Definition 3.1] The indicator of f P H pTq is:

ν p f q lim

rÑ1

lim sup

nÑ 8

} f

}

(1) where p f

q

is an arbitrary representative of f .

It is shown (c.f. [17, Proposition 3.6] that ν p f q is also given by ν p f q lim

rÑ1

#

lim sup

nÑ 8

sup

kPZ

p r

| p f

p k q|q

+

. (2)

Then we have:

Proposition 2.3. [17, Proposition 3.1] Let f, g P H pTq and λ P C

. Then the following holds.

(i) ν p f q ¥ 0 and ν p f q 0 iff f 0;

(ii) ν p λf q ν p f q ; (iii) ν p f g q ¤ ν p f q ν p g q ;

(iv) ν p f g q ¤ sup p ν p f q , ν p g qq ; (v) | ν p f q ν p g q| ¤ ν p f g q ;

(vi) ν p f

q ¥ p ν p f qq

if f P H

pTq . Setting

ω p f, g q ν p f g q , f, g P H pTq ,

we define a translation invariant ultrametric distance on H pTq . More- over addition and multiplication are continuous mappings from H pTq

to H pTq where H pTq

is endowed with the ultrametric distance D de- fined by

D rp f, g q , p u, v qs sup p ω p f, u q , ω p g, v qq .

The inverse fonction is a continuous operator of H

pTq (see [17, Propo- sition 3.4 and Corollary 3.2]). We end this section by the following result.

Proposition 2.4. [17, Corollary 3.5] The following holds:

(i) If f P ¯i p B pTqq and f 0, then ν p f q 1.

(ii) The mapping ν is surjective from H pTq to R . 3. Completeness of basic subalgebras

3.1. Completeness of the ultrametric space C. The subalgebra C of H pTq is endowed with the restriction of ν and then with the restriction of the metric ω.

Theorem 3.1. The ultrametric space p C , ω q is complete. Then it is a closed subspace of H pTq .

Proof. Let p λ

q

be a Cauchy sequence in C ; we denote by p λ

q

a representative of λ

. Then we have:

@ ε ¡ 0, D m

P N

, @ p, q P N

, p ¡ q ¥ m

, lim inf

nÑ 8

| λ

λ

|

¤ ε { 2.

Hence, for each p p, q q as above there exists η ¡ 0 such that | λ

λ

|

¤ ε. It follows that we can define two sequences p m

q and p η

q of positive integers both strictly increasing and such that:

@ k P N

, @ n P N

, n ¥ η

, | λ

λ

| ¤ 1

2

. (3) We define the sequence p µ

q

in C by

µ

λ

if n ¥ η

and µ

0 if n   η

.

Since the sequence p η

q is increasing, we have µ

0 if n   η

. Then it follows that

@ k P N

, @ n P N

, | µ

µ

| ¤ 1

2

. (4)

Hence, we have

¸

| µ

µ

| ¤ ¸

k1

1 2

1

2

1 .

It follows that for each n P N

, the sequence p µ

q

converges to ζ

where

ζ

µ

¸

µ

µ

.

This shows that p ζ

q is a moderate element, and then we set ζ cl p ζ

q . Using (4), we have for every p P N

:

| µ

µ

| ¤

p

¸

| µ

µ

| ¤

p

¸

1 2

¤ 1

2

1 2

1 . Letting p Ñ 8 , we get that

| ζ

µ

| ¤ 1

2

1 2

1 , from which it follows that

lim sup

nÑ 8

| ζ

µ

|

¤ 1

2

. This means that ν p µ

ζ q ¤

showing that p µ

q

converges to ζ in p C, ω q . But since µ

λ

for n ¥ η

, it follows that µ

λ

which implies that p λ

q

converges to ζ and concludes the proof. l 3.2. The ultrametric algebras H

pTq . For every r ¡ 1 we set X

pTq tp f

q

P X

pTq , D η P N , @ n ¡ η, f

P O

, lim sup

nÑ 8

} f

}

  8u and we define

H

pTq t f P H pTq , Dp f

q

P X

pTq , cl p f

q f u . Therefore, if R r 0, 8q , we get a well defined mapping

ν

: H

pTq Ñ R by setting

ν

p f q inf t lim sup

nÑ 8

} f

}

, p f

q

P X

pTq , cl p f

q f u . (5) Then, ν

satisfies to the following.

