On distribution spaces based on Jacobi polynomials

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Eijndhoven, van, S. J. L., & Graaf, de, J. (1984). On distribution spaces based on Jacobi polynomials. (EUT-Report; Vol. 84-WSK-01). Technische Hogeschool Eindhoven.

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ONDERAFDELING DER WISKUNDE EN INFORMATICA

DEPARlllliNT OF MATHEMATICS AND COMPUTING SCIENCE

On distribution spaces based on Jacobi polynomials

by

S.J.L. van Eijndhoven and J. de Graaf

EUT Report 84-WSK-OI ISSN 0167-9708 Coden: TEUEDE

Eindhoven March 1984

by

Summary

Preliminaries

Analyticity spaces and trajectory spaces Entireness spaces and ultra-trajectory spaces

Some notations

Survey of results

I. Classification of analyticity spaces and entireness spaces

2. The classification of analyticity spaces and entireness spaces generated by the Jacobi polynomials

3. The characterization of the analyticity spaces and entireness spaces based on the Jacobi polynomials, with applications to classical analysis 2 5 8 10 17 33 44 3.1. The characterization of 3.2. The characterization of

s (

)

and

_T(L2

([-n,n]),6V) L 2 [-norr] ,6v the spaces 46 2 _1 ( 2 . d2 d)V and L2 ( [-1 ,1] ,(I-x ) 2dx) ,I - (I-x ) - 2 +x dx \ dx ( 2

-!

(

2 d2 d

)V)

T,L

_{2 ([-I,I],(I-x) dx, -(I-x) dx2} +K _dx '

and related classical analytic results 61

4. Hyperfunctions and trajectory spaces 77

5. Ultra-hyperfunctions and ultra-trajectory spaces 98

6. Representation of some large continuous groups as groups of

continuous linear operators on spaces of (ultra-) hyperfunctions 120

Appendix 125

The Jacobi operator

A a,S

2 d2 d

- (I-x) - «S-a) - (a+S+2)x) dx ' dx2

a>-I,S>-I,

1S self-adjoint and positive 1n the Hilbert space

X

a,S

We study the analyticity domain DW(Av Q) and the entireness domain a,1-'

00 v

D (exp(A Q» of the v-th power, v ~

!,

of the operator A Qin X Q

a,1-' a,1-' a,1-'

It is shown that for fixed v the analyticity and entireness domains do not depend on a and S. Moreover,for each fixed v the mentioned domains are characterized as spaces of analytic functions of suitable well-described

(growth-) classes.

Next the spaces DW(Av Q) and Doo(exp(Av Q» are considered as test spaces for

a,1-' a,1-'

distribution theories. They are very special examples of general functional analytic constructions as g1ven by De Graaf and Van Eijndhoven.

The distribution spaces (dual spaces of

described in detail and many natural examples of continuous linear function-als and (extendible) continuous linear mappings are given. These examples are based on simple geometric and analytic considerations. Further, expansions of distributions in Jacobi polynomials and Jacobi functions of the second kind are studied and sharp estimates on the expansion coefficients are produced. For v

=

!

the relation with (ultra-) hyperfunction theory is discussed.

Finally, some large groups of analytic functions are represented as (extendi-ble) linear operators ,on the mentioned spaces.

In the past decennia generalized functions have been introduced in mAny different ways. We mention Schwartz's distribution theory in which general-ized functions are regarded as continuous linear functionals on some locally convex space of good functions, Jones' theory on generalised functions, in which generalized functions are weak limits of regular sequences of good

functions, and the theory on hyperfunctions in which generalized functions are 'boundary values' of analytic functions defined in a region of the complex plane. Probably the easiest way to introduce generalized functions is by means of formal series expansions with respect to some orthonormal basis in a Hilbert space. Here we shall clarify this method a bit more.

00

Let (vn)n=O be an orthonormal basis in a Hilbert space X. The Riesz-Fischer

00

I

a v where the sequences n=O n n

a class which contains

l2.

A candidate for the dual of E

00

considering sequences (b )00 0 for which

I

I~ b

I

< 00

n n= n=O n n

E.

Let 0 denote the vector space of the related series

00

(an)n=O belong to can be obtained by

00

for all (an)n=O ~n

00

theorem says that each f in X is represented in

l2

by the l2-sequence

N

«f,vn»noo=O' and Ilf -

I

(f,v)v 11+0 as N + 00. Now consider a vector space

n=O n n

E

the elements of which are formal series

I

b v . Then it is clear that 0 c X c E. Thus we get a functional analytic n=O n n

analogue of the so called pans ion theory [7] developed by Korevaar. We note that Korevaar looks at the Hilbert space L

2(IR) with the Hermite functions as an orthonormal basis.

Analyticity spaces and trajectory spaces

A simplified version of the theory [6] on generalized functions developed by De Graaf can be described as follows. The space

E

consists of all formal

00

series

I

a v where the sequences (an)noo=O satisfy n=O n n

(0. 1)

-;\ t n

Vt>O: sup (Ia Ie ) < 00.

nEIN n

of the positive self-adjoint operator

H

I

n=O spectrum

a

< A O~ Al ~ A2 ~ •.. and { A I n E IN u {a}} is the n

Here (An):=O is a fixed sequence of positive numbers with the property that 00 -;\ t

e n < 00 for all t > O. The set

loosely defined by (0.2)

Hv

_n Av

n n n=0,1,2, . . . .

00 00

Let (an)n=O be a sequence with property (0.1). Then to (an)n=O we let cor-respond the mapping F: (0,00) + X defined by

F(t) 00

I

n=O -A t n e a v n n

-TH

It is clear that the mapping F satisfies F(t+T) = e F(t), t,T >

o.

-TH

Conversely, if a mapping G: (0,00) + X satisfies G(t+T) = e G(t) for all

can be identified with the set of all So the set of all sequences

a v .

n n

t,T > 0, then there exists a sequence (a )00 0 which obeys condition (0.1) 00 -A t n n=

and for all t > 0, G(t) =

I

e n n=O 00

(an)n=O determined by condition (0.1)

mappings F from (0,00) into X with the mentioned property.

Since each positive self-adjoint operator

A

generates a one parameter sem1-group of so-called smoothing operators, the previous considerations lead to the following more general definition.

(0.4)

Definition

Let A be a positive self-adjoint operator in X. Then the space T~,A'

the trajectory space, consists of all mappings F: (0,00) + X with the

called property that

Heuristically, the space

T

_x

A can be regarded as the space of 'initial

con-,

du

ditions' u(o) to the evolution equation dt - Au, which give rise to a solution t ~ u(t), t > 0, through X.

The topology for the space

Tx

_,

A is generated by the seminorms

F t+ IIF (t)IIX' t > 0 , F E TX,A •

With these sem~norms

Tx,A

becomes a Frechet space. Moreover,

Tx,A

is ~10ntel

iff the operators e-tA, t > 0, are compact;

TX,A

is nuclear iff the operators -tA

e t > 0, are Hilbert-Schmidt. (Observe that

T

_X

H

is nuclear.)

00 _~ t

Let f E X, f

L

(f,v)v. satisfy (f,v ) = O(e n) for some t > O. Then

n=O n oon n for every F E

T

X

H'

F =

L

anvn, we have " n=O I N ( N 2 -2A

T)2(

N 2

I \;

(f,v

)1

~

I

la l e n \

L

1

(f,v

)1

n=O n n n=O n n=O n

If we take 0 < T < t the right-hand side converges if N + 00. So the series

00

a

(f,v ) is absolutely convergent. We remark that the order estimate on

n n

00

«f,v» 0 implies that f is contained in the domain of each

n n=

unbounded self-adjoint operator e

TH ,

0 < T < t. This leads to the general

I

n=O

the sequence

(0.5) Definition

u

t>O

-tA

{e

w.

I t > 0, W E xl .

It is clear that Sx A is a dense subspace of x. Since Sx A consists of

_,

_,

precisely all analytic vectors of A, we call the spaces of type Sx A

analyt-,

icity spaces.

