DOI: 10.1007/s00209-005-0870-1

Math. Z. (2005) Mathematische Zeitschrift

Compactness properties of certain integral operatorsrelated to fractional integration

Eduard Belinsky�, Werner Linde1

1Faculty of Mathematics and Computer Sciences, Friedrich-Schiller-Universitat Jena,Ernst Abbe Platz 2, 07743 Jena, Germany (e-mail: lindew@minet.uni-jena.de)

Received: 17 January 2005 /Published online: 16 August 2005 – © Springer-Verlag 2005

Abstract. Suppose 1 ≤ p, q ≤ ∞ and α > (1/p − 1/q)+. Then we investigatecompactness properties of the integral operator

(Sαf )(x) :=∫ 1

0(x + t)α−1f (t) dt , 0 ≤ x ≤ 1 ,

when regarded as operator from Lp[0, 1] into Lq [0, 1]. We prove that its Kolmogo-rov numbers tend to zero faster than exp(−cα n1/2). This extends former resultsof Laptev in the case p = q = 2 and of the authors for p = 2 and q = ∞.As application we investigate compactness properties of related integral operatorsas, for example, of the difference between the fractional integration operators ofRiemann–Liouville and Weyl type. It is shown that both types of fractional integra-tion operators possess the same degree of compactness. In some cases this allowsto determine the strong asymptotic behavior of the Kolmogorov numbers of Rie-mann–Liouville operators.

Mathematics Subject Classification (2000): 47B06, 46B28, 26A33

1. Introduction

In a basic paper A. A. Laptev ([9]) investigated compactness properties of the inte-gral operator T defined by

(Tf )(x) :=∫ 1

0

xβtγ

(x + t)αf (t) dt , 0 ≤ x ≤ 1 .

� In memoria of Eduard (University of the West Indies) who passed away in October2004.

E. Belinsky, W. Linde

Here the numbers α, β and γ are assumed to satisfy α > 0, β, γ > −1/2 andβ + γ − α + 1 > 0. It is shown in [9] that the singular values of T as operator inL2[0, 1] behave like e−c n1/2

. In the present paper we regard an integral operatortightly related to T in a more general situation, namely, as operator from Lp[0, 1]into Lq [0, 1] with 1 ≤ p, q ≤ ∞. More precisely, if α > (1/p − 1/q)+, then weinvestigate Sα defined by

(Sαf )(x) :=∫ 1

0(x + t)α−1f (t) dt (1.1)

as operator from Lp[0, 1] into Lq [0, 1]. Since singular values are only defined foroperators between Hilbert spaces we need some substitute for these numbers. Twodifferent quantities are used, namely, the Kolmogorov and the (dyadic) entropynumbers defined as follows.

Let (E, ‖ · ‖E) and (F, ‖ · ‖F ) be Banach spaces, and let S : E → F be acompact operator. The Kolmogorov numbers of S, denoted by dn(S), are defined as

dn(S) = dn(S : E → F) := inf

{sup

‖x‖E≤1dF (Sx, Fn) : Fn ⊆ F , dim(Fn) < n

}

where, as usual,

dF (y, Fn) := inf{‖y − z‖F : z ∈ Fn

}

denotes the distance of y ∈ F to the subspace Fn (w.r.t. the norm in F ). If E andF are Hilbert spaces, then the Kolmogorov numbers of S coincide with its singularvalues (cf. [16] or [17]). Recall that the singular values are the square roots of theeigenvalues of S∗S in decreasing order (cf. [6] for more information).

The (dyadic) entropy numbers of S are given by

en(S) = en(S : E → F) := inf

ε > 0 : S(BE) ⊆

2n−1⋃j=1

(yj + ε BF )

.

Here BE and BF denote the (closed) unit balls in E and F , respectively. In otherwords, en(S) is the infimum over all ε > 0 such that S(BE) can be covered byat most 2n−1 balls of radius ε > 0 in F . We refer to [17], [2] and [18] for moreinformation about Kolmogorov and entropy numbers.

With these notations we shall prove the following (weaker) version of Laptev’sresult in the Banach space setting.

Theorem 1.1. Suppose that the numbers α > 0 and p, q in [1, ∞] satisfy α >

(1/p − 1/q)+. Then there is a constant cα = cα(p, q) > 0 such that

dn(Sα : Lp[0, 1] → Lq [0, 1]) � e−cα ·n1/2. (1.2)

Theorem 1.1 combined with Theorem 2 in [15] leads to the following.

Compactness properties of certain integral operators

Corollary 1.2. Let p, q and α be as before. Then there exists a constant cα > 0,such that

en(Sα : L0p[0, 1] → Lq [0, 1]) � e−cαn1/3

.

