Approximation spaces

JOURNAL

OF APPROXIMATION

THEORY

32, 115-134 (198 1)

Approximation

Spaces

ALBRECHT PIETSCH Sektion Mathematik, Friedrich Schiller Universit&t, 69 Jena, German Democratic Republic Communicared by P. L. Butrer Received October

15. 1979

Using the concept of an approximation space we describe certain analogies between spaces of sequences, functions and operators. In order to illustrate the power of this method, some old and new theorems about distributions of Fourier coefficients and eigenvalues are established.

The aim of this paper is to describe some analogies between well-known spaces of sequences, functions and operators which have for the first time been observed by J. Peetre. For this purpose we develop a theory of so-called approximation spaces. Similar concepts were already investigated by P. L. Butzer and K. Scherer, J. A. Brudnij and N. J. Krugljak, as well as by J. Peetre and G. Sparr within the framework of their interpolation theory of abelian groups. For the convenience of the reader and in order to be selfcontained, we present some of their basic theorems with new and very simple proofs, and apply them to the main examples. From our point of view, the most interesting results are to be found in the last chapter, where we treat some significant applications. First, we investigate the influence of smoothness properties of periodic functions on the behaviour of their Fourier coefficients. This is a special case of a more general problem. If we remember that the complex Fourier coefficients coincide with the eigenvalues of the corresponding convolution operators, it suggests itself to ask, how do the distributions of eigenvalues of integral operators depend on certain properties of the generating kernels? Results can be obtained via a generalization of Weyl’s theorem which we prove by the method of related operators. The original proof (unpublished) of this remarkable theorem is due to B. Carl and H. Konig.

115 0021-9045/g l/O601 15-20$02.00/O 640/32/2-Z

Copyright C 1981 by Academic Press. Inc All rights of reproduction m any farm reserved.

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ALBRECHT

PIETSCH

1. DEFINITIONS AND ELEMENTARY PROPERTIES

In this chapter we introduce the concept of an approximation space and formulate some elementary properties. 1.1. Quasi-Banach Spaces A quasi-norm is a non-negative function (1./Ix defined on a (real or complex) linear space X for which the following conditions are satisfied: (1) If jlfllx = 0 for somefE X, then f= o. (2) llJJllx = (A) /fllx for fe X and all scalars A. (3) There exists a constant c, 2 1 such that

Ilf+ gllx G cxwllx + IIgllxl

for f,g EX.

The quasi-norms II.II:” and (1.IIF’ are said to be equivalent if

IlfllY’ G Cl llfll~’

and

IlfllY G c2 IlflP

for all fE X,

where ci and c2 are suitable constants. Every quasi-norm generates a metrisable Hausdorff topology on the underlying linear space. For two equivalent quasi-norms the corresponding topologies coincide. A quasi-Banach space is a linear space X equipped with a quasi-norm 11.[Ix such that every Cauchy sequenceis convergent. A quasi-norm )I. IIx is called a p-norm (0

IIf+ gllf: G IlfK + IIglli

for f,g E X.

Then condition (3) is satisfied with cx := 21’p-‘. Conversely, for every quasinorm 11.IIx there exists an equivalent p-norm 1).IIt, if we determine p by l/p := 1 + log* c,. Clearly, every p-norm is also a q-norm for 0 < q < p < 1. Let X and Y be quasi-Banach spaces. Then f!(X, Y) denotes the linear space of all (bounded linear) operators T acting from X into Y. Putting

IITll, := SUPUI Tflly: Ilfll, < 11 we get a quasi-norm on f?(X, Y). 1.2. Approximation

Schemes

An approximation scheme (X, A,,) is a quasi-Banach space X together with a sequence of subsets A,, such that the following conditions are satisfied:

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APPROXIMATION SPACES

(1) A, GA, G *** GX. (2) AA, c A, for all scalars A and n = 1,2,... . (3) A,+A,cA,+, for m,n= 1,2 ,... . We put A ,, := {o). Obviously, A := lJ,“=, A, is a linear subset. 1.3. Approximation Numbers Let (X, A,) be an approximation scheme.ForSE X and IZ= 1,2,... the nth approximation number is defined by a,(J;X) := inf{ljS-- all,: a E A,-,}. Now some elementary properties are listed: (1) (2)

llfllx= a,(f,X) > aAAf;x> > -a. >O f0r-E X. a,,(A. X) = 111a,,(f, X) for fE X, all scalars 1, and n = 1, 2,... , for f,gEX

am+n-l (f+g,~~c*[a,(f,X)+a,(g,X)l

(3)

and

m, n = 1, 2,... .