Proposition 3.2. Let f, g P H

pTq and λ P C

. Then we have:

(i) ν

p λ q ν p λ q ; (ii) ν

p λf q ν

p f q ; (iii) ν p f q ¤ ν

p f q ;

(iv) ν

p f q 0 if and only if f 0;

(v) ν

p f g q ¤ ν

p f q ν

p g q ;

(vi) ν

p f g q ¤ max p ν

p f q , ν

p g qq .

Proof. Assume that cl p λ

q and cl p µ

q are two representatives of λ.

Then, we have p λ

µ

q

P N

and consequently for every b P p 0, 1 q there is η P N such that | λ

µ

|   b

for n ¡ η. Therefore

| λ

|

¤ p| µ

| b

q

¤ | µ

|

b

and then lim sup

| λ

|

¤ lim sup

| µ

|

. It follows that lim sup

| λ

|

lim sup

| µ

|

which shows that

ν

p λ q lim sup

nÑ 8

| λ

|

ν p λ q

and proves (i). The proof of (ii) can be done following those of [17, Proposition 3.1], (see Proposition 2.3). To prove (iii), let α ¡ ν

p f q . Then, there exists a representative p f

q

of f in X

pTq such that lim sup

} f

}

  α. Since } f

}

¤ } f

}

for ρ   r, it follows that ν p f q lim

p lim sup

} f

}

q   α. Thus, ν p f q ¤ ν

p f q . We see that (iv) follows from (iii). Now take β ¡ ν p g q and choose a representative p g

q

of g such that lim sup

} g

}

  β. Since lim sup

} f

g

}

¤ lim sup

} f

}

lim sup

} g

}

, it follows that ν

p f g q ¤ αβ proving (v). Using the above notation, there exists η P N such that } f

}

  α

and } g

}

  β

for n ¡ η. It follows that

} f

g

}

¤ p α

β

q

. Assuming tha α ¥ β we get

p α

β

q

α

1 β

α

Ñ α as n Ñ 8 which proves (vi). The proof of the proposition is then complete.

l

Clearly H

pTq is a subalgebra of H pTq and H

pTq € H

pTq if r ¥ s ¡ 1 since ν

¥ ν

. Moreover we have H pTq Y

H

pTq . We introduce the ultrametric distances ω

on H

pTq and D

on H

pTq

as follows:

ω

p f, g q ν

p f g q and D

pp f, u q , p g, v qq max p ω

p f, g q , ω

p u, v qq . It is easily seen that addition and multiplication are continuous maps from H

pTq

to H

pTq , and the inverse map is a continuous operator on H

pTq

the group of invertible elements in H

pTq . Moreover, if r ¥ s ¡ 1 the embeddings u

: H

pTq Ñ H

pTq and u

: H

pTq Ñ H pTq are continuous. It follows that

H pTq ind lim

H

pTq ,

can be endowed with the inductive limit topology of the spaces H

pTq

which will be denoted by T . Then we have:

Proposition 3.3. The inductive limit topology defined by the ultramet- ric spaces H

pTq on H pTq is finer that the one induced by ν.

Proof. Let V be an open set in H pTq for the topology defined by ν and take f P V . Then, there exists an open ball centered at f such that B p f, α q € V . If r ¡ 1 is such that f P H

pTq , the corresponding open ball B

p f, α q for the topology induced by ν

satisfies B

p f, α q € B p f, α q since ν ¤ ν

. It follows that B

p f, α q € V X H

pTq which proves that V X H

pTq is an open set in H

for the topology induced by ν

. Hence V is an open set for the topology T , which concludes the proof. l For any bounded function g on T , we set

} g }

sup

zPT

| g p z q| . Then, the following holds:

Proposition 3.4. Let f P H pTq . If p f

q

and p g

q

are two represen- tatives of f , then

lim sup

nÑ 8

} f

}

lim sup

nÑ 8

} g

}

.

Proof. Since p f

g

q

P N

pTq , then for every b P p 0, 1 q there are r ¡ 1 and η P N such that f

, g

P O

and } f

g

}

  b

if n ¡ η.

Thus we have: @ b P p 0, 1 q , D r ¡ 1, D η P N , @ n ¡ η, } f

g

}

  b

, n ¡ η.

It folows that } f

}

¤ } g

}

b

for n ¡ η and then lim sup

nÑ 8

} f

}

¤ max p lim sup

nÑ 8

} g

}

, b q .