We shall use Theorem 1.1 to investigate compactness properties of some opera-tors related to Sα . For example, let Rα and Iα be the fractional integration operatorsof Riemann–Liouville and Weyl type, respectively (cf. (3.1) and (3.2) below forthe definition of these operators). In section 3.1 we will prove that dn(Rα − Iα)

tends to zero faster than e−cα n1/2. In particular, both types of fractional integration

possess exactly the same compactness properties. This allows us to prove strongasymptotic properties of dn(Rα) in section 4.

In the investigation of stochastic self–similar processes the operator

(Vαf )(x) := 1

�(α)

∫ ∞

−∞

[(x − t)α−1

+ − (−t)α−1+]f (t) dt , 0 ≤ x ≤ 1 ,

plays an important role. If we suppose 1/p < α < 1 + 1/p, then the operator Vα

maps Lp(−∞, ∞) into L∞[0, 1]. In section 3.2 we prove that Vα possesses thesame degree of compactness as the Riemann–Liouville operator Rα . For p = 2 thiswas proved in [1] and led to interesting consequences about the relation betweenfractional Brownian motions and so–called Riemann–Liouville processes.

Finally, we investigate the following question (also motivated by the investiga-tion of stochastic processes). Given numbers 0 < a < b, we compare the operatorRα from Lp[0, b] into Lq [0, b] with Ra

α defined as the composition of two “indepen-dent” Riemann–Liouville operators on Lp[0, a] and Lp[a, b], respectively. Againit turns out that dn(Rα − Ra

α) tends to zero exponentially, where, surprisingly, theappearing constants only depend on b − a, yet not on the number a. This will becarried out in section 3.3.

2. Proof of Theorem 1.1

Proof of Theorem 1.1. First we note that Sα is of finite rank provided that α ∈ N,hence (1.2) holds for those α’s by trivial reason. Furthermore, if 1 ≤ q ≤ p ≤ ∞,then Sα : Lp[0, 1] → Lq [0, 1] factors through Lp[0, 1], hence, if q ≤ p, then itsuffices to prove estimate (1.2) for q = p. Summing up, we have to verify Theorem1.1 for α, p and q satisfying

1 ≤ p ≤ q ≤ ∞ , α > 1/p − 1/q and α /∈ N .

To estimate the Kolmogorov numbers of Sα we have to find good approximationsof functions φ on [0, 1] representable as

φ(x) =∫ 1

0f (t)(x + t)α−1 dt , 0 ≤ x ≤ 1 ,

for some f ∈ Lp[0, 1] with ‖f ‖p ≤ 1. To this end we divide the interval [0, 1]into the n + 1 subintervals [0, 2−n] and [2−(k+1), 2−k], k = 0, . . . , n − 1 and on

E. Belinsky, W. Linde

each subinterval we approximate φ separately by elements of a linear subspace ofLq [0, 1] possessing dimension less than a multiple of n. We shall do this in differentways for α > 1/p, α < 1/p and α = 1/p, respectively.First case: α > 1/p.Here it suffices to approximate φ in the L∞–norm. We fix an integer n > α anddivide the interval [0, 1] as mentioned above.Let us start with the interval [0, 2−n]. We represent φ(x) for x ∈ [0, 2−n] as thesum

φ(x) =∫ 2−(n−1)

0f (t)(x + t)α−1 dt +

∫ 1

2−(n−1)

f (t)(x + t)α−1 dt . (2.1)

The first integral in (2.1) is small by Holder’s inequality. Indeed we easily obtain∣∣∣∣∣∫ 2−(n−1)

0f (t)(x + t)α−1 dt

∣∣∣∣∣ ≤ c · 2−(n−2)(α−1/p) .

The second integral in (2.1) we represent in the form∫ 1

2−(n−1)

f (t)tα−1(

1 + x

t

)α−1dt

and approximate(1 + x

t

)α−1 by the partial sums of its Taylor series. Doing so weobtain ∣∣∣∣

∫ 1

2−(n−1)

f (t)tα−1[(

1 + x

t

)α−1 − Pn−1

(x

t

)]dt

∣∣∣∣≤ |(1 − α)(2 − α) · · · · · (n + 1 − α)|

n!2−n

∫ 1

2−(n−1)

|f (t)|tα−1 dt

= O

(�(n + 2 − α)

|�(1 − α)| �(n + 1)2−n

)� 2−κ n

for any positive κ < 1. The last estimate may be derived from the asymptoticformula for the gamma function (cf. [8]) asserting

�(z + a)

�(z + b)= za−b

[1 + (a − b)(a + b − 1)

2z+ O

(|z|−2

)]. (2.2)

For each subinterval [2−(k+1), 2−k], 0 ≤ k ≤ n − 1, we decompose φ(x) into thethree parts

∫ 2−(k+2)

0f (t)(x + t)α−1 dt +

∫ 2−(k−1)

2−(k+2)

f (t)(x + t)α−1 dt

+∫ 1

2−(k−1)

f (t)(x + t)α−1 dt . (2.3)

The last integral in (2.3) we represent and approximate as above, with the sameorder of error.