The natural topology for the space Sx A is the inductive limit topology

,

induced by the spaces e-tA(x) with Hilbert space topology generated by the

. tA tA -tA

1nner product (f,g)t := (e f,e g), f,g E e (X). This inductive limit is not strict. The construction of seminorms which generate the inductive limit topology has led to a thorough description of several topological features of SX,A' For example, the sequence (fn)nElN is a null sequence in SX,A iff

tA

there exists t > 0 such that (e f ) :IN is a null sequence in x. Further, n nE

we note that Sx A is complete, bornological and barreled.

_,

On Sx A

_,

x

T

_X

_,

A we introduce the pairing <","> by

(0.6) <f,G>

The definition (0.6) makes sense for T > 0 sufficiently small and it does not depend on the choice of T. Through this pairing, all strongly continuous

functionals on SX,A are represented by the members of

Tx,A'

Conversely, all continuous functionals on

TX,A

are represented by the members of SX,A" The spaces Sx A and

_,

T

_X

_,

A are in duality.

Entireness spaces and ultra-trajectory spaces

Another distribution theory which is a considerable generalization of tHe theory of tempered distributions, has been developed by Van Eijndhoven, see

[2]. In order to introduce this theory along the lines of formal series expansions we start with a fixed sequence of positive numbers (~n):=O which are ordered, 0 < ~O ~ ~I ~ ••• and ~n + 00, n + 00. Further, there has to be

~ e-~ntO

to > 0 such that L < 00. Now for the space

E

we take the vector

n=O 00

space of all formal series

I

a v where the sequences (a )00 0 satisfy

n=O n n n n= (0.7.a) 3_t>0: sup la Ie-~nt < 00 nElNu{O} n or equivalently (O.7.b) 00 -2~ s 3 V •

I

la

1

2 e n t>O s~t' n=O n 00

To a sequence (an)n=O with the property (O.7.b) we associate a mapping ~

from [t,oo) into X as follows:

Note that Hs) 00

I

n=O s ~ t .

On the other hand, let ~: [t,oo) + X satisfy

Then there exists a sequence (b)oo which satisfies (O.7.b) and n n=O'

'P(s) 00

I

n=O -fl nS e b v n n s ?': t

We thus arrive at the following general definition.

(0.8) Definition

Let

A

be a positive self-adjoint operator in X. Let X

t' t > 0, denote the space of all mappings ~ from [t,oo) into X satisfying

Then the space a(X,A) is defined to be the inductive limit

a(x,A)

(We note that X c X for all t, 0 < t < T. )

t T

The spaee X_t IS a Hilbert space with inner product (iP,'P)(t) We note that X

t is a copy of X.

The space a(X,A) is called the ultra-trajectory space. Inspired by [6] explicit semihorms have been constructed which generate a locally convex topology equivalent to the inductive limit topology. It has been proved that a(x,A) is complete, bornological and barreled. Further, a(X,A) is Montel iff

-tA .

tA

e IS compact for some t > 0; a(X,A) is nuclear iff e- is

Hilbert-Schmidt for some t > O.

Suppose that the Fourier coefficients (f,v ), n E IN u {O}, of an element

n

-fl t

f E X satisfy Vt>O: (f,v

n) = O(e n) or, equivalently, Vt>O:

00 2 f l t 00

I

I

(f,v

n

)!

e n < 00. Then for any sequence (an)n=O satisfying (0.7) the

n=O 00

series

I

a

(f,v ) converges absolutely. So the following definition seems

n=O n n

(0.9) Definition

Let A be a positive self-adjoint operator in x. Then the space T(X,A) is defined by

T(X,A) A

00

D(ee ) ) •

The space T(X,A) is called the entireness space, because it contains all entire vectors of A. We note that wis an entire vector of A iff

V 3 V IIAn II :<=::: n'. anb.

a>O b>O nEIN: w. With the seminorms

tA

wi+ lie wll

X ' W E TeX,A), t > 0 ,

Tex,A) becomes a Frechet space.

The pairing between the spaces Tex,A) and aex,A) is defined as follows. Let w E TeX,A) and let ~ E a(X,A). Then

eO.IO)

where s > 0 has to be taken so large that ~ EX. We note that (0.10) does

s

not depend on the choice of s. With this pairing a(X,A) is a representation of the strong dual of TeX,A) and, conversely, T(X,A) is a representation of the strong dual of aex,A).

From a topological point of view the spaces Tex,A) and

T

_X

_,

A'

and aex,A) and Sx A have similar properties. So the theory [2] can be considered as a kind

_,

of reverse of the theory [6]. For instance, a sequence e~) IN in aex,A) is

n nE a null sequence iff there exists s > 0 such that (~) IN c X and

n nE s

II~ (s)1I -+ 0 as n -+ 00. Finally we mention the following inclusion scheme

n

Tex,A) c Sx A c X c

T

x

A c aex,A) .

SOME NOTATIONS

In this paper we consider the Hilbert spaces

and the positive self-adjoint operator A S 1n X

a, a,S

2 d2 d

A

= -

(I-x) --- - «S-a) - (a+S+2)x) dx

a,S dx2

where we take a and S larger than -I. The ouerator A_{. a,S} has a discrete

spectrum: {n(n+a+S+l)

I

n E IN u {O}}. Its normalized eigenvectors are the normalized Jacobi polynomials

R~a,S),

1

ra + 8 + 2n + I f(n+l)f(n+a+S+I)]2 p(a,S) L 2a+S+1 f(n+a+l)f(n+S+I) n

where

(d. [8], p. 209).

In this report we shall work out the following program: Classification of the space

take v 2

!

and a,S> -I. We

and (y,o) and for all

v

Sx

(A )V and T(X S,(A S)) where we

B' S a, a,

a, a,

get the following result: For all pairs (a, S)

and, also, V T(X_a,B' (A_a,s)) \! T(X 0' (A 0) ) y, y,

Y Cl,S and C a,S d2 1 d d

de 2 + (8-0.) sin e

ere -

(0.+13+1) cot e de It is clear that

n(n+a+S+l)R(a,S)(cos e) •

n

Moreover, for all v

and V T(Y Q , (C (3) )

=

0.,1-' Cl, V T(Y_y,<5'(C_y,<5) )

Characterization of the spaces S (C _l)V and Y_1 _1' I

2 ' 2 2, 2

for each v _~

!,

1n classical analytic terms. Here we employ the relations

. and

(_1 _1)

W

R 2, 2 (cos e) = - cos ne

n I T ' n E 1N

By means of this characterization we can also describe the spaces

Relations between the spaces

T

1

X Q'(A (3)2

0.,1-' a, some classes of hyperfunctions, respectively

1

and 0"(X Q , (A (3) 2) wi th

0.,1-' a, ultra-hyperfunctions.

SURVEY OF RESULTS

This University Report is a contribution to our research project of making a

link between our general functional analytic theory on analyticity spaces,

trajectory spaces, entireness spaces and ultra-trajectory spaces on one hand

and "classical" analysis and distribution theory on the other hand. Similar

results have been achieved in the papers [I], [3], [4] and [5] where test

spaces have been introduced in which the Hermite functions and the Laguerre

functions serve as bases in the way pointed out in the preliminaries. In the

present paper the main emphasis is on the test spaces in which the Jacobi

polynomials establish bases. So the elements of these test function spaces

are determined by sequences of expansion coefficients of certain growth

orders.

We give a detailed classification of such test spaces and characterize them

as classes of analytic functions which satisfy specific growth conditions.

This characterization enables to describe a variety of continuous linear

functionals and continuous linear operators on these spaces in classical

analytic terms. As a further result we find conditions on the asymptotics of

the coefficients in the Jacobi series expansion of an analytic (entire)

function which belongs to such a growth class. Finally, we show that the

duals of some of our test spaces can be represented by spaces of (ultra-)

hyperfunctions. Each (ultra-) hyperfunction can be expanded in so-called

Jacobi functions of the second kind. We describe the convergence of these

expansions in terms of complex analysis.

Now we go into more detail.

In Chapter I we study the general classification problem which can be stated

as follows. Let there be given a separable Hilbert space X and a positive

,(X,A)

=

,(Y,B), both set theoretically and topologically. For instance, if

W

is a continuous bijection on Sx A' respectively ,(x,A), then we may take

_,

Y=

x

W

and

B

=

W-1AW.