It is worthwhile mentioning that a,(& X) is in general not a continuous function of f: This unpleasant situation can be avoided if we use an equivalent p-norm on X.1.4. Approximation

Spaces

Let 0 < p < 00 and 0 < u < co. Then the approximation space X;, or more consists of all elements fE X such that precisely (X, A,,);, (F”“a,(f; X)) E I,,, where n = 1, 2,... . We put

llfllxe := IIW l/UUI.LX)Il,u

for fE X;.

The proof of the following result is straightforward. PROPOSITION

1. Xi is a quasi-Banach space.

If X is a Banach space and we have A,, + A, = A,, for n = I, 2,..., then XE with 1 < u < co is a Banach space, as well. This case was treated in detail by Butzer and Scherer [6]. Remark.

Now we formulate a very useful criterion. PROPOSITION 2.

(2k”a&

An element f E X belongs X)) E I,, where k = 0, l,... . Moreover,

to X;

Ilf II7 := ll(2kP~zr(f; x>>lL, defines an equivalent quasi-norm on XE.

if and only

if

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ALBRECHT PIETSCH

Next we state that the scale of approximation ordered. PROPOSITION 3.

spaces is lexicographically

There holds

X”I 3xp2 UI-- 4

for

O
xc, 52 x;,

for

O
and

Finally, we mention a deep result of Peetre and Sparr [ 141, stating that the approximation spaces form a real interpolation scale. See also [6, Korollar 2.3.11. PROPOSITION 4.

Let 0 < 8 < 1 and p = (1 - 0) p0 + @, , where POf PI.

Then

2. EXAMPLES In this chapter we introduce some important examples of approximation spaces. 2.1. Sequence Spaces As is usual, 1, with 0

n. It follows easily that (lp,f,) and (I,, 0,) are approximation schemes. We Put sp p,u :=
and

b;,, := (1,) 0,);.

Later on we shall show that s,“,, coincides with the well-known Lorentz sequence space I,,, , where l/r =p + l/p. The sequence spaces bg,, can be considered as the discrete counterpart of the Besov function spaces which will be defined in the next section; cf. also [ 191. 2.2. Function Spaces As is usual, L, with 0

APPROXIMATION

119

SPACES

Let F, denote the set of all trigonometrical polynomials a(s) = a, + 7

El

[a,, sin 2nms + asm+ 1cos 2nms ]

possessing at most n coefficients (x, # 0. We also consider the subset 0, which consists of all trigonometrical polynomials such that a,,, = 0 if m > n. It easily turns out that (Lp, F‘,) and (LP, 0,) are approximation schemes.We Put

SLI := (L,, F,)P,

and

Bp”u := (LP, 0,);.

A detailed theory of S;,, spaceshas not been developed until now. However, there is an important result of SteEkin [20] which states that Si!: consists precisely of those periodic functions having an absolutely convergent Fourier series. On the other hand, it can be seenfrom approximation theory that B;,U are the famous Besov function spaces; cf. 12, 3, 6, 13, 2 11. Note. In order to simplify our notation we will not indicate by an additional symbol that all spaces under consideration consist of periodic functions. The above definitions extend to the case of X-valued functions, where X is a quasi-Banach space. Then we denote by [L,, X] the quasi-Banach spaceof all measurable and absolutely p-integrable X-valued functions f defined on the unit interval; cf. [22]. If the sets [F,, X] and [O,, X] are introduced canonically, we obtain the approximation spaces ls;..Jl

:= ([L,Jl>

lI;,,~l)p,?

[B;,,,X]

:= ([L,J],

[O,,X])::.

and

2.3. Operator Spaces

Let f!(E,F) denote the Banach space of all (bounded linear) operators acting from the Banach space E into the Banach space F. An operator S E P(E, F) is called absolutely p-summing (0

0 such that (2, llsxill~)“pCOsuP

] ($,

ICxiy

alp)

I”:

for all finite systems of elements x, ,..., x, E E. We put (1SII,,p =: inf 0.