Compactness properties of certain integral operators

The first integral in (2.3) we write as

∫ 2−(k+2)

0f (t) xα−1

(1 + t

x

)α−1

dt

and approximate(1 + t

x

)α−1 by the partial sums of its Taylor series. Then weobtain

∣∣∣∣∣∫ 2−(k+2)

0f (t)xα−1

[(1 + t

x

)α−1

− Pn−1

(t

x

)]dt

∣∣∣∣∣≤ |(1 − α)(2 − α) · · · · · (n + 1 − α)|

n!2−n

∫ 2−(k+2)

0|f (t)|xα−1 dt

= O

(�(n + 2 − α)

|�(1 − α)| �(n + 1)2−n

)� 2−κ n

for arbitrary κ < 1.The second integral in (2.3) we consider as function of x and approximate it by

the partial sum of its Taylor expansion of order n in the neighborhood of the pointxk = 2−(k+1) + 2−(k+2). We obtain for the error of the approximation

∣∣∣∣∣∫ 2−(k−1)

2−(k+2)

f (t)(x + t)α−1dt − Pn−1(x − xk)

∣∣∣∣∣≤ �(n + 2 − α)

|�(1 − α)| �(n + 1)|x − xk|n

∫ 2−(k−1)

2−(k+2)

|f (t)|(θx + t)α−n−1 dt

≤ �(n + 2 − α)

|�(1 − α)| �(n + 1)2−(k+2)n2(k+1)n

∫ 2−(k−1)

2−(k+2)

|f (t)|tα−1 dt

= O

(�(n + 2 − α)

|�(1 − α)| �(n + 1)2−n

)� 2−κ n .

Taking into account the dimension of the subspaces we obtain dn2(Sα) � 2−cαn forany positive cα < min {α − 1/p, 1} or, equivalently,

dn(Sα : Lp[0, 1] → Lq [0, 1]) � 2−cα n1/2

as asserted.Second case: 1/p − 1/q < α < 1/p ≤ 1, hence q < ∞We start again with the interval [0, 2−n]. For numbers x in this interval we applyHolder’s inequality and obtain

∣∣∣∣∫ 1

0f (t)(x + t)α−1 dt

∣∣∣∣ ≤ xα−1/p‖f ‖p ≤ xα− 1/p .

E. Belinsky, W. Linde

For each interval [2−(k+1), 2−k], 0 ≤ k ≤ n − 1, we decompose φ(x) into thethree integrals

∫ 2−(k+2)

0f (t)(x + t)α−1 dt +

∫ 2−(k−1)

2−(k+2)

f (t)(x + t)α−1 dt

+∫ 1

2−(k−1)

f (t)(x + t)α−1 dt. (2.4)

The last integral in (2.4) we represent in the form

∫ 1

2−(k−1)

f (t)tα−1(

1 + x

t

)α−1dt

and approximate(1 + x

t

)α−1 by the partial sums of its Taylor series. We obtain

∣∣∣∣∫ 1

2−(k−1)

f (t)tα−1[(

1 + x

t

)α−1 − Pn−1

(x

t

)]dt

∣∣∣∣≤ (1 − α)(2 − α) · · · · · (n + 1 − α)

n!2−n

∫ 1

2−(k−1)

|f (t)|tα−1 dt

= O

(�(n + 2 − α)

�(1 − α)�(n + 1)2−n2k(1/p−α)

)� 2−κ n 2k(1/p −α)

for any κ < 1.The first integral in (2.4) we represent in the form

∫ 2−(k+2)

0f (t)xα−1

(1 + t

x

)α−1

dt

and approximate(1 + t

x

)α−1 by the partial sums of its Taylor series. This leads to

∣∣∣∣∣∫ 2−(k+2)

0f (t)xα−1

[(1 + t

x

)α−1

− Pn−1

(t

x

)]dt

∣∣∣∣∣≤ (1 − α)(2 − α) · · · · · (n + 1 − α)

n!2−n

∫ 2−(k+2)

0|f (t)|xα−1 dt

= O

(�(n + 2 − α)

�(1 − α)�(n + 1)2−n2k(1/p−α)

)� 2−κ n 2k(1/p −α) .