- If lim sup

} g

}

0, then lim sup

} f

}

¤ b for every b P p 0, 1 q which implies that lim sup

} f

}

0.

- If lim sup

} g

}

¡ 0, taking b   lim sup

} f

}

gives lim sup

} f

}

¤ lim sup

} g

}

.

We have proved that in any case we have lim sup

nÑ 8

} f

}

¤ lim sup

nÑ 8

} g

}

.

The converse inequality can be shown to be true in the same way. l This allows us to define

ν

p f q lim sup

nÑ 8

} f

}

(6)

where p f

q

is any representative of f . It is easy to see that properties

(i), (iii) and (vi) of Proposition 3.2 are satisfied for r 1 and ν

¤ ν.

Theorem 3.5. For every r ¡ 1 and for every f P H

pTq we have:

(i) ν

p f q ¤ max p ν

p f

q , ν

p f qq ; (ii) ν

p f

q ¤ a

ν

p f q ν p f q .

Proof. For z P C

set z

z {| z | . If p f

q

is a representative of f , we have

f

p z q

»

rz¹,zs

f

p ξ q dξ f

p z

q and then

| f

p z q| ¤ | z z

|} f

}

} f

}

.

Since | z z

| ¤ max p r 1, 1 1 { r q r 1, it follows that

| f

p z q| ¤ p r 1 q} f

}

} f

}

. Finally we obtain

lim sup

nÑ 8

} f

}

¤ max p lim sup

nÑ 8

} f

}

, lim sup

nÑ 8

} f

}

q from which (i) follows.

Now let a P T and choose s ¡ 0 such D p a, s q € C

where D p a, s q t z P C , | z a |   s u . Recall that the remainder after the term of degree m in the Taylor expansion of f

about a is

R

p z q p z a q

2iπ

»

Γs

f

p ξ q dξ p ξ z qp ξ a q

where Γ

t ξ P C , | ξ a | s u . It follows that if | z a | ¤ ρ   s, then

| R

p z q| ¤ s s ρ

ρ s

m 1

} f

}

.

Thus, if | z a | ρ and z P T , writting f

p z q f

p a q p z a q f

p a q R

p z q and using the above inequality with m 1 gives

} f

}

¤ 2 } f

}

ρ

ρ

s p s ρ q } f

}

. (7) Set ρ ts with t P p 0, 1 q . Therefore (7) becomes

} f

}

¤ 1 s

2 } f

}

t

t

1 t } f

}

. (8)

Let α 2 } f

}

and β } f

}

. We let ϕ denote the function ϕ p t q α

t

βt 1 t where t P p 0, 1 q . A simple calculation gives

ϕ

p t q p β α q t

2αt α

t

p 1 t q

.

For β α 0, the value of the reduced discriminant of the polynomials p β α q t

2αt α being equal to ?

αβ, we find that it has two roots t

and t

given by

t

α ? αβ

β α and t

α ? αβ β α . If β ¡ α, we find that

t

  0 and t

? α

? α ? β , If β   α, we find that

t

¡ 1 and t

? α

? α ? β . If α β, ϕ

p t q vanishes for t 1

2 and ϕ p 1

2 q 3α.

Therefore, in any case ϕ p t q reaches its minimum at t

? α

? α ? β in p 0, 1 q and we find that

ϕ

?

? α

α ?

β

α 2 a αβ.

This equality is also true when β α. Finally we obtain } f

}

¤ 2

s p 2 } f

}

b

2 } f

}

} f

}

q . It follows that

ν

p f

q ¤ max p ν

p f q , a

ν

p f q c

lim sup

nÑ 8

} f

}

q .

Making r Ñ 1 and using ν p f q lim

p lim sup

} f

}

q gives (ii) and concludes the proof. l

Using Theorem 3.5, (ii) we get straightforwardly:

Corollary 3.6. Let f P H pTq . If ν

p f q 0, then for every m P N

we have ν

p f

q 0.

3.3. Continuity of the differential operators d { dz and B

. To establish the continuity of these differential operators we state and prove the following.

Theorem 3.7. Let f P H

pTq for some r ¡ 1. The following holds:

(i) ν

pB

f q ν

p f

q ¤ ν

p f q , @ ρ P p 1, r q ; (ii) ν pB

f q ν p f

q ¤ ν p f q ;

(iii) If f ˆ p 0 q 0, then ν pB

f q ν p f

q ν p f q .