IlallEc

<

11

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ALBRECHT PIETSCH

The quasi-Banach space of all absolutely p-summing operators from E into F will be denoted by ‘$,(E, F). It is convenient to define ‘Q,(E, F) := Q(E, F), and ]]S]] Y’m:= ]]S](,. Let s,(E, F) be the set of all operators A E 2(E, F) having an image M(A) := (Ax: x E E} which is at most n-dimensional. It easily turns out that (‘Q,E FL S,(E,F)) is an approximation scheme. We put

13$‘,,E F> := (‘J$,(E,F), S,(E, F)):. In the special casep = co we also use the notation P;(E, F).

3. THEOREMS

In the sequel we prove some fundamental theorems. 3.1. Representation Theorem

First of all we establish the famous Representation Theorem which, in the case of function spaces, goes back to Besov [3] and Amanov [ 11. The analogous result for operator spaces is due to Pietsch [ 16, 171. We also mention that a very general theorem of this type was recently proved by Brudnij and Krugljak [4,5]. See also the Equivalence Theorem in [6]. REPRESENTATION THEOREM. Let (X, A,) be an approximation scheme. Then f E X belongs to Xt if and only if there exist ak E A 2ksuch that

f=

2 k=O

ak

and

(2kp 11ak

IiX>

E

ha

Moreover,

where the infimum is taken over all possible representations, dejines an equivalent quasi-norm on Xi. Necessity.

Let f E X;. Then we choose ai E Azk-, such that

IV- 4 /lx< 2d.L XI. Put a, := o, a, := o, and ak+z := a,*+, - at for k = 0, l,... . Obviously we have ak E Azk, and m

f=lipaz=

2: ak. k=O

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APPROXIMATION SPACES

Moreover. it follows from

I/ak+*IIXG cxlllf- a,*+1IIX+ IV- 4 II*1G 4CX%(f;

a

that (2kp Ila,llx) E I,, and we obtain the estimate

Ilf IILy< 2zp+*cxIlflliy * Suflciency. Without loss of generality we may suppose that /I. [IXis a pnorm with 0

In the case 0 < u < co we put q := u/p, and choose (x such that pp > a > 0. Then

<


i.

zhpU

F

( kgh

2h(pu-as)

2-k”q’)dq’(

,“-

kzh

2knq

IbkiI:)

2kw Il”kili

he0

keh


2h(pu--aq)

h=l


Therefore we have fE X; and Ilfllr;zP< c Ilf/ly[. treated analogously.

The case u = co can be

As an immediate consequenceof the Representation Theorem we get PROPOSITION

5. If 0 < u < co, then the linear subset A is dense in X;.

Proof: It easily turns out that every seriesf = CpCo uk with uk E Azk and (2kp 1)uk [lx) E I, is evtu convergent relative to the quasi-norm I(I IIXp. An approximation scheme (X, A,) is called linear if therUeexists a uniformly bounded sequence of linear projections P, mapping X onto A,.

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ALBRECHT PIETSCH

Then it follows that

Ilf- P,- ,fL < ca,(f,Xl for allfE X and n = 1, 2,..., where c := cx[ 1 + sup ]]P,]]J. Remark. In this case Butzer and Scherer [6] speak of an idempotent approximation procedure.

With the help of the projections Qk := Pzkit-, - Pzk-, we can formulate the LINEAR REPRESENTATION THEOREM. Let (X,A,,) be a linear approximation scheme. Then f E X belongs to XE if and only if

GkpIIQ/JIM E 6,. In this case we have k=O Moreover,

Ilfll!$ := 1/(2kp11 Qkfllx)ll~, is an equivalent quasi-norm on XE. EXAMPLE 1. The approximation scheme (I,, 0,) is linear, since the sequenceof the canonical projections has the required property. EXAMPLE 2. The approximation scheme (LP, 0,) with 1

3.2. Reiteration

< 00 is required 03. For from L,

Theorem

For every approximation scheme (X, A,) we have A, G Xz. Therefore it follows that (X;, A,) is an approximation scheme, as well. This means that we can reiterate the construction of approximation spaces. LEMMA

1.

There exists a constant c > 0 such that

npazn - ,(f, X> < 9J.L

X)

for all

f E XE and n = 1, 2,... .

APPROXIMATION

123

SPACES

We choose a,, a2 E An-, satisfying the estimates

Proof:

llf-

a, Ilx; ,< 2Q.L$3 and IV- a, - a,ll, <

2a,df-a,, x).