The second integral itself we consider as function of x and approximate it by thepartial sum of its Taylor expansion of order n in the neighborhood of the pointxk = 2−(k+1) + 2−(k+2). Then the error of approximation can be estimated by

Compactness properties of certain integral operators

∣∣∣∣∣∫ 2−(k−1)

2−(k+2)

f (t)(x + t)α−1 dt − Pn−1(x − xk)

∣∣∣∣∣≤ �(n + 2 − α)

�(1 − α)�(n + 1)|x − xk|n

∫ 2−(k−1)

2−(k+2)

|f (t)|(θx + t)α−n−1 dt

≤ �(n + 2 − α)

�(1 − α)�(n + 1)2−(k+2)n2(k+1)n

∫ 2−(k−1)

2−(k+2)

|f (t)|tα−1 dt

= O

(�(n + 2 − α)

�(1 − α)�(n + 1)2−n2k(1/p−α)

)� 2−κ n 2k(1/p −α) .

Thus in view of the dimension of the subspaces we obtain

dn2(Sα) �{∫ 2−n

0xq(α−1/p) dx + 2−κ n

n∑k=1

2kq(1/p−α)−k

}1/q

,

hence

dn(Sα : Lp[0, 1] → Lq [0, 1] � 2−cαn1/2

for any positive cα < α − 1/p + 1/q. This proves (1.2) in this case as well.Third case: α = 1/p, hence q < ∞.Following the same proof as in the previous case, we obtain

dn2(Sα) �{∫ 2−n

0log q x + 1

xdx + 2−κn

n∑k=1

kq2−k

}1/q

,

consequently

dn(Sα : Lp[0, 1] → Lq [0, 1]) � 2−cαn1/2

for any positive cα < 1/q. Thus Theorem 1.1 has been proved completely.

3. Applications to Other Operators

3.1. Fractional Integration Operators

Let f be an integrable function on [0, 1]. Then, if α > 0, its Riemann-Liouvillefractional integral Rαf is defined by the formula

(Rαf )(x) = 1

�(α)

∫ x

0(x − t)α−1f (t) dt . (3.1)

This type of fractional integration is very useful in different branches of mathemat-ics, e.g. in probability theory or in approximation theory, yet it is not convenientfor the investigation of periodic functions. Observe that Rαf of a periodic functionf will be in general non–periodic.

E. Belinsky, W. Linde

When dealing with periodic functions the Weyl operator of fractional integra-tion is the adequate type of integration. It is defined on each exponential by theformula (see, for example [24], ch.12)

(Iα e2πik · )(x) = e2πikx

(2πik)α, k �= 0. (3.2)

Here, if α is fractional, then (ik)−α means |k|−α exp(−i πα2 sign(k)). This defi-

nition is extended by linearity to trigonometric polynomials and by continuity tointegrable (complex valued) periodic functions of the space.

L01[0, 1] :=

{f ∈ L1[0, 1] :

∫ 1

0f (t)dt = 0

}.

Since Iα maps real valued functions into real ones, here and later on we may regardIα as operator acting on the real space L0

1[0, 1]. We refer to the encyclopedic book[19] for further properties of these as well of other operators of fractional integra-tion.

A basic question is how far are Rα and Iα . To that purpose let us introduce theirdifference

Qα := Rα − Iα . (3.3)

Since Qα has finite rank for integer α, it is very likely also for non–integer α > 0the operator Qα is small in the sense of its compactness properties. Indeed, whenregarding Qα as operator from L2[0, 1] into L∞[0, 1], then it was proved thatdn(Qα) tends to zero exponentially (cf. [1]). Our aim is to extend this result to allpossible cases of indices. Before proving this let us state some properties of Qα forlater use.

Proposition 3.1.

(1) If α > (1/p − 1/q)+, then Qα is bounded as operator from L0p[0, 1] into

Lq [0, 1]. Here L0p[0, 1] := Lp[0, 1] ∩ L0

1[0, 1].(2) Given α, β > 0, then it follows that

Qα+β = Qα ◦ Iβ + Rα ◦ Qβ. (3.4)

(3) If α < 1, then

(Qαf )(x) = 1

�(α)

∫ 1

0f (t)rα(x − t) dt (3.5)

where

rα(s) := 1

�(α)lim

n→∞

{n∑

k=1

(s + k)α−1 − nα

α

}.

Compactness properties of certain integral operators

Proof. Property (1) is well–known and follows from the corresponding propertiesof Rα and Iα , respectively.Equation (3.4) may easily be derived from the semi–group properties of Rα andSα .Finally, representation (3.5) is stated and proved in [24], Chapter 12, n. 8.