Here

x

W

denotes the completion of Sx A (,(x,A»

_,

with respect to the norm II'II

W= IIW'IIX' Special attention is given to the case

X

=

lZ and A

=

Awith Aa diagonal operator in lZ' Also we mention the case that B is a perturbation of A. This has been investigated in an earlier paper

[4] and quoted in Chapter I for completene·ss. Further, we !lay attention to the interrelation between the spaces

Tx,A

and

Ty,B

which both are representa-tions of the dual S~

_,

A' if Sx A

_,

=

Sy

_,

B'

In Chapter Z we apply the abstract results on classification of analyticity spaces and entireness spaces to the following concrete case.

Let Xa,s be the Hilbert space Lz([-I,I],(I-x)a(l+x)Sdx) and let Aa,S be the positive self-adjoint operator

A

a,S

Z d7 d d

- (I-x) --- - (S-a) -- + (a+S+Z)x dx

dxZ dx

as introduced in the Preliminaries. We show that for all pairs (a,S), (y,o) with a,S,y,o > -I and for all v 2

!

and

The coordinate transformation x cos e transforms the Hilbert space X ~n

a,S

a

S

Ya,S

=

LZ([O,n],(1 - cos e) (I + cos e) sin e de)

and the operator A becomes a,S

C

a,S

d2 1 d d

de 2 + (S-a) sin e

de -

(a+S+I) cot e de We have for all pairs (a,S), (y,o), a,S,y,o > -I, and all v ~

and

V

T(y 0' (C 0) )

y, y,

In order to apply the general theorems of Chapter 1, we derive rather subtle estimates for the matrix entries of the operators

d:

and x

d~

with respect to the Jacobi polynomial bases. (See the Appendix to this paper.)

Chapter 3 deals with the characterization problem. We look for a description

in terms of classical analysis, of Sy (C )V and T(Y (3'(C a)v).

a' (3 a, a,fJ

a,fJ a,

tion of the spaces

v the spaces

Sx

(A )v, T(X a,(A a))'

a' (3 a,fJ a,fJ

a,fJ a,

To this end, we start with a

characteriza-,(_£)V

de2

and

These spaces can be presented as classes of 2TI-periodic analytic functions of well-defined growth behaviour dependent on v. Since

S v

Y_a,fJa,(C_a,fJa) S_Y (C I)V

I I ' I -2'-2 -2,-2

the space S ~s the subspace of

S

(

d2)V which consists

Y_{a, fJ}a'(C_{a, fJ}a)V

L ([

_{2 -TI , TI}J) _{, - --2}

de

of all even functions in

S

(

d2 )V' Similarly, T(Ya ,S' (CC':,(3)v) L2 ([ -TI , TI J), - - 2

characterization of the spaces

Sx

(A )V and T(Xa,~,(Aa,B)V) 18 then

(X,r~ (X,(~

obtained by means of the conformal mapping w

=

cos z.

Important consequences of this characterization are theorems on approxima-tion of analytic (entire) funcapproxima-tions by means of (normalized) Jacobi poly-nomials. We mention the following.

Let f be an entire function satisfying 2v

If(w)

I

s A eXP(B(log Iwl)2V-I) , Iwl 2 I ,

where A,B > 0 and v >

!.

Let a,S> -I. Then

f 00

I

n=O a(a'S)R(a,S) n n

where the coefficients a(a,S) satisfy

n

sup

la~a,S)

I exp(nv(t-s» < 00 nE1N

2v-1

for all 0 < s < t with t

=

(2v)-2V(2~-I)

The series

exp (-BOOg 00

I

n=O a(a,S)R(a.,S)(w) n n

for some v >

!

and some t > O. Then the func-b(a'S)R(a,S)(w) is entire analytic and

n n

converges uniformly on ~. (a,S)

Let a.,S > -I, and let (bn )nElNu{O} be a sequence for which sup (lb(a,S)/ exp(nVt» < 00

nElN n

tion f defined by f(w)

=

I

n=O

2v If(w) I S A

I (

2v)-2 V)2V-I Here A_f > 0, and for B

f any number larger than (2v-l) t can be taken.

Let f be an analytic function in an open neighbourhood of the interval a(a,S)R(a,S) where the coefficients

n n

00

< 00 for some t > 0 depending on a

on a sufficiently small

neighbour-I

n=O exp(nt)) [-1,1]. Let a,S> -1. Then f

(a S ) .

I

(a S)

I

a ' sat1sfy sup (a '

n nElN n

and S. The series converges uniformly

hood of [-1,1]. (Cf. [9], where sharper results have been obtained.)

Let a,S> -I and let (b(a,S))oo c ~lNu{O}. Then

n n=O

represents an entire function iff for all t > 0, < 00.

since

e5

z

for all v >

!,

we get:

3 A ,.0 B,w 00

I

I

n=O 2v s

~,w

eXP(B(IOg(max(l, [z 1)))2V-I)

With the aid of these classical analytic descriptions and with elementary geometrical considerations we introduce natural classes of continuous linear

f~nctionals and continuous (extendible) linear mappings on all

S-

and

In Chapters 4 and 5 we limit ourselves to the case v ~. The duals

s'

_{Y Q'(C Q)}~ a,1-' a,1-' and

s'

1 X

Q,(A

0)" a,I-' a,1-'

are linked to three classes of hyperfunctions. The representation of the considered linear functionals as hyperfunctions is by means of contour integrals.

In Chapter 4 we introduce a natural Frechet topology on each of these hyper-function spaces. As a consequence we get the following classical result (see

[9], p. 250): A function 8 which is analytic on the region ~, [-I,J] with

8(00) = 0, can be expanded in a series of associated Jacobi functions, Q(a,S)

n defined by 'Q(a,s)(w) I n

=

21fi I

f

-J

have also been obtained

I

(a S) -nt

I

sup a ' e < 00

nEIN n

outside each open converges uniformly

series satisfy: Vt>O (a,S). h" a ~n t ~s n a(a'S)'Q(a,S) n n 00

I

n=O

of [-J,I]. Results in this directions and the series

neighbourhood The coefficients

in [J0] .

In Chapter 5 the _{duals T'(L2([-1f,n]), - de 2} },(} d2\4\ _T'(Y ! a,S,(Ca,S)2) and

are treated in the same way as the corresponding S'-spaces in the previous chapter. These duals can be represented by classes of so-called ultra-hyperfunctions. Each space of ultra-hyperfunctions can be regarded as an inductive limit of Banach spaces. It follows that the spaces o(L₂([-1f,1f]), (-

d:)~\),

o(Y o,(C

Q)~)

and o(X o,(A 0)4) are homeomorphic

de a,I-' a,I-' a,1-' a,I-'

to certain ultra-hyperfunction spaces. Finally, classical results are ob-tained concerning the expansion in associated Jacobi function series of functions which are analytic at infinity.

In Chapter 6 we deal with two large continuous groups of analytic functions.

These groups can be represented as groups of continuous linear mappings on

1. CLASSIFICATIONS OF ANALYTICITY SPACES AND ENTIRENESS SPACES

In this chapter, we discuss two methods to classify analyticity spaces

and

entireness spaces. First, we consider the case of a general positive self-adjoint operator in a Hilbert space. Next, we turn to a more concrete situation.

Let X denote a Hilbert space and let A be a positive self-adjoint operator 1n X. In our paper [5] we have proved the following classification theorems,

in which a central role is played by perturbations.

(1.1) Theorem

Let P be a linear operator in X with O(P) ~

Sx

AV

where we take v > 0 fixed.

,

Let the following conditions be satisfied:

(i) There exists a Hilbert space Y such that the operator exp(-tAv) maps X into Y for all t > O.

(ii) The operator A + P defined on

Sx

_,

AV

can be extended to a positive self-adjoint operator in Y (denoted by A + P also).

(iii) There exists a monotone non-increasing function ~ on the open interval (0,I) such that

Remark. In fact we proved that under the conditions of Theorem (I. I) the following result is valid: V 0 3~ 0 3 0 such that the operator

t> t> T>

V ~ V V .

exp(TA )exp(t(A+P) )exp(-tA ) is bounded on X.