Then %n-l(f, x> G Ilf-

(a, + %Ilx < 2a,(f-

u,, x) <

2n-O IIf-

u, llxc

< COnepIlf- a, Ilxu”< 2conePa,(f, Pu). LEMMA

2.

There exists a constant c > 0 such that for all

II4po4 Proof

a E A, and n = 1, 2,... .

We consider the case where 0 < u < co. Then I/U

( ?’ [kPmVuak(a,X)]”

ll4:,=

k%I

G (g, ~pUl)liYll~llXCC~Dll~llX. Remark. Within the framework of [6, Definition und Lemma 2.3.11 the results of Lemmas 1 and 2 are said to be a Jackson inequality and a Bernstein inequality, respectively.

Now we are in a position to establish the basic theorem of this section. Similar results are well known within the context of interpolation spaces,cf. [2, 3.5.31, and [21, 1.10.21. See also [6, Korollar 2.3.21. REITERATION

(x;);

THEOREM.

Let (X, A,,) be an approximation

scheme. Then

= x; to.

Proof: If fE (X;);, we have (2k”a,kdf, X;)) E 1,. Lemma 1 yields k(Pi-o)a2k(f; X)) E I,. H encef E X; +O.This proves that (X;); c X; ’ O.Now (2 let fE X;+O, and consider a representation f = CpZOuk such that ak E Azk and (2k(pf”) ]]akljx) E I,. Then we obtain from Lemma 2 that (2ko /]a,ll,$ E

I,. Consequently fE (Xg): . So we have XEt UG (Xt):. EXAMPLE 1. Since (a,@, I,)) is the non-increasing rearrangement of xE&, we get s$,, = I,,, with l/r = p, where l,,, is the well-known Lorentz sequencespace. Now the Reiteration Theorem yields

Sk4= (SZJ::= szp = l,,,

with

l/r =p +

l/p.

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ALBRECHT PIETSCH

3.3. Transformation

Theorem

The next result is very simple. However, it often happens in mathematics that trivial theorems can be extremely powerful. TRANSFORMATION THEOREM. Let (X, A,) and (Y, B,) be approximation schemes. Let T E 2(X, Y). If there are constants A > 0 and c > 0 such that

WA

whenever

c B,

n > cm’,

then T(Xtp) c c. Proof:

We consider the case 12 1 and put

N,:=(n:c(m-l)*+l
for

m = 1, 2,... .

Then card(N,) < c, rn’-‘, ?fJ- l/U


UP - l/U)

for

nEN,,

and

Therefore, if 0 < u < co, we obtain 11~fll~~=

( g1 F [n’-‘“a,(Tf,

m

c,m*-‘[

QY)“”

c2m’(p-y”) 11 Tll,.a,(f,

X)1”) ” < CIlfllxe.

This proves the assertion for A > 1 and 0 < u < co. The remaining casescan be treated similarly. Many applications of the Transformation Theorem will be given in the next chapter. 3.4. Embedding Theorem

In the sequel we consider a couple of quasi-Banach spacesX and Y which are continuously embedded into some linear topological Hausdorff space. Furthermore, let (X, A,) and (Y, A,) be approximation schemes built from the same sequenceof subsets. LEMMA 3. Let 0 < p < o < co and 0 < u < co. Suppose that Y can be vnormed with 0 < v < 1. Then the following conditions are equivalent:

APPROXIMATION

(1)

for all a E A,, and n = 1,2,... .

con0 IMIx

There exists a constant c,., > 0 such that

II4 G cp.uno-pl141xf (3)

125

There exists a constant c, > 0 such that Il4~

(2)

SPACES

for all a E A,, and n = 1, 2,... .

There exists a constant c~,~ > 0 such that

I141y~~,,,I141x~ forallaEA. ProoJ The implication (2) + (1) follows immediately from Lemma 2. Because of X; = (X;)geP and Lemma 2 we have

IIu Ilxp
Hence (1) 3 (3). Now we are ready to establish the famous EMBEDDING THEOREM. Let X and Y be quasi-Bunuch spaces which are continuously embedded into some linear topological Huusdorff space. Furthermore, let (X, A,,) and (Y, A,,) be approximation schemes built from the same sequence of subsets. Suppose that there exist constants o > 0 and c > 0 such that

Il~ll,~c~“ll4x Then Xp ’ a c y”, . In particular, have X”C YT

for all a E A, and n = 1, 2,... . if Y can be v-normed with 0 < v < 1, we

Proof: Since the linear subset A is dense in c, condition (3) of the preceding lemma yields XE G Y. Now it follows from the Reiteration Theorem that XEto z (Xg); c y”, . EXAMPLE

1. Let 0 < q ( p < co. Becauseof Holder’s inequality we have

I141r,< n“9-LlpllullrD for all a E f, and n = 1, 2,... .