Remark. Let us note that for any f ∈ L1[0, 1]

Rα(f ) = Rα(f − Mf ) + Rα(Mf ), Mf =∫ 1

0f (t) dt.

This means that the operator Rα defined on the whole space Lp[0, 1] differs fromthat on the subspace L0

p[0, 1] by a one dimensional subspace only. Therefore anyestimates of Kolmogorov or entropy numbers for Rα : L0

p[0, 1] → Lq [0, 1] willimply the same estimates for Rα : Lp[0, 1] → Lq [0, 1].

Now we are in position to state and to prove the main result of this section.

Theorem 3.2. Suppose α > (1/p − 1/q)+. Then there is a constant cα = cα(p, q)

> 0 such that

dn(Qα : L0p[0, 1] → Lq [0, 1]) � e−cα n1/2

. (3.6)

Proof. Let us start with the case α < 1. Then Qα may be represented as stated in(3.5). Next we split the function rα into two parts as rα = r ′

α + r ′′α where

r ′α(s) := 1

�(α)(s + 1)α−1 and r ′′

α(s) := 1

�(α)lim

n→∞

{n∑

k=2

(s + k)α−1 − nα

α

}.

Consequently, Qα = Q′α + Q′′

α where

(Q′αf )(x) :=

∫ 1

0f (t) r ′

α(x − t) dt and (Q′′αf )(x) :=

∫ 1

0f (t) r ′′

α(x − t) dt,

respectively. An easy change of variables transforms Q′α into Sα defined in (1.1).

Hence, Theorem 1.1 applies and leads to

dn(Q′α : Lp[0, 1] → Lq [0, 1]) � 2−cα n1/2

. (3.7)

Therefore it suffices to estimate dn(Q′′α) suitably. This will done in a more general

setting in the next lemma.

Lemma 3.3. Let 0 < α < 1. Then it follows that

dn(Q′′α : L0

1[0, 1] → L∞[0, 1]) � �(n + 3 − α)

�(1 − α)�(n + 2)2−n. (3.8)

E. Belinsky, W. Linde

Proof. By the definition of Kolmogorov numbers we have to find a good approx-imation of Q′′

αf by elements of a linear subspace in L∞[0, 1]. Fix f ∈ L1[0, 1]and approximate Q′′

αf by its Taylor polynomial Pn(Q′′αf ; x) of order n in the

neighborhood of the point x0 = 1/2. Then∣∣(Q′′

αf )(x) − Pn(Q′′αf ; x)

∣∣≤ 2−n (1 − α)(2 − α) · · · · · (n + 2 − α)

(n + 1)!

∫ 1

0|f (t)|

∞∑k=2

1

(x − t + k)n+2−αdt

� �(n + 3 − α)

�(1 − α)�(n + 2)2−n‖f ‖1.

Thus the lemma is proved. Combining (3.8) with (2.2), by (3.7) we obtain (3.6) provided that (1/p −

1/q)+ < α < 1.Suppose now, that 1 < α < 2 and 1 ≤ p ≤ q ≤ ∞. Recall that the case q ≤ p

follows from this by the same reason as mentioned at the beginning of the proofof Theorem 1.1. Then we choose β = 1 − min{ 1

2 (1 − 1/p + 1/q), 12 (2 − α)} and

obtain 1/p − 1/q < β < 1 as well as 0 < α − β < 1. Using the multiplicationformula (3.4) and the algebraic properties of the Kolmogorov numbers, togetherwith dn(Iα) ≤ ‖Iα‖ and dn(Rα) ≤ ‖Rα‖, we obtain

d4n−3(Qα : Lp → Lq) ≤ dn(Qβ : Lp → Lq) · dn(Iα−β : Lp → Lp)

+dn(Rβ : Lp → Lq) · dn(Qα−β : Lp → Lp) � e−cα n1/2.

One case is not covered by this proof, namely p = 1 and q = ∞. Suppose1 < α < 2. By the multiplication formula (3.4) and the algebraic properties of theKolmogorov numbers (and using the same obvious bounds for dn(Iα) and dn(Rα))we obtain

d4n−3(Qα : L1 → L∞) ≤ dn(Qα/2 : L2 → L∞) · dn(Iα/2 : L1 → L2)

+dn(Rα/2 : L2 → L∞) · dn(Qα/2 : L1 → L2) � e−cα n1/2.

Iterating this procedure we obtain (3.6) for the whole range of α. This completesthe proof. Corollary 3.4. Let p, q and α be as before. Then there exists a constant cα > 0,such that

en(Qα : L0p[0, 1] → Lq [0, 1]) � e−cαn1/3

.