Proof. Let p f

q

denote a representative of f in X

pTq and let z P C

with ρ P p 1, r q . We have pB

f qp z q izf

p z q with

¤ | z | ¤ ρ, and then 1

ρ } f

}

¤ }B

f

}

¤ ρ } f

}

which gives

lim sup

nÑ 8

}B

f

}

lim sup

nÑ 8

} f

}

. It follows that ν

pB

f q ν

p f

q and ν pB

f q ν p f

q .

Let ρ P p 1, r q and take r

such that ρ   r

  r. Hence, for all z P C

we have

f

p z q 1 2iπ

»

|ξ|r¹

f

p ξ q dξ ξ z 1

2iπ

»

|ξ|1{r¹

f

p ξ q dξ ξ z and then

f

p z q 1 2iπ

»

|ξ|r¹

f

p ξ q dξ p ξ z q

1

2iπ

»

|ξ|1{r¹

f

p ξ q dξ p ξ z q

. It follows that

| f

p z q| ¤ r

} f

}

p r

ρ q

1 r¹

} f

}

p

q

. Simple calculation gives

| f

p z q| ¤ r

r

ρ

p r

ρ q

} f

}

and then

} f

}

¤ r

r

ρ

p r

ρ q

} f

}

. Using } f

}

¤ } f

}

and letting r

Ñ r yields

} f

}

¤ r rρ

p r ρ q

} f

}

.

It follows that ν

pB

f q ν

p f

q ¤ ν

p f q and ν pB

f q ν p f

q ¤ ν p f q which proves (i) and (ii).

Since pB {

f

qp k q ik f p

p k q for all k P Z , it follows from (2) that ν p f

q lim

ρÑ1

#

lim sup

nÑ 8

sup

kPZ

p ρ

| k || p f

p k q|q

+

.

Hence, if ˆ f p 0 q 0, we can choose p f

q

such that f p

p 0 q 0 for every n and we will have

sup

kPZ

p ρ

| k || p f

p k q|q ¥ sup

kPZ

p ρ

| p f

p k q|q .

This leads to ν pB

f q ¥ ν p f q and then ν pB

f q ν p f q , proving (iii). l Thus, the following corollary is a straightforward consequence of The- orem 3.7.

Corollary 3.8. The differential operators d { dz and B

are continuous in each of the following cases:

(i) from H pTq to H pTq ; (ii) from H

pTq to H pTq ;

(iii) from H

pTq to H

pTq with 1   s   r.

Consequently H pTq is a topological differential algebra.

3.4. Completeness of the topological algebras H

pTq .

Theorem 3.9. The ultrametric algebra p H

pTq , ω

q is a complete one.

Proof. Let p F

q

be a Cauchy sequence in H

pTq . It follows from the definition of ν

that there exist m

, m

P N

with m

¡ m

and two representatives p F

q

and p F

q

of F

and F

respectively such that:

lim sup

nÑ 8

} F

F

}

  1

2

. (9)

Then, we set

F

F

and F

F

. (10) In the same way we get m

P N

with m

¡ m

and two representatives p F

q

and p F

q

of F

and F

respectively such that:

lim sup

nÑ 8

} F

3,n

F

2,n

}

  1 2

. Then, for each n P N

, we set

F

F

F

F

.

Hence, by induction, we get a subsequence p F

q

along with repre- sentatives p F

k 1,n

q

and p F

k,n

q

of F

k 1

and F

k

respectively such that for every k P N

,

lim sup

nÑ 8

} F

k 1,n

F

k,n

}

  1

2

. (11)

Then, for every p k, n q P N

N

we set F

F

k 1,n

F

k,n

F

. (12)

It follows that

F

F

F

j 1,n

F

j,n

for 1 ¤ j ¤ k, and summing up we find that for every k ¥ 2:

F

F

k 1,n

¸

p F

F

q . (13) Since p F

j,n

q

and p F

j,n

q

are both representatives of F

, it follows that °

j2

r F

F

s

n

P N

pTq and then p F

q

is a repre- sentative of F

. Using (12), we get F

F

F

k 1,n

F

k,n

and then using (11) we find lim sup

nÑ 8

} F

F

}

  1

2

. (14)

Then, there exists a sequence p η

q

of positive integers which is strictly increasing and such that

@p k, n q P N

N

, n ¥ η

, } F

F

}

¤ 1

2

. (15) For each k P N

, we define the sequence of functions p G

q

as follows:

G

F

if n ¥ η

and G

0 otherwise.

It follows that p G

q

is a moderate sequence, and if G

rp G

qs , then G

F

. We also have:

@p k, n q P N

N

, } G

G

} ¤ 1

2

. Using successively the above inequality, we get for every p P N

:

} G

G

}

¤ } G

G

}

} G

G

}

¤

¤

1 2ⁿ

1 } G

G

}

¤

2ⁿ1

.

It follows that for each n P N

, the sequence p G

q

is a Cauchy se- quence in O

and then it converges to an element g

in O

. Letting p Ñ 8 in the above inequality gives

} g

G

}

¤ 1

2

1

2

1 . (16)

This shows that p g

q is a moderate element; in fact we have:

} g

}

¤ } G

}

1 2

.

Then we set g rp g

qs . Using (16), we have for every p P N

: } g

G

}

¤

1 2

1 2

1

which gives

ν

p g G

q ¤ lim sup

nÑ 8

} g

G

}

¤ 1

2

and proves that

kÑ 8

lim ν

p g G

q 0.

Hence, p F

q

converges to g in H pTq , and since p F

q

is a Cauchy sequence, it converges to g which concludes the proof. l

4. Functional calculus and applications

All the results stated in this section for the algebra H pTq are also true for the subalgebras H

pTq and C.

4.1. Exponential, logarithm and power functions.

4.1.1. The exponential of a generalized hyperfunction. Let u P H pTq and let p u

q be a representative of u such that u

P O

for some r ¡ 1.

If z P C

, then | exp p u

p z qq| exp p <u

p z qq and consequently } exp p u

q}

exp p sup

zPCr

<u

p z qq .

It follows that p u

q satisfies } exp p u

q}

¤ a

for some positive constant a if and only if sup

<u

p z q ¤ n ln a.

Definition 4.1. A generalized hyperfunction u is said to be real sublin- ear if it admits a representative p u

q

such that u

P O

for some r ¡ 1 and sup

<u

p z q ¤ λn for a real constant λ and n large enough.

We have the following:

Proposition 4.1. For a generalized hyperfunction u, the condition to be real sublinear does not depend on the chosen representative.

Proof. Let p u

q

and p v

q

be two representatives of u where p u

q

is real sublinear; we set

α

sup

zPCr

<u

p z q and β

sup

zPCr

<u

p z q . It follows that

| e

e

| |} e

}

} e

}

| ¤ } e

e

}

and then using | e

1 | ¤ | z | e

, we get

| e

e

| ¤ } e

p e

1 q}

¤ } e

}

} e

1 }

¤ e

e

} v

u

}

.

Since p v

u

q

is negligible, for every ε ¡ 0 there exists η

P N such that e

} v

u

}

¤ ε if n ¡ η

. It follows that e

¤ p 1 ε q e

for n ¡ η

. Hence, if α

¤ λn for n ¡ η ¡ η

, then we have β

¤ r λ ln p 1 ε qs n for n ¡ η which proves the proposition l

We notice that if u is bounded, i.e. } u

}

¤ α for some α ¡ 0 for n large enough, then it is real sublinear. Clearly, if u is real sublinear then λu is also real sublinear if λ is a nonnegative real number. It is easily seen that if u, v P H pTq , then

exp p u v q exp u exp v.

Moreover, since sup

p <u

p z qq inf

<u

p z q , it follows that p u q is real sublinear if and only if inf

<u

p z q ¥ µn for some µ P R when n is large enough. Thus u and p u q are both real sublinear if and only if there are λ, µ P R such that

µn ¤ inf

zPCr

<u

p z q ¤ sup

zPCr

p <u

p z qq ¤ λn.

Under this condition exp p u q and exp p u q are invertible with r exp p u qs

exp p u q .

4.1.2. The exponential of u for ν p u q   1.

Theorem 4.2. If u P H pTq is such that ν p u q   1, then exp p u q is well defined in H pTq and is given by

exp p u q ¸

k0

u

k! .

Proof. Let u P H pTq satisfy ν p u q   1 and choose any representative p u

q

of u. Then we have:

ν p u q lim

rÑ1

p lim sup

nÑ8

} u

}

q   1.

Hence, for every α such that ν p u q   α   1, there exists ρ ¡ 1 such that

ν

p u q lim sup

nÑ8

} u

}

  α, and there exists n

P N

such that for every n ¥ n

:

} u

}

  α

  1.