(1.2) Theorem

Let P be a linear operator in X such that O(P) ~ exp(-tAV) (X) for some t > 0 and some v > 0 fixed. Let the following conditions be satisfied:

(i) There exists a Hilbert space Y and there exists t > 0 such that the

operator exp(-tAv) maps X into Y.

(ii) The operator A + P defined on T(X,Av) is entendible to a positive and

self-adjoint operator in Y.

(iii) There exist positive constants d and r

Oand there exists q, 0 < q < v such that

v -I v q

Ilexp(rA )PA exp(-rA)1I < dr .

A

v v

Then T(X, ) C T(Y,(A+P) ).

Remark. In fact we proved that under the conditions of Theorem (1.2) the following result is valid: V~ 0 V 0 3 0 such that the operator

t> T> t>

v ~ v v

exp(TA )exp(t(A+P) )exp(-tA ) is bounded on X.

Next we consider another way of classifying analyticity spaces and entireness spaces. We prove that each continuous bijection

W

on Sx A

_,

(T(x,A)) gives rise to a Hilbert space xWand a positive self-adjoint onerator AWin xW

such that S A

=

S

X, XW,AW

W W

(T(x,A)

=

T(X ,A )).

We denote the inner product of X by (.,.). Let

W

be a continuous bijection -I

on the analyticity space

Sx

A. Since also its inverse mapping

W

is

con-,

tinuous on

Sx

_,

A the sesquilinear form

turns Sx

_,

A

into a pre-Hilbert space. By x

W

we denote the Hilbert completion of Sx A with respect to ( ,

_,

)W.

Then we have

(1.3) Theorem -I

The operator

W AW

with domain Sx A is positive and essentially self-adjoint

,

in

XW.

If

AW

denotes its unique positive self-adjoint extension

~n

X

W

,

then

SX,A

=

S

W W·

X ,A

Proof

Let f

_W

-Ig E Sx A' Since g

_,

E Sx A there exist a,b

_,

> 0 such that

Hence

n E IN.

for all n E IN. Thus it follows that the analyticity domain of the operator

W-IAW

in x

W

contains Sx

_,

A'

Since Sx A

_,

~s

dense in

x

W

,

the analyticity domain

-) W -I

of

W AW

is dense in X . So following [II], Theorem 8.31,

W AW

is

essen-• 11 If d" .

x

W • 1 h

W-IA(t)'

. .

S

x

W

t~a y se -a Jo~nt ~n • It ~s c ear t at ~s pos~t~ve on XAC •

,

Hence the unique self-adjoint extension

AW

is positive. Since S contains x

W

A(JJ

precisely all analytic vectors of

A

W

,

we obtain SX,A C

SXW,AW'

on'SX,A we have for all t > 0

Now observe that the linear mapping (JJ on Sx

_,

A

can be extended to a bounded

W

linear operator from X onto X. We denote this extension also by

W.

It follows that for all X E x(JJ

The assertion

S

~

S

is then obtained from the observation that

x,A

xW

AW

W '

1

Wx

E X for all X E X and that

W-

is continuous on

SX,A.

n

Since

Sx A

_,

elements G

S W W'

X

,A

E T

_x

_,

A

the elements f E

Sx A

can be paired both with the

,

TA

<f,G> = (e f,G(T», T > 0 sufficiently small, and with the elements H (

<f,H>W

A

W

(eP f,H(p»W' p > 0 sufficiently small.

Let £ be a continuous linear functional on

Sx

A.

Then there exists

,

Tf£} E

Tx,A

such that £(f) = <f,T(£», f E

SX,A'

and also

TW

(£) E

TxW,AW

such that £(f) =

<f,TW(£»W'

f E

Sx A.

SO the antilinear mappings T: £ ~ T(£)

,

and

_TW:

£ ~

_TW

(£) are isomorphisms from S~,A onto

TX,A

and from S~,A onto

T W W'

respectively.

X ,A

We shall investigate the relationship between T(£) and

TW

(£). To this end, we define the mapping

j

on T

_x

_,

A by

-I

j(F): t ~

W

F(t) , t > 0, F E T

_x

_,

A •

W

It is clear that (j(F»(t) E

Sx,A

C X for all t > 0, and also that for all

t,T > 0

(j(F»(t+T)

W

-I F(t+T)

Hence j(F) E

T W W.

Further it follows that j 1S a continuous linear mapp1ng

X ,A

and that j is invertible with

-I

Now let f E S~

_,

A'

Then for all f E

Sx

_,

A'

-I '

<f,j(T«W

)

f))>W

We thus obtain

In the same way

It leads to the following result

-I '

T

_W

=

joT

0

(W

)

T

-I

j 0 T

W 0 W'

which expresses the relation between the representations

T

and

_TW'

We now repeat the above considerations for the entireness space T(X,A).

-)

Let

V

be a continuous bijection on the Frechet space T(X,A). Then

V

is also continuous on T(x,A). With the inner product

T(x,A) becomes a pre-Hilbert space. Let

xV

denote the Hilbert space

comple--I

tion of T(X,A) with respect to

(·,·)V'

It is clear that the operator

V AV

with domain T(X,A) c

xV

is symmetric and positive. Similarly to Theorem (1.3) we have

(1.4) Theorem

-I V

The operator

V AV

~s essentially self-adjoint and positive. Let A denote its unique self-adjoint extension in

XV.

Then

Proof

T(X,A) = T(X

V V

,A ) .

By standard arguments it can be proved that T(X,A) is contained in the

-I V

analyticity domain of the operator

V AV.

Since T(X,A) is dense in X , it follows that

V-1AV

is essentially self-adjoint. The operator

V

defined on T(X,A), can be extended to a bounded operator from xV into X. We denote this

extension also by

V.

Conversely, for each g E X the linear functional

o f (Vf ) . . V h . h xV h h

~: ~ ,g ~s cont~nuous on X . So t ere ex~sts E sue t at

g

(Vf,g)

=

(f,h)V •

Hence g

Vh.

Therefore we write h

-I V g E X# V g E X . -I V g, and We thus obtain V-I(e-tA(X) ) and finally

T(X,A) V-I (n e-tA (X) )

t>O

n

V-I (e-tA(x)) t>O

n

t>O (V- I

e-tAV) (XV) =

n

t>O

v

V T(X ,A ) •

o

Next we apply Theorems (1.3) and (1.4) to the following concrete case. Let

H

be a positive self-adjoint operator in the separable Hilbert space X, and let

H

have a discrete spectrum. Then in X there exists a complete

< 00

for all t > O. The spaces ,(X,H)

ex> -Akt

t > 0 such that

I

e k=O

We define the unitary operator

U

from X into i₂ by such that He = A e • The spaces

n n n

orthonormal basis (en):=o and there are positive numbers A_{k , k} E IN u {OJ,

00 -Akt

Sx

Hand

T

_X

H

are nuclear iff

I

e

" k=O

and o(X,H) are nuclear iff there exists

Uw = «~:I,

e ))

00 0 '

n n= W EX.

Then

and

Here A denotes the diagonal operator with matrix

k,l E IN u {OJ •

So, instead of the spaces SX,H and ,(x,H), we can study the spaces Si

2,A and ,(i2,A) as well. For instance, in Si A there exists a natural identification

2'

between continuous linear mappings on this space and infinite matrices. In the sequel we make no distinction between a linear operator in

i

₂ and its corresponding matrix. Let A be a diagonal 00 -A t and where

I

e k k=O

V

_kk > 0 and with matrix with A kk = Ak, k E IN u {OJ, where 0 < AO ~ Al ~ •••

for positive constants c and a. Then we have

(1.5) Lemma

Let v > O. Then

Sf

AV 2' Proof

For each t > 0, we define the subspaces R](t) and R

2(t) as follows:

{(f

J.) E f2

I

_jElNu{O}sup

(ILl

_J exp(A~ t»_J < oo}

{(f

J.)

E f

2

I

_jElNu{O}sup

(ILl

_J exp«V .. )_JJ via t» < co}.