Proof. This result follows immediately from the previous theorem, and the relationbetween entropy and Kolmogorov numbers as stated in ([15], Theorem 2). Remark. Theorem 3.2 and Corollary 3.4 tell us that the Kolmogorov numbers aswell as the entropy numbers of Rα and Iα possess the same decreasing order. Asurvey of the known results may be found, for example, in [4], [12], [21] and [22].

Compactness properties of certain integral operators

3.2. An Operator Related to Self–Similar Processes

Let us state another application of Theorem 1.1. Suppose 1/p < α < 1 + 1/p.Then the operator Vα : Lp(−∞, ∞) → L∞[0, 1] with

(Vαf )(x) := 1

�(α)

∫ ∞

−∞

[(x − t)α−1

+ − (−t)α−1+]f (t) dt , 0 ≤ x ≤ 1 ,

is well–defined and bounded. The operator Vα plays (for p ≥ 2) an importantrole in the investigation of self–similar symmetric stable processes (cf. [20] or [11]for more information). In particular, if p = 2, then Vα generates the fractionalBrownian motion with Hurst index α − 1/2.

The next result shows that Vα is in fact very “near” (in the sense of compactness)to the Riemann–Liouville operator Rα . For p = 2 this observation was importantto verify certain properties of fractional Brownian motions (cf. [1], [14] or [10]).

Proposition 3.5. Assume 1/p < α < 1+ 1/p. Then there is a certain cα > 0 suchthat

dn(Vα − Rα : Lp(−∞, 0) → L∞[0, 1]) � e−cα n1/2.

Proof. We may represent Vα − Rα as V ′α + V ′′

α − Fα where

(V ′αf )(x) := 1

�(α)

∫ −1

−∞

[(x − t)α−1 − (−t)α−1

]f (t) dt,

(V ′′α f )(x) := 1

�(α)

∫ 0

−1(x − t)α−1f (t) dt

and

(Fαf )(x) := 1

�(α)

∫ 0

−1(−t)α−1f (t) dt.

The operator Fα is of rank 1, hence it does not influence the behavior of dn(Vα−Rα).Furthermore, an easy change of variables shows that V ′′

α can be isometrically trans-formed to Sα defined in (1.1). Consequently, Theorem 1.1 applies and leads todn(V

′′α ) � e−cα n1/2

.Thus in order to complete the proof it remains to estimate dn(V

′α) suitably. To this

end suppose that the function φ on [0, 1] can be written as

φ(x) = 1

�(α)

∫ ∞

1

[(x + t)α−1 − tα−1

]f (t) dt

with some f ∈ Lp(1, ∞). Let Pn(φ; x) be the nth Taylor polynomial of φ at thepoint x0 := 1/2. Then it follows that

|φ(x) − Pn(φ; x)| ≤ 1

2n· (1 − α)(2 − α) · · · (n + 2 − α)

(n + 1)!

∫ ∞

1t−n−1+α |f (t)| dt

≤ c · �(n + 3 − α)

�(n + 2)· 2−n ‖f ‖p

which gives the desired estimate for dn(V′α) as well and completes the proof.

E. Belinsky, W. Linde

3.3. Splitting Riemann–Liouville Operators

For some b > 0 let us regard Rα as operator from Lp[0, b] to Lq [0, b]. Of course,we again assume α > (1/p − 1/q)+. Given a number a in (0, b) we split now Rα

into two “independent” pieces as follows:

(Raαf )(x) :=

{1

�(α)

∫ x

0 (x − t)α−1f (t) dt : 0 ≤ x ≤ a1

�(α)

∫ x

a(x − t)α−1f (t) dt : a < x ≤ b

Our aim is to show that Raα differs not very much from Rα where surprisingly the

estimate of dn(Rα − Raα) does not depend on a, only on b − a. More precisely, the

following is true.

Theorem 3.6. Let Rα and Raα be as before. Then there are constants c, cα > 0

independent of a and b such that for n ≥ α + 1

dn(Rα − Raα) ≤ c · (b − a)α−1/p+1/q e−cα n1/2

. (3.9)

Proof. The operator Saα := Rα − Ra

α maps Lp[0, a] into Lq [a, b] as follows:

(Saαf )(x) = 1

�(α)

∫ a

0(x − t)α−1f (t) dt , a ≤ x ≤ b .

Changing variables and setting δ := b−a, there exist isometries �p from Lp[0, a]into Lp[0, a/δ] and q from Lq [0, 1] into Lq [a, b] such that

Saα = (b − a)α−1/p+1/q q ◦ T a

α ◦ �p

where T aα : Lp[0, a/δ] → Lq [0, 1] is defined by

(T aα f )(x) := 1

�(α)

∫ a/δ

0(x + t)α−1f (t) dt , 0 ≤ x ≤ 1 .