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ALBRECHT PIETSCH

This yields the embeddings if p+ l/p=a+

s;,. s s;,ld

l/q.

From the formula Z,,, = s& with l/r = p + l/p we see that even identity holds. Analogously we get if p + l/p = 0 + l/q.

b;,u E 4.u EXAMPLE

2. Let 1

states that

l14Lq~cn1’p-Uql141Lp - for all a E 0,

and n = 1, 2,... .

This yields the embeddings

%,uG%,u

if p-

l/p=a-

l/q.

Nessel and Wilmes [ 121 observed that the above estimate remains true also for a E E;, if 1

IP IIL,< cn”q-l’pIP lLp

for all A E s,(E, F) and n = 1, 2,... .

This yields the embeddings

‘Q;,,(E, F) E Fp;,,(E,F)

if p + l/p = u + l/q.

In particular, we get the inclusion gr* G V2, which was first proved by Konig [9]. Remark. Another result of this type is due to the author [ 15, 18.6.31.Let ‘$l(E, F) denote the Banach space of all nuclear operators T E P(E, F)

equipped with the norm I(. ](.fl. Then

IIAIL G n IV IL

for all A E g,(E, F) and n = 1, 2,...,

and it follows that 2:(E, I;) E W(E, F). Furthermore, it turns out that many inclusions proved in the theory of operator ideals can be understood as embeddings within the context of approximation spaces. 3.5. Composition Theorem

Now we prove a so-called

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APPROXIMATION SPACES

COMPOSITION THEOREM. Let (X,A,), (Y,B,), and (Z, C,) be approximation schemes. If M is a bounded bilinear map from Xx Y into Z such that M(A,, Y) E C, and M(X, B,) c C, for n = 1, 2,..., then

M(XP, , YE) c z;,+, Proof

whenever

l/u + l/v = l/W.

The assertion follows immediately from a ,+.-l(~(f;g),Z)~lIMlla,(f,X)a,(g,

0

and Holder’s inequality. Let x E I, and y E I,. If M(x, y) is defined to be the coordinatewise product x +y = (<,q,,J of the sequences x = (&,) and y = (v,,J, then Holder’s inequality yields M(x, y) E I,. with l/r = l/p + l/q. Consequently we have EXAMPLE

1.

EXAMPLE 2. Let fE L, and g E L, such that l/p + l/q > 1. If M(f, g) is defined to be the convolution

f * g(s) = 1’ f (t> ds - t) dt n of the functions f and g (periodically extended on the real line), then it is well known that M(f; g) E L, with l/r = I/p + l/q - 1; cf. [ 23, Chap. II,1 1. Consequently we have

EXAMPLE 3. Let T E VP,@, F) and S E Vp,(F, G) such that l/p + l/q < 1. If M(S, T) is defined to be the product ST of the operators S and T, then it is well-known that STE ‘p,(E, G) with l/r = l/p + l/q; cf. [ 15, 20.2.41. Consequently we get

3.6. Commutation Theorem Let (X, A,) be an approximation scheme, and consider the quasi-Banach space [L,, XE]. On the other hand, if [L,,A,] consists of all AU-valued functions f E [L,, X], we get an approximation scheme ([L,, X], [L,, A,,]). The corresponding approximation spaces will be denoted by [L,, Xl:.

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ALBRECHT PIETSCH

There holds

COMMUTATION THEOREM.