The following relations are not hard to prove:

U R] (t) t>O

Let g E R1(t). Then there exists K] > 0 such that

Ig·1 v t) Ig·1 v -v K] ~ exp(A. exp«cA.) t c ) ~ J J J J Ig.1 I v / a - v ~ exp (z(V •• ) t c ) J JJ

for all _J > jo with _jo E IN so large that

(D .• )via :s: JJ v 2(cA.) J I -v I -v Hence g E R 2(ztc ), and R1(t) c R2(ztc ). Conversely, let h E R

I I

h . exp

«

V. . )vIa t):<::;

J JJ

Moreover, there exists JO E 1N such that for all J > jo

V•. ~! CA~

JJ J

So for J > jo we obtain

I I

_{h . exp}

« .. )

V vIa _{t ) ~}

I I

_{h . exp(A.}v₍₂I_c)V _{t ) .}

J JJ J J

v v

Thus we find that h E R]«!c) t) and R

2(t) c R] «!c) t). Now our assertion

is proved by taking intersections or unions.

o

A. = j. Observe that for all v,t > 0,

J

in

l2'

We assume that the entries

K

_mn

In the remaining part of this chapter we consider the diagonal matrix Awith

00 •V t

I

e-J < 00, It implies that the

j=O

and the entireness space

,(l2,A

v) are both nuclear. analyticity space

Sl

A

_V

2'

Let K

=

(K ) be a linear operator mn

satisfy the following conditions:

(I .6) K = 0 for m > n

mn

3

_c

>0 V_m,n: IK_mm - Knnl > C

3_D>0 3_y>O V 0< V : _{IKmnl :<::; Dn}Y ,

m, -m<n n

It is clear that the numbers K , n E 1N u {a}, are eigenvalues of Kwith

nn

eigenvectors u(n), say,

u~n)

= 0 for j > n. If we take u(n)

=

I, then the

J n

(n)

u , m

=

O,l, ••.,n, have to satisfy

m

(K - K )u(n) + K u(n) + ... + K u(n) 0

mm nn m m,m+1 m+] mn n

for m _{O,I, •.. ,n-l. Next, we define the linear operator $ ($ ) by} mn

S mn

if m > n

if 0 :::; m :::; n •

Then algebraically we have the relation KS =

SM,

where

M

denotes the diagon-al matrix

Mk

= ~k' k E IN u {a}. We note that

M

is injective on finite sequences.

In the remaining part of this chapter we take ~ > I fixed. We want to prove that S is a continuous bijection on Sf A~ and on T(f2,A~). To this end,

2' first note that

S mn For t > 0 we put n-m

K

_-

K

_L

\'

K_m,m+k S_m+,nk • mm nn k=l t

Then for 0 the following recurrent relation is valid mn

t n-m t

o K K

I

exp (-t«m+k)f.I-m~))K +k 0

mn mm - nn k=l m,m m+k,n

We take a fixed Nt E IN so large that for all n > Nt

%

ny+1 exp(-(p-l)t nP-1) < 1 where we set p

(1.7) Lemma

min(~,2). It leads to the following result.

~-l

Proof

t

Let n > Nt' Since ann I, the inequality is valid for m = n. Hent~, with 'backward induction' it follows that

exp(tm~-I(n-m» lot

I

~ mn • exp(tm~-I (n-m».exp(-t(m+k)~-I (n-m-k»} $ n-m $

f

nY

I

exp(-nt«m+kl~-l

- mll-I)) $ k=l n-m

~

%

nY

I

exp (-ntk (p-1)(m+k) p-2) $ k=I n-m $

%

nY

I

exp(- (p-I)t n P-Ik) k=I $

f

ny+I exp(-(p-I)t nP-I) < I • I f we put Lt := max

la~nl,

O$n$Nt O$m$Nt Then Lemma (1.7) gives

(1.8) Lemma

n

exp ( - (nll-mll)t)

I

S

I

~ C

t • mn

The same techniques as used above apply in the proof of the following lemma: (1.9) Lemma Let V= (V ) satisfy mn V mn and, also, if m > n if m = n Vt>O 3

B >0: sup

I

exp (- t (nil-mil))Vmn

I

< Bt

t n,mE'lN u{O}

Th_en V _{~s ~nvert~}• ·bl_{e an}dV-1 _satis' f '_~es

(V-I)

mn

o

if m > n ,

and

Proof

Vt>O VE>O: sup

(I

(V-l)mn

_l

exp(-nll(t+E) + milt)) < 00 •

n,m

Let (W ) denote the inverse matrix of (V ) (which exists algebraically!).

mn mn Then we have W mm W mn W mn Now we put

o

i f m > n n

I

Vmt Wn9, if 0 ~ m < n • 9,=m+1 t w mn and t ermn

We want to estimate wt • To this end note first that mn t w mn O:;:;m<n. Hence by assumption

IW~nl

where we may as well suppose that B t > 1. We assert that

Iw

t

I :;:;

2n-m+1(B

t)n-m, 0 :;:; m :;:; n. To show this we use back-mn ward induction: 1 < 2 n 00

I

B~~+m+1 2-~+m:;:;

I

~=m+1 k=l (1.10) Corollary

Let

V

be as in Lemma (l.9). Then

-k 2 1 •

n

v

V • t>O r::>0' and < 00

Proof

Let t > 0 and let E > O. Let N,M E IN. Then we estimate as follows:

(

M~N

2\4

\m

;=0

IV_mn

exp(-(t+E)n~

+

tm~)

I }

~

,

So, by Lemma (1.9), Also, we have M N 1

(

~ I 2\2 1..

I

(V- )mn exp(- (t+E)n]l + tm~)

I )

:;;

m,n=O

<; sup (I(V-I)

exp(-(t+E)n~

+

(t+48)m~)

• mn

n,mElN

Hence, IIIetA~ V-I e-(t+E)A~111 < co

If we apply the previous results to the diagonalizing operator S, we get

(1.11) Theorem

-I

The linear operator

S

and the inverse

S

satisfy

and

o

y V •

and any t >

a.

From the characterization of continuous linear mappings on analyticity

spaces and the entireness spaces (cf. [6c] and [2]) it follows that

(1.12) Corollary

I. The operator

S

is a continuous bijection on S with continuous f.₂,All

-I

inverse

S .

II. The operator

S

is a continuous bijection on T(l2,All ) with continuous inverse

S .

-1

Now we assume in addition that the matrix entries

K

n E IN u {a} of the nn'

matrix (K ) introduced in (1.6), satisfy mn K nn a = cn (1 + 0(1)) -1

In the first part of this section the Hilbert space (f. 2)S

for some positive constant c. If we define the diagonal operator Mby

M_k = Kkk' k E" IN u fa}, then Sf. All = Sf. MlJ/a by Lemma (1.5). Moreover,

2' 2'

the operator

S

is continuous on Sf. Mll/a (T(f. 2

,M

ll/a_{)) and has a continuous} 2'

inverse. We have the relation

K

=

SMS-

1•

has been

intro--1

duced. It follows that K is a positive self-adjoint operator in (f. 2)S which satisfies Kll/a

=

SMll/aS- 1• It leads to the following classification

result.

(1.13) Theorem

II.

Proof

2. THE CLASSIFICATION OF ANALYTICITY SPACES AND ENTIRENESS SPACES GENERATED BY THE JACOBI POLYNOMIALS

The two classification methods for analyticity and entireness spaces dis-cussed in the previous chapter, lead to the classification of the spaces

S and

X Q,(A Q)V

ex,I-' ex,I-'

the Hilbert space

v

T(X Q,(A Q) ) with ex,S> ex,I-' ex,I-'

L

z([-1,1],(1-x)ex(1+x}S dx}

-1 and v ~ ~. Here X Q denotes

ex,I-'

and

A

Q the positive

self-ex,1-' adjoint operator

2 dZ d d

- _{(l-x ) dxZ + (ex+S+Z)x dx - (S-ex) dx •}

It is well known that the Jacobi polynomials p(ex,S). n E IN u {O}, are the

n

eigenfunctions of A Q with eigenvalues n(n+ex+S+1}. (For the definition of

ex,I-' (ex S)

P , we refer to [8], p. Z08.}

n

Instead of p(ex,S) we rather consider the normalized Jacobi polynomials

n

1

{zn + ex + S + 1 f(n+1)f(n+a.+S+1}}2 p(a.,S)_{Zex+S+1 f(n+ex+1)f(n+S+l) n}

which constitute an orthonormal basis in X S. ex,

The normalized Chebyshev polynomials T

R-!'-!

establish a special class

n n

of Jacobi polynomials. They satisfy

(Z. I) T (cos 8)

n

w

cos n8 , n ~ I , 8 E [O,7T] ,

Yr,

2 -!