Thus, in order to prove (3.9) (first for α < 1) it suffices to verify the followinglemma.

Lemma 3.7. For � > 0 define T (�)α from Lp[0, �] into Lq [0, 1] by

(T (�)α f )(x) := 1

�(α)

∫ �

0(x + t)α−1f (t) dt , 0 ≤ x ≤ 1 .

Then, if (1/p − 1/q)+ < α < 1, it follows that

dn(T(�)α ) ≤ c · e−cα n1/2

with c, cα > 0 independent of � > 0.

Compactness properties of certain integral operators

Proof. Write T (�)α as Sα + Tα + Fα where Sα is defined by (1.1),

(Tαf )(x) := 1

�(α)

∫ �

1[(x + t)α−1 − tα−1]f (t) dt

and

(Fαf )(x) := 1

�(α)

∫ �

1tα−1f (t) dt.

Then rk(Fα) = 1 and dn(Tα) ≤ dn(T α) where

(T αf )(x) := 1

�(α)

∫ ∞

1[(x + t)α−1 − tα−1]f (t) dt.

We may estimate dn(T α) exactly as dn(V′′α ) in the proof of Proposition (3.5) leading

to

dn(Tα) ≤ dn(T α) ≤ c · e−cα n1/2

with c, cα > 0 independent of � > 0. Consequently, Theorem 1.1 implies

d2n(T(�)α ) ≤ dn(Sα) + dn(Tα) + d2(Fα) ≤ c · e−cα n1/2

completing the proof of the lemma and that of Theorem 3.6 in the case α < 1. Suppose now α ≥ 1. If α is an integer, then rk(Sa

α) ≤ α, hence (3.9) is trueprovided that n ≥ α + 1. Thus let us suppose now α > 1 and α /∈ N. Set k := [α]and β := k − α. Observe that 0 < β < 1, hence (3.9) is satisfied for Sa

β regardedas operator from Lp[0, a] into Lp[a, b]. Yet note that in this case the appearingpower of b − a is β. Let Rk be the ordinary Riemann–Liouville operator mappingLp[a, b] into Lq [a, b]. As shown in [13], there exists an operator Fα of rank lessthan k = [α] such that

Saα = Rk ◦ Sa

β + Fα.

Consequently, by∥∥Rk : Lp[a, b] → Lq [a, b]

∥∥ = (b − a)k−1/p+1/q · ∥∥Rk : Lp[0, 1] → Lq [0, 1]∥∥ ,

for each n ≥ 2 it follows that

dn+k(Saα) ≤ dn(Rk ◦ Sa

β) + dk+1(Fα)

≤ ∥∥Rk : Lp[a, b] → Lq [a, b]∥∥ · dn(S

aβ)

≤ c · (b − a)k−1/p+1/q · (b − a)β e−cβ n1/2

= c · (b − a)α−1/p+1/q e−cα n1/2

completing the proof.

E. Belinsky, W. Linde

4. Strong Asymptotic

The aim of this section is to show how Theorem 3.2 leads to certain strong asymp-totic properties of dn(Rα). In a first example we strengthen a result in [23] and [3]by estimating the second term in the asymptotic approximation.

Theorem 4.1. Suppose α > 0. Then it follows that

dn(Rα : L2[0, 1] → L2[0, 1]) = 1

πnα+ O

(log2 n

nα+1

).

Proof. Recall that for operators on Hilbert spaces the Kolmogorov numbers coin-cide with the singular values. Yet these numbers are known for the Weyl operatorIα as Fourier coefficients of the kernel, i.e., we have

dn(Iα : L2[0, 1] → L2[0, 1]) = 1

πnα. (4.1)

Given a natural number m < n it follows that

dn(Rα) = dn(Iα − Qα) ≤ dn−m(Iα) + dm+1(Qα)

with Qα defined by (3.3). Thus, from (4.1) and Theorem 3.2 we get

dn(Rα) ≤ 1

π(n − m)α+ c · e−cα m1/2

which leads to

dn(Rα) ≤ 1

πnα+ C · m

(n − m)α+1 + c · e−cα m1/2. (4.2)

Choosing now m of order κ · log2 n with κ > 0 sufficiently large, from (4.2) wederive

dn(Rα : L2[0, 1] → L2[0, 1]) ≤ 1

πnα+ O

(log2 n

nα+1

).

Starting with

dn(Rα) ≥ dn+m(Iα) − dm+1(Qα)

we obtain

dn(Rα : L2[0, 1] → L2[0, 1]) ≥ 1

πnα+ O

(log2 n

nα+1

)

by similar methods.

Compactness properties of certain integral operators

For the sake of completeness we also investigate the strong asymptotic behaviorof the singular values of Rα,α in L2[0, 1]2. Here Rα,α := Rα ⊗ Rα denotes thetwofold fractional integration operator of order α > 0, i.e.,

(Rα,αf )(x, y) := 1

�(α)2

∫ x

0

∫ y

0(x − t)α−1(y − s)α−1f (t, s) ds dt,

0 ≤ x, y ≤ 1 .

Similarly, Iα,α := Iα ⊗ Iα is the twofold integration operator of Weyl type. It hasbeen proved in [1] that

dn(Rα,α − Iα,α : L2[0, 1]2 → L∞[0, 1]2) ≤ c n−α, (4.3)

thus the asymptotic behavior of the singular values of Rα,α is similar to those ofIα,α . Consequently, first we have to investigate the singular values of Iα,α morethoroughly.

Lemma 4.2. For α > 0 and Iα,α in L2[0, 1]2 holds

dn(Iα,α) = 1

π2

(log n

n

)α

+ O

(logα−1 n

nα· log log n

). (4.4)

Proof. The singular values of Iα,α are of the form k−α l−α

π2 for certain k, l ≥ 1.Hence, using a number theoretic estimate in [5], for x ≥ 1, it follows that

#

{n ∈ N : dn(Iα,α) ≥ x−α

π2

}= #

{(k, l) ∈ N × N :

(k l)−α

π2 ≥ x−α

π2

}

= x log x + O(x1/3).

To prove the upper estimate in (4.4) we use

#

{n ∈ N : dn(Iα,α) ≥ x−α

π2

}≤ x log x + c0 · x1/3 (4.5)

for a certain c0 > 0. Given m ∈ N sufficiently large we define x(m) as

x(m) := m

log m(1 + f (m))

where

f (m) := log log m

log m.

Some elementary estimates show that x(m) log x(m) + c0 · x(m)1/3 ≤ m for m

large, hence (4.5) implies

dm(Iα,α) ≤ x(m)−α

π2 = 1

π2

(log m

m

)α

· (1 + f (m))−α

= 1

π2

(log m

m

)α

+ O

((log m

m

)α

f (m)

)

E. Belinsky, W. Linde

proving the right hand side of (4.4).The lower estimate in (4.4) follows in similar way by setting f (m) := (1 +

η)log log m

log mfor some η > 0 starting this time with

#

{n ∈ N : dn(Iα,α) ≥ x−α

π2

}≥ x log x − c0 · x1/3 .

We are now in position to prove the announced strong asymptotic for the sin-

gular values of dn(Rα,α).

Theorem 4.3. Let α > 0. Then it follows that

dn(Rα,α : L2[0, 1]2 → L2[0, 1]2) = 1

π2

(log n

n

)α

+ O

(logβ n

nα

)(4.6)

with β := α2

α+1 < α.

Proof. Using the additivity properties of the Kolmogorov numbers, by Lemma 4.2and by (4.3) for 1 ≤ m < n it follows that

dn(Rα,α) ≤ dn−m(Iα,α) + dm+1(Rα,α − Iα,α)

≤ 1

π2

(log(n − m)

n − m

)α

+ c

(logα−1(n − m)

(n − m)αlog log (n − m) + 1

mα

)

= 1

π2

(log n

n

)α

+ 1

π2

{(log(n − m)

n − m

)α

−(

log n

n

)α}

+c

(logα−1(n − m)

(n − m)αlog log (n − m) + 1

mα

).

Choosing m := nlogγ n

with γ := αα+1 some elementary estimates show that the

error terms in the last estimate are less than c· logβ nnα where as above β = α2

α+1 . Thisgives the upper estimate in (4.6).

To prove the lower estimate we use

dn(Rα,α) ≥ dn+m(Iα,α) − dm+1(Rα,α − Iα,α)

and proceed as in the proof of the upper estimate. The last example gives the asymptotic behavior of the Kolmogorov numbers of

Rα from L1[0, 1] to L2[0, 1]. The exact behavior of these numbers in the periodiccase was found by Ismagilov ([7]).

Theorem 4.4. Let α > 1/2. Then it follows that

dn(Rα : L1[0, 1] → L2[0, 1]) = (2π)−α(2α − 1)−1/2n−α+1/2 + O

(log2 n

nα+1/2

).

Compactness properties of certain integral operators

Proof. The proof follows as that of Theorem 4.1, but based on the following resultin ([7]).

dn(Iα : L01[0, 1] → L2[0, 1]) = 1

(2π)α

( ∞∑k=n+1

k−2α

)1/2

, α > 1/2 .

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