(1) bp, Xl s v&x

if

o
(2) I& Xl 2 Fp, Xl”,

if

O
and Proof. (1) We consider a measurable step-function f = Cr= I f,,~,,, where f, ,...,f, E PU, and x, ,..., x,,, are characteristic functions of pairewise disjoint measurable subsets 0, ,..., J2,. Choose ahn E A,-, such that llfh - ah,,IL < 2dfi, XL and put %I

:=

s,

‘hnxh-

Let 0

it follows that

%I&L&w QIlf- h?nll,L,,x, =

( 2 h=l

llfh

-

ahniIiph)

I”

VP 2 h=l

.

a,(fhyX)P&,

Here ph denotes the Lebesgue measure of 0,. Therefore we get

IIfll [L,.XIp, = g, [nP-“Ua,(S[Lp,Xl)]” 1l” I nP- vu [

(

h$

a.(fi~X)p~h)vp]"~

'l" P/U

(

VP

$ [nP-"Uan(fhTx)]" iuh n=l 1 I

ilfhll&

ph

1 I”

=

2 Ilfil,Lp,X;]*

So we have

llfll[L,,XIp. G 2 llfllrr,,*p,, for all X;-valued measurable step-functions & Now the assertion follows from the fact that these functions are dense in [L,, Xc]. (2) Let f E [L,, Xl”,. Then there exists a representation f= CpzO g, such that g, E [L,,A,,] and (2kp]]g,]),, ,x,) E I,. It can be shown that f(s) = C& gk(s) almost everywhere. So, i”f 0 < u

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APPROXIMATION SPACES

and therefore 1”’ < 1j’ ( f 0

[2kp ,,a,(s),,.ly)p’Y ds 1I”

k=o

This implies that fE [L,, X;]. Remark. If u = co, then we get into trouble with respect to the measurability of f as an X:-valued function. Remark. An analog of the above Commutation Theorem is well known within the framework of interpolation spaces; cf. 12, 5.8.61 and [21, 1.18.41.

4. APPLICATIONS

Now we apply the theory of approximation spaces in order to get results about the distribution of Fourier coefftcients and eigenvalues. 4. I. Fourier Coeflcients

For every scalar function fE L, the sequence of Fourier coefficients is defined by

<,(f) :=j1f(s) & 0

&,,(f)

:= 2 ,f’ f(s) sin 2zms ds, 0

and &,,, + 1(f) := 2 !’ f(s) cos 27cmsds. 0

There are many classical results concerning the problem how the

asymptotic

behaviour of (c,(f)) depends on certain properties of the functionf; cf. [ 181. All these facts are summarized in the following theorems. Let I/r = p + l/2. are equivalent. E lr,u

THEOREM

(L(f)>

1.

Then

the assertions SE ST-. and

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ALBRECHTPIETSCH

ProoJ: According to the Fischer-Riesz theorem Tf := (r,(f)) defines an isomorphism between L, and I,. Hence the Transformation Theorem tells us that T is also an isomorphism between S;,, and s;,,. But the latter space coincides with I,-, . If 1

2. Let l/r =p + l/p+. Then f E Sg,, implies (t,,,(f)) E lT,U.

Proof: In the case 1

3. Let l/r =p + l/p’.

Then f E B;,U implies f{,(f))

E f,..

Remark. Using examples constructed in [23, Chap. V,2 + 4 and VI,3 ] we can see that the preceding results are best possible. 4.2. Integral Operators Let K be a scalar kernel defined on the unit square. By k(s) we denote the function K(s, .) where s is fixed. Then k: s + k(s) is an abstract function. We say that the kernel K belongs to [B,P,,,Bz,u] if this is true for the B;,,-valued function k. Kernels of type [Lp, L,], [Lp, Bt,r,], and [Bg,,, L,] are defined analogously. In the following we investigate integral operators

s,: g(t) -f(s) = [I K(s, t) g(t) cft -0

generated by a kernel K. A first result is well-known from the theory of absolutely p-summing operators; cf. [8]. LEMMA

4. Let l/p + l/q < 1. Then

(0) K E IL,, L,] * S, E %&,,> LJ Now we are in a position to establish THEOREM 4.

true:

Let l/p + l/q < 1. Then the following implications are

APPROXIMATION SPACES

131

Proof. We consider the operator T transforming every kernel K into the corresponding integral operator S,. Then the Transformation Theorem, the Commutation Theorem, and Lemma 4 yield

Applying once again the Transformation (2) and (3) respectively. 4.3. Convolution

Theorem to (0) and (1), we obtain

Operators

Let f be a l-periodic scalar function defined on the real line. Then we put K,(s, t) :=f(s - t). The well-known LEMMA

(0)

theorem about the continuity

of the shift operator yields

Let 0 < q < co. Then

5.

fE L,

*

K,E IL,,

Ld

Now we get THEOREM 5.

(1)

Let 0 < q < co. The following

fE fcy,t~T

(2) fE

Vfco

3 =s

B:,;"

implications are true:

K.fE [LCD, Kt.1~ K,E PP,,,,B&J.