Lz([-l.l].(l-x) dx) and, also, that

A_I

_1 =

2, 2

d Z

x dx • The eigenvalues of

A

I I are the numbers n ,

-2,-2 We note that X I I - 2 '-2 Z dZ (I-x) - + dxZ n E IN u {a}.

If we consider the transformation x forms into

cos e, then the operator A Q

trans-a, ...

C a,S

1 d d

(S-a) sin e de - (a+S+I) cot e

d8

So, with the aid of (2.1) it follows that the matrix of A S with respect to

a,

the orthonormal basis (T )00

a

satisfies

n n=

o

if n > m (2.2) _{(Aa,S T ,T} )x = n(n+a+S+l) n n I -~ -2, I(Aa,S T ,T )x ~ Dn

H

a

~ 'rn < n n m I I -2,-2

for some positive constant D. So we can apply the theory of the previous chapter. It yields an operator S with the property

a,S

A

=

S A (S )-1

a,S a,S a,S a,S

where

A

S denotes the diagonal operator

A

T = n(n+a+S+l)T •

a, a,S n n

Moreover, for all ~ > 1 the operator Sa,S is a continuous bijection on the as well as on the entireness space

-1

inverse

S

u.

a, ...

We note that we have proved in chapter 1 the following sharper result:

(2.3) < 00

< 00

We find that

(2.4) p(a,S)

n Tn + ( 5 )a,S n-],n Tn-] + ••• + (5a,1JQ)1,n T] + ($a,iJ0)0,n TO

1S an eigenfunction of Aa,S. In order to arrive at the spaces

Sx

(A )v/2

/2

a,S' a,S

and T(X Q,(A Q)V ) we have to normalize the polynomials p(a,S) with

a,1J a,1J n

respect to the norm of X Q. To put it differently, we have to compute the

a,1J constants d(a,S) which satisfy

n

p(a,S)

n

d(a,S)R(a,S)

n n

Following [8], p. 210. the following relations are valid for all n E IN:

T (x) n and n-] 1(2 n n-] 2

V;

x + ••• x + ••• So we obtain T (x) n Therefore 2n-] 2 n-] x + ••• • p(a,S) n with T + ( $ ) T + ••• n a,S n-],n n-] 2n-] 2 22n-] I

With the aid of Lemma (a.II) and the result (a.13.ii) of the Appendix it follows that there are constants q Q and p Q > 0 such that

a,fJ a,fJ -I q Q(n+l) a,fJ and

Id

(0.,S)

I

~

p S (n+ I) • n 0.,

So the diagonal operator V

S

with entries dCa,S) and its inverse V-I

0., n a,S

satisfy the following

(2.5) V

E>O·

IIIV

a , S exp (-E:(A- 2 ' - 2I ,)]J./2)III < 00

V_E>O·

_II'V-

IS exp (- E:(A I I ) ]J. /2)

III

< 00

0., -2,-2

for each ]J. > I.

So if we put

W

S

V

Q' then

W

Q is a continuous bijection both on

a,S a,S a,fJ a,fJ

-I

S

A

and on T(X I

"A

I I ) with continuous inverse W Q.

X 1 I I I -2,-2 -2'-2 a,fJ

- 2 ' - 2 ' -2·-2

Further we obtain from (2.3) and (2.5):

(2.6) Lemma

with 11 > I.

Since W T = R(a,S) n E IN u {OJ, and since

A

a,S n n ' a,S

obtGin from Theorem (1.3) and Theorem 1.4)

W A W-I we

(Z.7) Theorem

Let ~ > 1. Then for all a,S> -I

and

s

X , (A )~/Z a,S a,S

- s

/

- Z _1 ( Z dZ d

)~

2

L

_Z( [-1 , 1] , (l-x ) 2 dx), - (l-x )

-Z

+x dx dx .

If we apply the transformation x = cos 8, we get

(Z.8) Corollary

Let ~ > I. Then for all a,S> -I

and

s

y (C )~/Z a,S' a,S ~/Z ,(y o,(C 0) ) a,I-' a,I-'

s

( dZ

)f1/Z

L Z([O,rr],d8), - - Z d8 '

(

(

dZ)~/Z)

, LZ([O,rr] ,d8), - - Z . de .

Here Ya,S denotes the Hilbert space Lz([O,TI],(l - cos 8)a(1 + sin 8)S de).

The classification method, based on the use of continuous bijections is not refined enough to obtain the classification of the spaces

S

I and

X 0' (A 0)2

I a,I-' a,I-'

,(X o,(A 0)2). For this case we use the other classification method, based a,I-' a,I-'

on perturbations. With Theorems (1.1) and (l.Z) we will obtain the results of Theorem Z.7 for ~

=

1. We note that the spaces

T

_x

(A

)4

and

!

a,S' a,B

o(x_a,1-'o,(A_a,B) ) will be related to the spaces of (ultra-) hyperfunctions in the Chapters 6 and 7.

In the Appendix we have snown that the matrix elements of the operators

d

(2.9) n3 / 2(n_k) ! (k+I)2 if k ~ n ifO~k<n

where G Q is some positive constant;

a,I-'

a

if k > n (2.10)

_I

«XV)R(a,S) R.(a,S))

I

~ n n '-1c a.,S if k

=

n (n)3/2 H Q 1 a,1-' (k+I)2 if

a

~ k < n

for some positive constant H Q.

a,I-'

For each a,S> -I we put

A

Q

=

A Q + 1, and further

a,I-' a,I-'

pY,O {(o-y) - (S-a)}V - {(o+y) - (S+a)}(xV) • a,S

Then we have the relation

A

A

+

pY,o

y,o a,S a,S

With the aid of the estimates (2.9) and (2.10) we prove the following auxiliary results.

(2. II) Lemma

Let a,S> -I and let

!

~ v ~ I. Then there exists a positive constant e Q

a,1-' such that for all r >

a

~ v ~ -I

II exp (r (A Q) )V(A Q)

a,I-' a,I-'

~ v

exp(-r.{A Q}}II Q

a.,I-' a,I-'

-3/2

~ e r .

a,S Proof

o

i f k :f:. t+n, t E 1N u {O}

v v

• exp -r(«nH)(nH+IX+S+))+ I) - (R, (t+IX+8+ 1)+ I) )

First we shall prove that W(IX,S)(r) is a bounded linear operator on

X

Q for

n IX,~

all r > O. Therefore, note first that

«n+R,)(n+R,+M8+1 )+1)v - (t(R,+a+8+1 )+1)v

v-I = v[«n+t) (n+i+IX+8+1)+I) - (i(t+IX+8+1)+I)]~(n,R,)

for some number ~(n,t) E [R,(t+a+8+1)+I,(n+i)(n+i+a+8+1)+I]. We get the following estimation

v-I ~ vn(n+i+a+S+I) «n+i) (n+i+a+S+I)+I) ~ vn In addition, for all n E IN we have by the estimate (2.9)

{

(nH) 3/2 n

sup G

iEINU{O} IX,S (R,+I)!

I

(n+ t) (n+ R,+a+S+ I)

(IX S) 1 -vrn

So the norm of the operator

W '

(r) is smaller than G Q n2 e for

n n,~

each n E ill. Hence the series

I

IIW(n,S) (r)1I converges. Thus we obtain the

n=1 n following estimation

00

<; G I n ! exp (-vrn)

a,S n=I

for some positive constant e Q.

a,I-'

-3/2 <; e r

a,S

o

Compairing the bounds for the matrix elements of the operator xV and of the operator

V,

the reader will see that the same method as used in the proof of Lemma (2.11) applies in the proof of the following result.

(2.12) Lemma

Let a,S> -I and let! <; v <; I. Then there exists a positive constant f Q

a,I-' such that for all r > 0

-I

:<; f r + I .

a,S

Remark. From the preceding lemmas it follows that

V

and xV are well-defined continuous linear mappings on the analyticity spaces

S

~A v and on

X Q' ( Q)

~ V a,I-' a,I-'

the entireness spaces ,(X Q,(A Q) ) with! <; v <; I and a,S> -I.

a,1-' a,I-'

Remark. The Lemmas (2.11) and (2.12) also hold for v > I. However, for simplicity in the estimation we have given the proof in case of ! :<; v :<; only.

We want to apply Theorem (1.1) and Theorem (1.2) to the perturbations

Py,o.

_a,S

Therefore we need the following lemma.

(2.13) Lemma

Let

a,S,y,o

> -1 and let v > O. Then for every t > 0 the linear operator

~ v

exp(-t(A_a,Q) ) maps X .Q continuously into X ~.

Proof

By [8], p. 217, there exist q > 0 and c > 0 such that

max

-I$;x$;1

So with the aid of (a.13.i) we derive

Now let t E IN be so large that t > q+3j2. Then for each f E X Q

a,I-'

1

J

~ - t 2 y 0

I«Aa,S) f)(x) 1 (I-x) (I+x) dx$;

-I 1 $;

J

(l-x)[y]+I(I+x)[o]+I!«A )-t f)(x)\2(I_x)-I+Y-[Y](I+x)-I+o-[o] dx $; a,S -I 1

$; 2[y]+[o]+2 max (1«A

a S)-t f)(x)1 2)

f

(l_x)-I+y-[y](I+x)-I+o-[6] dx $; -1$;x$;1 '

-I

$; h max

I

«A Q)t f)(x)12 y,o -1$;x$;1 a,I-'

for some constant h > 0 which does not depend on f E X Q.

y,o a,I-'

Consider the following estimation

max -1$;x$;1 I «A Q)-t f) (x) 1 2 a,I-' max -1$;x$;1

I

I

(f R(a,S)) ( 1 )t R(a,s)(X)12 $; , n a,S n(n+a+S+I) + 1 n n=O

$;

(~

I (f Ra,S) If I )t max IRn(a,S) (x) 1)2 $; n__LO ' n a,S \n(n+a+S+I) + 1 -)$;x$;1

:0; Ilf

II~,

S 00 ( 2!L

I

I I ) ( I +n) 2q+ I n (n+a+ s+I) + n=O := k IIfl12 Q

y,o a.,I-'

Thus we have found that there exists a constant oy,o > 0 independent on the a.,S

choice of f such that

:0; cry,

°

Ilfll Q '

a,S a.,I-'

~ -£

SO (A Q) maps X Q continuously into X

a,I-' a,I-' y,o

~ 'J

Finally we note that exp(-t(A Q) ) can be written as

a,I-'

~ 'J exp(-t(A Q»

a.,I-'

Since the operator between {.} maps X Q continuously into itself the proof

a.,I-' is complete.

(2.14) Theorem

Let a.,S,y,o > -I and let ~ :0; 'J :0; I. Then

n

and, also

Proof

Following Lemma (I.S) it is sufficient to prove that

Sx

(A )'J a,S' a,S

=

Sx

(A )'J and that T(X S,(A S)'J) = T(X o,(A o)'J)· By the Lemmas

Y,o' y,o a, a, y, y,

(2.1), (2.12) and (2.13) the following assertions are valid:

The perturbation Py,o =«o-y) - (S-a»D - «o+y) - (S+a.)xD satisfies a,S

D(PY'o) _:>

_Sx

~ v and

A

A

+ pY,o is a positive

self-a,S _{a S,(Aa}

_,

_,

(3) y,o a, 13 a,S

adjoint operator in X 0'

Y,

~ v

X Scontinuously in X

For all t > 0, exp(-t(A (3)) maps _Y,o'

a, a,

Then by Theorem (1.1),

Sx

(A

)v

a,S' a,S

~ v ~ v

T(Xa,S,(Aa,s) ) C T(Xy,o,(Ay,o) ). If we interchange the roles of (a,S) tained.

Application of the transformation x

(2.15) Corollary

C

Sx

(A

)V, and by Theorem (1.2), y,o' y,o

and (y,o), the wanted result is

ob-o

cos 6 yields

Let a,S,y,o > -I and let 4 ~ v ~ I. Then

and

v T(Y o,(C ~))

y, y,u

For a = 13 = -4, we have seen that

R~4,-4(cos

6) =

~

and R: 4,-4(cos 6) =

=

W

cos n6, n E: IN. So it becomes rather easy to characterize the spaces

S . and

Y, 1,(C _I .)V

-2,-2 -2,-2

Theorem (2.7) and Theorem (2.14) yield the other values of a,S> -1.

same analytic description for the

Further, by means of the conformal mapping w

=

cos z, we get the

characteri-zation of the spaces

S

and

X_a,S,(A_a,S)V be carried out in the next chapter.

T(X (3,(A S)v). This program will

3. THE CHARACTERIZATION OF THE ANALYTICITY SPACES AND ENTIRENESS

SPACES BASED ON THE JACOBI POLYNOMIALS, WITH APPLICATIONS TO CLASSICAL ANALYSIS

In this chapter we intend to derive characterizations in classical analytic

terms of the elements in the analyticity spaces Sy (C )v and in the a,S' a,S

entireness spaces T(Y S,(C S)v) where we consider a,S> -I and v ~

!.

a, a,

Following the results obtained in the previous chapter we only have to

consider one pair Y

Q,C

.0. It is rather natural therefore to study the

a,I-' a,I-'

case a = 13

=

-2,I ~.e.,• the so-called Chebyshev case.

For convenience we put Y= Y_!,_! = LZ([O,TI]) and

C

=

C_!,_!

The normalized eigenfunctions c of the operator

C

given by

n c (8) n

'fli

cos n8 ,

\ I

.!.

V

'TT ' 8 E [O,TI] , n E 1N,

establish an orthonormal basis ~n Y. We want to determine the analytic behaviour of the series

00 Z t+

L

n=O a c (z) n n 00

for all coefficients (an)n=O satisfying the following order estimate

O(exp(-tnZv) , n E 1N u {a}, v ~

!

Since cos nz

z it is necessary and sufficient to characterize

the analytic behaviour of the type of series

00 n=-oo b n ~nz e where b 11

O

Zv

consists of all even functions in S

L

Z( [-'IT, 'IT]) ,~v

entireness space T(Y,CV) consists of all even functions in T(L

Z([-'lT,n]),6 v_). 1 ina

We observe that the functions e : 8 t+ - - e ,

e

E [-'IT, 'IT], nEll, establish

n

If.IT

an orthonormal basis for the Hilbert space L₂([-'lT,'IT]). They are the

eigen-2

functions of the positive self-adjoint operator

~

= -

~

in LZ([-'lT,n])

Z

de

with eigenvalues n , n E~. It is clear that the analyticity space Sy

_,

c

_v and, similarly, that the

The program of this chapter is the following. In a separate section we characterize

These spaces

the elements of

SLz([-'lT,'IT]),~V

_{and T(L 2 ([-n,n]),6}v), v

~ ~.

correspond to classes of Z'IT-periodic, analytic functions which mostly satisfy certain growth conditions. In a rather natural way we define norms in these classes of analytic functions. This leads to an alternative description of sequential convergence in the corresponding analyticity spaces and the corresponding entireness spaces.

As we have seen all results carryover to the spaces

Sy,c

_{v and T(Y,C}V), v ~ ~. We thus obtain vector spaces of even, 2n-periodic, analytic functions with their natural norms.

We next employ the conformal mapping w = cos z to the function classes v

associated to Sy

_,

CV

and T(Y,C ). The thus obtained classes of analytic functions lead to a characterization of the spaces Sx (A )V and

v a.,S' a.,S

T(X Q,(A Q» for v ~ ~ and a.,S > -1. Moreover we derive a classical

a.,1J a.,1J

description of sequential convergence in these spaces. As a result we are able to connect the growth behaviour of analytic (entire) functions with the growth of their expansion coefficients with respect to bases of Jacobi

polynomials (R(a.,S».