Proof: We consider the operator T transforming every scalar function f into the corresponding convolution kernel K,. Then the Transformation Theorem, the Commutation Theorem, and Lemma 5 yield (1) Applying

W;,,,K

[L,J,l:G

[LB;,,].

once again the Transformation

Theorem to (1) we obtain (2).

4.4. Eigenvalues of Operators In the sequel let E be a complex Banach space. Since every operator S E Q;,,(E, E) can be approximated by operators of finite rank, it follows from Riesz’s theory that S possesses a set of eigenvalues which is at most countable. Let (1,(S)) denote the sequence of these eigenvalues counted according to their multiplicities and ordered such that IA,(S)1 > I&(S)/ > . . . > 0. If S has less than n eigenvalues, then we put n,(S) = 0. Now we

132

ALBRECHT

PIETSCH

state an interesting generalization of Weyl’s theorem which goes back to Carl and K&rig [IO]. THEOREM 6. Let 2 Qp < co and l/r =p + l/p. Then S E !j$,(E, F) implies (I,(S)) E lr,u.

Proof: First we treat the case where S E V’;,,(E, E). Then there exists a representation S = CFZOS, such that Sk E g&E, E) and (2kp1)S,l]J E I,. As shown in the theory of absolutely 2-summing operators [ 15, 17.3.71,we can find factorizations Sk = X,A, with A, E g(E, I’:“) and xk E r?(li’, E) such that

Consider I, as the orthogonal sum of the spaces1:” with k = 0, l,... . Let J, E f?(l:“, /,) and Qk E II!&, r:“) denote the canonical injections and surjections, respectively. Then it follows from (2kp(IA,((>J E I, and (2kp][xk(($) E 1, that A := 2 JkAk E $$;,(E, 1,) k=O

and

X := f

XkQk E I!?;&, E).

k=O

Since S = XA is related to T = AX, we have n,(S) = 1,(T); cf. [ 15, 27.3.31. Moreover, using qz2(12,1,) = II:‘*&, I,), the Reiteration Theorem, and the Composition Theorem, we get TE Y;!:,(E, 1,) 0 Q&l,, E) = 13;,,(1,, 12)= 2:’

y2(12,

12).

Hence the classical Weyl theorem yields (A,(T)) E l,,, with l/r =p + l/2. This proves the assertion for p = 2. Now we suppose that 2

l/2. Then it follows from the Embedding Theorem that Ip&(E, E) E ‘$;,,(E, E), where u :=p + l/p l/2. Consequently, for S E Tp;,,(E, E) we get (/I,(S)) E I,, with l/r = o + l/2 = p + I/p. Unfortunately the above method does not work for p f l/p < l/2. However, in this case we can apply K&rig’s interpolation procedure as presented in [8,9]. Remark. If p = co and p < l/2, then it is possible to choose a natural number m such that mp > l/2. Now we obtain from S E IE(E, E) that S” E I$&?& E), and therefore (n,(Sm)) C lr,m,lJmwith l/r =p. Hence, by the Spectral Mapping Theorem, it follows that (n,(S)) E I,,. Summarizing Theorems 4 and 6 we get some interesting results about the distribution of eigenvalues of integral operators.

APPROXIMATION

THEOREM

(1)

7. Let I/p + l/q < 1. The following

KE [L,,Bg,l,],p
133

SPACES

implications are true:

*

(&(S,))EIFs,

with l/r=a+ with l/r=p

l/q+,

(2)

KE [%uJql

*

(W,>> E L,,

(3) 1/q+.

KE P;,u~~,“J~l

3

(A,(S,)) E I,,, with l/r =p + u $

t l/q+,

Remark. Using another method the author [ 191 was able to show that the condition l/p t l/q < 1 can be replaced by the weaker assumptions l/pt l/q
Finally we recall that the (complex) Fourier coefficients

A,(f) := j?(s) ds, 0 A,,,,(f) := f f(s) exp(-2zims)

ds,

0

3,Zm+,(f) := ,f ’ f(s) exp(2nims) ds, 0

coincide with the eigenvalues of the convolution operator generated by the kernel K,. Moreover, we have h(f>=&(fh

~2,+,(f)+~zmdf)=r*mt,df>~

A2m+d.0 - J2Af>

=

it2Af).

Therefore, summarizing Theorems 5 and 7 we once again get the statement of Theorem 3. Remark. The counterexamples constructed in the theory of trigonometrical series show that the results stated in Theorem 7 are best possible.

ACKNOWLEDGMENTS The basic ideas of this theory were presented in the Kudrjavcev-Nikolskij-Seminar (November 1978) during a stay at the Steklov Mathematical Institut in Moscow. I take this opportunity to thank my Soviet colleagues for their hospitality. Moreover, I am extremely grateful to P. L. Butzer for bringing to my attention the very interesting booklet [6] which was not available to me before. His valuable remarks have also led to several other improvements in this paper.

134

ALBRECHT PIETSCH REFERENCES

1. T. I. AMANOV, Representation and embedding theorems for functional spaces (Russian), Trudy Mat. Inst. Steklov. 17 (1965), 5-34. 2. J. BERGH AND J. L&STRehl, “Interpolation Spaces,” Springer-Verlag. Berlin/Heidelberg/New York, 1976. 3. 0. V. BESOV,Investigation of a family of functional spaces in connection with embedding and extension theorems (Russian), Trudy Mat. Inst. Steklov. 60 (1961), 42-81. 4. Ju. A. BRLJDNIJ,Approximation spaces (Russian), Collection of papers on “Geometry of linear spaces and operator theory,” pp. 3-30, Jaroslavl, 1977. 5. Ju. A. BRUDNIJ AND N. JA. KRUGLJAK, About a family of approximation spaces (Russian), Collection of papers on “Theory of functions of several real variables,” pp. 15-42, Jaroslavl, 1978. 6. P. L. BUTZERAND K. SCHERER,“Approximationsprozesse und Interpolationsmethoden,” Mannheim/Ztirich, 1968. 7. B. CARL, Inequalities between absolutely (p, q)-summing norms, Studia Math. 69 (1980). 143-148. 8. W. B. JOHNSON,H. K~NIG, B. MAUREY, AND J. R. RETHERFORD,Eigenvalues of psumming and /,-type operators in Banach spaces, J. Functional Anal. 32 (1979), 353-380. 9. H. K~NIG, s-Zahlen und Eigenwertverteilungen von Operatoren in Banachraumen,

Habilitationsschrift, Bonn, 1977. 10. H. KOENIG,Weyl-type inequalities for operators in Banach spaces, in “Proc. Paderborn Conf. Functional Analysis (1979),” pp. 297-3 17, North-Holland, Amsterdam, 1980. 11. D. R. LEWIS,Finite dimensional subspaces of L,, Studia Math. 63 (1978), 207-212. 12. R. J. NESSELAND G. WILMES, Nikolskii-type inequalities for trigonometric polynomials and entire functions of exponential type, J. Austral. Math. Sot. Ser. A 25 (1978), 7-18. 13. M. S. NIKOLSKIJ, Approximation of functions of several variables and embedding theorems (Russian), Moscow, 1969. 14. J. PEETREAND G. SPARR,Interpolation of normed abelian groups, Ann. Mat. Pura Appl. 92 (1972), 217-262. 15. A. PIETSCH, “Operator

Ideals,” Dt. Verl. Wiss., Berlin, 1978, and North-Holland, Amsterdam, 1980. 16. A. PIETSCH,Einige neue Klassen von kompakten linearen Abbildungen, Rev. Romaine Math. Pures Appl. 8 (1963), 427447. 17. A. PIETSCH,Factorization theorems for some scales of operator ideals, Math. Nachr. 97 (1980), 15-19. 18. A. PIETSCH, ijber die Verteilung von Fourierkoefftzienten und Eigenwerten, Wiss. Z. Univ. Jena 29 (1980), 203-211.

19. A. PIETSCH,Eigenvalues of integral operators I., Math. Ann. 247 (1980), 169-178. 20. S. B. STEEKIN, About the absolute convergence of orthogonal series (Russian), Dokl. Akad. Nauk SSSR 102 (1955), 37-40. 21. H. TRIEBEL, “Interpolation Theory, Function Spaces, Differential Operators.” Dt. Verl. Wiss., Berlin, 1978, and North-Holland, Amsterdam, 1978. 22. D. VOGT, Integrationstheorie in p-normierten Riiumen, Math. Ann. 173 (1967), 2 19-232. 23. A. ZYGMUND,Trigonometric series, Cambridge Univ. Press, London, 1968.