3.1. The characterization of SL [ ] v and T(L

2([-n,n]),6v)

2 ( -n, n ),6

First we discuss the case v

(3. 1• I) Lemma

~. To this end, consider the following lemma:

Let (a)oo E

~

Z. Then for the expression n n=-oo 00 f(z) -

L

n=-oo 1nz a e n Proof 00 I ~) f(x+iy)

I

a_n n=-oo {x+iy Iyl ::; T < t}. For y

the following assertions are valid.

I. f is an analytic function on an open strip 11m zl < t, t > 0, and it

tends to boundary functions x ~ f(x±it) in L

2([-n,n])-sense iff (an et In

I

):=-00 E 12'

II. f is an analytic function on an open strip 11m zl < to' to > 0, iff for

t1nl)00 f)

all t with 0 < t < to the sequence (an e n=-oo E ~2 (or equivalently,

l ).₀₀

e-ny einx converges un1formly on a. 1 1 'str1ps

t t (or y ~ -t) the sequence (an e-ny):=_oo tends

• f) h (a e-nt)oo (or (a ent)oo ).

1n ~2-sense to t e sequence n n=-oo n n=-oo

I ~) Both (a_n ent)oo_{n=-oo an}d (_{an e}-nt)oo_n=-oo h_{ave to} b_e l_2-sequences. Hence (an e1n1t):=_00 is an 12-sequence.

l~) Similar to I ~).

For all y E (-t,t) we have

II) ~) (a n open. E 12 and hence n

f

/f(x+iy)12 dx < 00. Therefore -n

(a e 1n1Y) E l since the interval (-t,t) is

n 00

We introduce the following notation: St denotes the symmetric open strip of width Zt around the real axis, i.e.

St

= {

Z E II:

I

I

1m Z

I

< t} .

(3.].2) Definition

Let t > O. Then the class A(St,27f-per) consists of 27f-periodic functions for which there exists £ > 0 such that f is analytic on St+£.

The class A(~,27f-per) consists of all 27f-periodic entire analytic functions.

(3.1.3) Theorem

Let t > O. Then f E L₂([-7f,7f]} can be extended to f E A(St,27f-per) iff

!

f E exp(-t~2)(SL2([_7f,7f]),~~). Proof

Immediately from Lemma 3.1.1.

We note that

o

!

U exp (- T~2) (L 2 ( [ -7f , 7f]) ) T>t

We obtain the following characterization:

(3.14) Theorem

1. iff there exists t > 0 such that f E A(St,27f-per).

Next we shall show that the 'functional analytic' topologies of the spaces

I

SL2([-7f,7f]),~~ and T(L2([-7f,7f]),~2) are equivalent to topologies which are related to the analytic properties of the functions in these spaces.

denote the Hilbert space t6!

t6~

(e f,e g)L where ( , )L • 2

(D

2 the norm 1n L₂

_,

t([-rr,rr]) product (f,g)t 1 = , 2 L₂([-rr,rr]). We denote denotes the inner product of

To this end, let

L~ ~ ~

([-rr, rr]), t > 0,

I '

-tl:l2

e (L₂([-rr,rr])) with inner

by II'lit 1

, 2

In the space A(St,2rr-per) we define the following norm

sup If(z)

I ,

11m z

I

~t

Note that with the usual identifications

U A(St,2rr-per) t>O and

n

A(St,2rr-per) t>O as sets. ( I ) Now let f E L 2 2

t([-rr,rr]) for some fixed t > O. From the characterization 1n

,

Lemma (1.3.1) it follows that f E A(S ,2rr-per) for all T, 0 < T < t. Let T, T

with 0 < T < t be fixed, and consider the following estimation

p (f) T sup 11m _ZI~T 00

I

L

n=-oo (f,e )e (z) I ~ n n e21nlt (f,e ) n

Thus we obtain with a

=

f!rr

-rr (I - e-2(t-T) _1) 2 the inequality t,T

p (f) ~ at IIf lit I

Conversely, let f c A(St,2n-per)

(1)

have f E L22 ([-n,n]). Write f =

, t

for some t > O. Then by Lemma (1.3.1) we

00 a e . Then a is given by n n n n=-oo a n n = _1

f

ffiT

-n f(x)e-inx dx , n E: 7l •

We have the following relations

a n n

=

_1 Jf

/fiT

-n f(x)e-inx dx f (xt~t. )e-~n x_~t. ( +' )dx tnt e -n 1T

J

f(xtit)e-inx dx • So for all n E IN

Thus we obtain for all T,

a

< T < t,

(**) IIfll I T,2

= 2n at P (f) •

,T t

The inequalities (*) and (**) yield the following result.

(3. 1.5) Theorem

1.

1

-t112

Let t > O. Le; (fn)nEIN be a sequence in e (SL

2([-n,n]),I1!)' The sequence (etI12_{f n )nEIN is} a null sequence in SL

2([-n,1T]),I1! iff there exists £ >

a

such that pt+£(f_n) + 0, i.e., fn(z) +

a

uniformly on

I

II. The sequence (g) 'IN is a null sequence in T(L2([-n,n]),~2) iff for all nnE.

t > 0, pt(gn) ~ 0, i.e., gn(z) ~

°

uniformly on each strip 11m zl ~ t.

Remark. The sequence (f) IN is a null sequence in

S

! iff

n nE L2([-n,n]),~2

(fn ) c A(St,2n-per) for some t >

°

and pt(f_n) ~ 0, i.e., fn(z) ~ 0 uniformly on 11m

zl

~ t.

In the remaining part of this section we investigate the case v >

!.

The next two lemmas are of main importance.

(3. 1 •6) Lemma

00 Z

Let (a )_{n n=-oo} E ~ . Then for the expression

f (z) 00 n=-oo inz a e n z = x+iy E (C ,

the following assertions are valid:

I. Let v ~ I. Then f is a 2n-periodic analytic function of growth behaviour

-n iff n

f

If(x+iy) 1 2 dx 2v

~

A2 exp (2Bly!2V-I) , A ~ 0 , _{B >}

°

for t ( 1) 2V-I (2v)-2 v\2v ; II. Let v >

!.

If (a n 2v 00 exp(tInl

»

_n=-oo

then f is a 2n-periodic function of growth behaviour (*). Moreover, if f satisfies the growth condition (*), then the sequence (an):=-oo satisfies

Proof

I .) Consider the following estimation

2v A2

~

exp (- 2BIy ,2v-1 ) 'IT

r

If(x+iy) 1 2 dx J -rr 2v = _1 exp (- 2BIy12v-] \

r

?'1T ) n=-OO 2v ] _{ooI ({ -2v-1} -2V(2V - 1)2V-I 2V} 2 1 1 - 2ny - 2(2v) B i nI •

=

2TI'

n=-oo exp - B Y

If we now take

( )

2V-]

2v - 1 2v-1

Yk = - 2vB. sgn(k)lkl , k Ell,

then in the k-th term the factor between curly brackets equals I. Hence

(

) 2V-I for all k E ll. This proves that for t = (2v) -2v\ 2v ; 1

2v 00

(a exp(tlnl» belongs to f .

n n=-oo 00

the sequence

I ~) The analyticity of f is trivial. We estimate as follows:

-'IT 'IT

f

1f(x+iy) 12 dx = _I

r

2'IT

I

~ -2_1f lIa_n

The proof is complete if we can show that the series between {.} is uniformly bounded for all y E JR. Then just take

I

(

2V)-2V\2V-I

B (2v-l) J

t .'

In order to show this uniform boundedness, consider the function 2v x ~ exp(-2tlxl - 2xy) , X E JR, at the 2v 2t (2v-1 )(lxl)2V-I exp 2vt

y. This function attains its maX1mum

I

I

y 12v-1

- 2vtl sign(y). Left to this point the function increases, point x =

for fixed

right to this point it decreases. Then we get the estimate

00 2v

I

e-2tlnl -2ny n=-oo 2v

~

exp

(2t(2V_I)(lxl)2V~I)

+ 2 \ \2vt 00

f

2v -2tlxl +2xy e dx . - 0 0

Finally, standard asymptotic techniques yield

00 n=-oo 2v -2tlnl -2ny e I - v 2v

~

C(I + lyl)2v-1 exp (2t(2V_I)fJ.xl)2v

-l\

\2vt /

So for v ~ I the series {.} is uniformly bounded in y.

II. In order to prove the first part of the statement we estimate as follows: