Non–determinism of Linear Operators and Lower Entropy Estimates

The Journal of Fourier Analysis and Applications

Non–determinism of Linear Operators and Lower Entropy Estimates Werner Linde Communicated by Hans G. Feichtinger ABSTRACT. Let u be a (bounded, linear) operator from a Hilbert space H with values in the Banach space C(T ), the space of continuous functions on the compact metric space T . We introduce and investigate numbers τn (u), n ≥ 1, measuring the degree of determinism of the operator u. The slower τn (u) decreases the less determined are functions in the range of u by their values on a certain set of points. It is shown that n−1/2 τn (u) ≤ 2 en (u) where en (u) are the (dyadic) entropy numbers of u. Furthermore, we transform the notion of strong local non–determinism from the language of stochastic processes into that of linear operators. This property together with a lower entropy estimate for the compact space T leads to a lower estimate for τn (u), hence also for en (u). These results are used to prove sharp lower entropy estimates for some integral operators, among them Riemann–Liouville operators with values in C(T ) for some fractal set T . Some multi–dimensional extensions are treated as well.

1.

Introduction and Main Results

Let u be a (bounded, linear) operator from a (real) Hilbert space H with values in the Banach space C(T ), the space of continuous real functions on the compact metric space (T, ρ). Typical examples are integral operators defined on some L2 –space. Assume that u is known to be compact. Then an important question is to describe its degree of compactness. One way to do this is to investigate its (dyadic) entropy numbers en (u) as defined in (1.5) below. The faster en (u) tends to zero (as n tends to infinity) the higher Math Subject Classifications. Primary: 47B06 ; Secondary: 60G15 Keywords and Phrases. Strong local non–determinism, entropy numbers, Riemann– Liouville operators.

c 2004 Birkh¨auser Boston. All rights reserved ° ISSN 1069-5869

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Werner Linde

is the degree of compactness of u. Most interesting are general results on the behavior of en (u), i.e., those which only depend on the size of T and some inner property of the operator u which, moreover, is easy to verify. To illustrate this approach, let us shortly recall a basic result in [3] (cf. also [4] and [22]). Let εn (T ) := inf {ε > 0 : ∃ an ε–net of T with cardinality ≤ n}

(1.1)

be the usual entropy numbers of the compact space (T, ρ). Suppose that εn (T ) ≤ f (n) for some regularly varying function f . If u : H → C(T ) is H–H¨older for some H ∈ (0, 1], i.e., if |(uh)(t) − (uh)(s)| ≤ c · khk · ρ(t, s)H

(1.2)

for all h ∈ H and t, s ∈ T , then it follows that en (u) ≤ c0 n−1/2 f (n)H . In many cases this result leads to sharp upper entropy estimates, yet, of course, it is not optimal in general. For example, take any finite rank operator u satisfying (1.2). This observation motivated our investigation. More precisely, our main objective is to find an inner property of u (similarly as in (1.2)) which, together with a lower estimate for εn (T ), implies a lower estimate for en (u). It turned out, that such an inner property of u already existed within the theory of stochastic processes, namely the so–called property of ”strong local non–determinism” (cf. the survey paper [25] and the references cited there). Because of the well–known tight relation between Gaussian stochastic processes and operators on Hilbert spaces (cf. [8] and [12]), it is not surprising that this property can be transformed into the language of operators, then leading to lower entropy estimates. In order to make this more precise, let us introduce the following notation. Let u be an operator from the Hilbert space H into C(T ). Given sets A, B ⊆ T , we define the quantity λu (A|B) depending on the sizes of the subsets A and B, on their distance and on a certain determinism property of u: λu (A|B) := sup {|(uh)(t)| : t ∈ A , khk ≤ 1 and (uh)(s) = 0 , s ∈ B} . (1.3) With this notation we introduce numbers τn (u) which describe in a certain sense the degree of determinism of the operator u, more precisely, of the set {u(h) : h ∈ H , khk ≤ 1}. The definition was motivated by the technique used in the proof of Lemma 3.3 in [14]. Given n ∈ N we set ( ) ³ ¯ j−1 ´ ¯[ τn (u) := sup min λu Aj ¯ Ai : A1 , . . . , An ⊆ T . (1.4) 1≤j≤n

i=1

Non–determinism of Linear Operators and Lower Entropy Estimates

3

S Here and later on we set 0i=1 Ai := ∅. Furthermore, in view of λu (A|B) = λu (A \ B|B), we can always suppose the sets Aj in (1.4) to be disjoint. Before stating the main result, let us shortly recall the definition of the (dyadic) entropy numbers of an operator u between two Banach spaces E and F . For n ∈ N set   n−1 2[   en (u) := inf ε > 0 : ∃ y1 , . . . , y2n−1 ∈ F, u(BE ) ⊆ (yj + ε BF ) (1.5)   j=1

where BE and BF denote the (closed) unit balls in E and F , respectively. We refer to [4] and [19] for more information about these numbers. Then we shall prove the general lower entropy estimate for operators u from H into C(T ).

Theorem 1. For all u : H → C(T ) it follows that n−1/2 τn (u) ≤ 2 en (u) ,

n = 1, 2, . . . .

Maybe it is of interest to demonstrate on an easy and known example how Theorem 1 applies. For α > 1/2 let Rα be the Riemann–Liouville operator from L2 [0, 1] to C[0, 1] defined by Z t 1 (Rα h)(t) := (t − x)α−1 h(x) dx . (1.6) Γ(α) 0 If A := [a, b] ⊆ [0, 1] and B := [0, a) it is straightforward that n o λRα (A|B) ≥ sup kRα hkC[a,b] : khk ≤ 1 , h(x) = 0 , 0 ≤ x < a . The right hand side equals c · (b − a)α−1/2 , hence we get τn (Rα ) ≥ c · n−α+1/2 which by Theorem 1 implies the well–known estimate en (Rα : L2 [0, 1] → C[0, 1]) ≥ c · n−α . The advantage of this technique is that it does, as we shall see later on, not only apply for Rα with values in C[0, 1], yet also when Rα maps L2 [0, 1] into C(T ) where T is any compact (small) subset of [0, 1]. In order to define the property of ”strong local non–determinism” (SLND) mentioned above, let us fix the notation. For ε > 0 and t ∈ T let Bε (t) := {s ∈ T : ρ(t, s) < ε} be the open ε–ball in the metric space T with center in t. Let ψ : [0, ∞) → [0, ∞) be a non–decreasing continuous function with ψ(0) = 0. An operator u : H → C(T ) satisfies the ψ–SLND property provided that for each t ∈ T and all 0 < ε < ε0 it follows that ¯ ¡ ¢ λu {t} ¯Bε (t)c ≥ ψ(ε). (1.7)

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In different words, for t ∈ T and ε small there is an element h = ht,ε ∈ H with khk ≤ 1, (uh)(s) = 0 if ρ(t, s) ≥ ε and with |(uh)(t)| ≥ ψ(ε). Then we shall prove the general lower entropy estimate announced earlier. Here εn (T ) is as in (1.1).

Theorem 2. Suppose the operator u : H → C(T ) satisfies the ψ–SLND property for some function ψ. Then this implies ψ(εn (T )) ≤ τn (u)

(1.8)

for n ≥ n0 with n0 sufficiently large. Consequently, we get ¡ ¢ n−1/2 ψ εn (T ) ≤ 2 en (u)

(1.9)

for those n ∈ N. Again we demonstrate the usefulness of this result on an easy example. Given H ∈ (0, 1), the operator SH : L2 (R) → C[0, 1] is defined by Z t h i H−1/2 (SH h)(t) := (t − x)H−1/2 − (−x)+ h(x) dx. (1.10) −∞

It is known (cf. [20] or Proposition 12 below) that for each a > 0 the operator SH satisfies the ψ–SLND property on [a, 1] with ψ(ε) = c εH for a certain c > 0. Hence, Theorem 2 applies and leads to the well–known estimate en (SH : L2 (R) → C[0, 1]) ≥ en (SH : L2 (R) → C[a, 1]) ≥ c n−H−1/2 .

2.

Properties of

λu

and

τn

We start with rewriting λu (A|B) as defined in (1.3). To this end denote by u∗ : C(T )∗ → H the dual operator of u and let δt ∈ C(T )∗ be the Dirac measure at point t ∈ T . Then we set HA := span {u∗ (δt ) : t ∈ A} and H∅ := {0}. Clearly, it follows that ⊥ HA = {h ∈ H : (uh)(t) = 0 , t ∈ A} . ⊥ , reBy PA and PA⊥ we denote the orthogonal projection onto HA and HA spectively. Furthermore, if A ⊆ T , define

CA := {f · 1A : f ∈ C(T )} and note that CA = C(A) for closed sets A. Recall that T is assumed to be compact. Thus, if uA h := (uh) · 1A , h ∈ H ,

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Non–determinism of Linear Operators and Lower Entropy Estimates

the operator uA is well–defined from H into CA . With these notations we can easily rewrite λu (A|B) as follows.

Proposition 1. If A, B ⊆ T , then it follows that λu (A|B) = sup dist(u∗ δt , HB )

(2.1)

t∈A

where the distance is taken in H. Moreover, we have ° ° ° ° ° ° ° ° λu (A|B) = °uA PB⊥ : H → CA ° = °uA : PB⊥ (HA ) → CA ° .

(2.2)

Proof. The first equality in (2.2) is obvious and (2.1) then follows from ° ° ° ° ° ° ° ° ° ° ° ° °uA PB⊥ ° = °PB⊥ u∗A ° = sup °PB⊥ u∗ δt ° = sup dist(u∗ δt , HB ). t∈A

t∈A

To ° ⊥prove °the right hand side identity°in (2.2) ° choose t0 ∈ A such that °P u∗ δt ° ≥ δ for some given 0 < δ < °uA P ⊥ °. Setting 0 B B PB⊥ u∗ δt0 ° h0 := ° °P ⊥ u∗ δt ° 0 B we easily get h0 ∈ PB⊥ (HA ), kh0 k ≤ 1 as well as |(uh0 )(t0 )| ≥ δ. Of course, this completes the proof. Let us state a first consequence of the preceding proposition.

Corollary 1. Let u1 and u2 be two operators from Hilbert spaces H1 and H2 into C(T ), respectively, such that hu∗1 δt , u∗1 δs i = hu∗2 δt , u∗2 δs i for all t, s ∈ T . Then for all A, B ⊆ T we have λu1 (A|B) = λu2 (A|B). Consequently, τn (u1 ) = τn (u2 ) and, moreover, u1 possesses the ψ–SLND property if and only u2 does so.

Proof. Observe that

° ) (° n ° ° X ° ° ∗ ∗ λi u δsi ° : λi ∈ R , si ∈ B . dist(u δt , HB ) = inf °u δt − ° ° ∗

i=1

(2.3)

H

Since the inner norm in (2.3) only depends on scalar products of u∗ δt and u∗ δs for suitable t, s ∈ T , this proves the corollary in view of (2.1).

Example 1. Suppose that u : L2 (X , σ) → C(T ) is defined by Z (uh)(t) = K(t, x)h(x) dσ(x) , t ∈ T , X

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with a suitable kernel K on T × X . Then (2.1) implies that in this case λu (A|B) coincides with  ¯  ¯2 1/2   ¯ n  Z ¯¯  X ¯ ¯ ¯   sup inf λj K(tj , x)¯ dσ(x) : tj ∈ T, λj ∈ R . ¯K(t, x) −   t∈A  X¯ ¯  j=1 The next lemma summarizes some obvious properties of λu (A|B) for later use.

Lemma 1. 1. If A1 ⊆ A2 , then for each B ⊆ T we have λu (A2 |B) ≥ λu (A1 |B). 2. On the other hand, for all A ⊆ T and B1 ⊆ B2 it follows that λu (A|B1 ) ≥ λu (A|B2 ).

(2.4)

λu (A|B) = sup λu ({t} |B).

(2.5)

3. It holds t∈A

We now turn to investigating the numbers τn (u) introduced in (1.4). A first result tells us that it suffices to investigate singleton sets A1 , . . . , An in the definition of τn (u).

Proposition 2. For each n ∈ N we have ½ ¾ τn (u) = sup min λu ({tj } | {t1 , . . . , tj−1 }) : t1 , . . . , tn ∈ T 1≤j≤n

(2.6)

where as before the right hand set is supposed to be empty for j = 1.

Proof. Take any δ < τn (u) and choose sets A1 , . . . , An ⊆ T with ³ ¯ j−1 ´ ¯[ min λu Aj ¯ Ai > δ.

1≤j≤n

i=1

Of course, we may assume these sets to be disjoint. Using (2.5), in a first step we find t1 ∈ A1 with λu ({t1 } |∅) > δ. Next observe that (2.4) implies λu (A2 | {t1 }) ≥ λu (A2 |A1 ) > δ , hence using (2.5) again we find a t2 ∈ A2 such that λu ({t2 } | {t1 }) > δ.

Non–determinism of Linear Operators and Lower Entropy Estimates

7

Proceeding further in the same way until n it follows that the right hand side in (2.6) is greater or equal than τn (u). Since the opposite inequality holds by the definition, this completes the proof. In view of Proposition 1 the preceding proposition shows that τn (u) describes some linear separation property of the set Au := {u∗ (δt ) : t ∈ T } in H, i.e., the following is valid.

Corollary 2. The quantity τn (u) coincides with ½ ¾ ¡ ¢ sup min dist {gj } , span {g1 , . . . , gj−1 } : g1 , . . . , gn ∈ Au . 1≤j≤n

(2.7)

Remark 1. A similar expression as in (2.7) (with the convex hull instead of the linear one) was recently used very successfully in [1]. Our next goal is to verify certain properties of the numbers τn (u). In particular, it turns out that they are tightly related with the Gelfand numbers cn (u) which in our situation coincide with the approximation numbers an (u). We refer to [17] or [18] for the definition of these numbers.

Proposition 3. Let u : H → C(T ) be some operator. 1. It holds kuk = τ1 (u) ≥ τ2 (u) ≥ · · · ≥ 0

(2.8)

and, if rank(u) < n, then this implies τn (u) = 0. 2. We have an (u) = cn (u) ≤ τn (u) ,

(2.9)

yet, in general, these quantities are not equivalent. 3. The operator u is compact if and only if limn→∞ τn (u) = 0.

Proof. Property (2.8) is obvious, thus we omit its proof. Suppose now τn (u) > 0. By Corollary 2 this implies rank(u∗ ) ≥ n, hence rank(u) ≥ n as well. This proves the second assertion. To verify (2.9) suppose first that τn−1 (u) > τn (u) for some n ≥ 2. Note that by τ1 (u) = kuk = c1 (u) there is nothing to prove for n = 1. By the assumption, there are t1 , . . . , tn−1 ∈ T such that for 1 ≤ j ≤ n − 1 we have λu ({tj } | {t1 , . . . , tj−1 }) > τn (u) and, moreover, for any t ∈ T it follows that λu ({t} | {t1 , . . . , tn−1 }) ≤ τn (u). This implies ° © ° ª⊥ ° ° → C(T )° ≤ τn (u) , °u : u∗ (δt1 ), . . . , u∗ (δtn−1 ) hence, by the definition of the Gelfand numbers (cf. [17], 11.5.6), we derive cn (u) ≤ τn (u) as asserted.

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Let now n ≥ 2 be arbitrary and let n0 ≤ n be the minimal number with τn0 (u) = τn (u). If n0 = 1, then we obtain τn (u) = kuk, hence the assertion follows by cn (u) ≤ kuk = τn (u). For n0 > 1 we get τn0 (u) < τn0 −1 (u), hence, by the first step this implies cn (u) ≤ cn0 (u) ≤ τn0 (u) = τn (u) and this completes the proof in that case as well. It remains to show that, in general, τn (u) cannot be estimated by a multiple of cn (u). To this end let T be a discrete set of cardinality m. Then m , i.e., with Rm endowed with the sup–norm. Let u C(T ) coincides with l∞ m m where `m is the usual Euclidean be the natural embedding from `m into ` ∞ 2 2 Rm . By a result of B. S. Stechkin (cf. [18], 2.9.11) it holds ¶ µ m − n + 1 1/2 , n = 1, . . . , m , cn (um ) = m while τ1 (um ) = · · · = τm (um ) = 1. This being true for any m ≥ 1 shows that there is no universal constant κ > 0 such that τn (u) ≤ κ cn (u) for all operators u. To verify the third property, let us first assume limn→∞ τn (u) = 0. Because of (2.9) this implies limn→∞ an (u) = 0, hence u is compact. If, conversely, τn (u) > η > 0 for all n ≥ 1, then by Corollary 2 for each n ≥ 1 there are g1 , . . . , gn ∈ Au = {u∗ (δt ) : t ∈ T } such that kgi − gj k ≥ η, 1 ≤ i < j ≤ n. Consequently, u∗ cannot be compact, hence u also cannot be a compact operator. This completes the proof.

Remark 2. As shown above, in general τn (u) cannot be estimated by a multiple of cn (u) = an (u). Using more involved properties of approximation numbers of diagonal operators (cf. [17] 11.11.7 and 11.7.4) it is not very difficult to construct a diagonal operator D from `2 to `∞ satisfying lim supn→∞ τn (D)/an (D) = ∞. Note that τn (D) = σn for a diagonal operator D : `2 → `∞ generated by a non–increasing positive sequence (σn )n≥1 tending to zero.

3.

Proof of the Main Theorems

We start with two auxiliary results. First we construct orthogonal subspaces of H generated in natural way by disjoint subsets in T . Thus take arbitrary disjoint subsets A1 , . . . , An of T and set B1 := ∅ and Bj := A1 ∪ · · · ∪ Aj−1 , 2 ≤ j ≤ n. Then the following holds.

Lemma 2. If Hj := PB⊥j (HAj ) ,

1≤j≤n,

Non–determinism of Linear Operators and Lower Entropy Estimates

9

then H1 , . . . , Hn are orthogonal subspaces of H.

Proof. Assume k < j. Then by the construction we have Ak ⊆ Bj as well ⊥ ⊆ H⊥ as well as H⊥ ⊆ H⊥ , hence as Bk ⊆ Bj , which clearly implies HB Ak Bj Bk j ⊥ as well as H ⊆ H⊥ . Thus, if t ∈ A and h ∈ H , it follows that Hj ⊆ HA j j k Bk k E D PB⊥k u∗ δt , h = hu∗ δt − PBk u∗ δt , hi = 0 because u∗ δt ∈ HAk and PBk u∗ δt ∈ HBk . By the definition of HAj this completes the proof.

Remark 3. In view of (2.2) the preceding lemma implies the following. Suppose that τn (u) > δ. Then there are disjoint subsets A1 , . . . , An ⊆ T and orthogonal subspaces H1 , . . . , Hn in H such that ° ° °u : Hj → CA ° ≥ δ , 1 ≤ j ≤ n , j and, moreover, (uh)(s) = 0

whenever h ∈ Hj

and s ∈ Ai ,

1 ≤ i < j ≤ n.

The following result seems to be well–known, at least to experts. Yet, since we could not find a reference, we include its short proof. ¡ ¢n Proposition 4. Let S = σij i,j=1 be an n × n–matrix of real numbers such that σij = 0 whenever 1 ≤ i < j ≤ n, i.e., S is a triangular matrix. Regard S as operator from `n2 into `n∞ . Then it follows that  1/n n Y ≤ 2 en (S). n−1/2 ·  |σjj | j=1

Proof. First regard S as operator from `n2 into `n2 . Let B2n be the closed unit ball and suppose that for some N ≥ 1 and ε > 0 we have S(B2n )

⊆

N [

{yj + ε B2n }

j=1

with certain y1 , . . . , yN ∈ voln (B2n )

·

n Y

`n2 .

This implies

¡ ¢ |σjj | = voln S(B2n ) ≤ N · εn · voln (B2n ) ,

j=1

¢1/n ¡ Qn Q . thus we necessarily have nj=1 |σjj | ≤ N · εn . Set ε := 2−1 j=1 |σjj | n n n−1 Then it follows that N ≥ 2 , i.e., S(B2 ) cannot be covered by 2 balls of radius ε as chosen before. Thus we finally arrive at  1/n n Y en (S : `n2 → `n2 ) ≥ 2−1 ·  |σjj | . (3.1) j=1

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Werner Linde

To complete the proof write [S : `n2 → `n2 ] = [idn : `n∞ → `n2 ] ◦ [S : `n2 → `n∞ ]. Here idn is the identity map from `n∞ to `n2 . Then the desired estimate follows from (3.1), from kidn k = n1/2 and by en (S : `n2 → `n2 ) ≤ kidn k · en (S : `n2 → `n∞ ).

Proof of Proposition 1. Let δ > 0 be an arbitrary number with δ < τn (u) which was defined in (1.4). By Remark 3 we find orthogonal subspaces Hj of H and disjoint subsets Aj ⊆ T , 1 ≤ j ≤ n, possessing the properties stated there. Choose hj ∈ Hj with khj k ≤ 1 and tj ∈ Aj such that |(uhj )(tj )| > δ ,

1 ≤ j ≤ n.

(3.2)

Note that (uhj )(ti ) = 0 provided 1 ≤ i < j ≤ n. Next we define operators Jn : `n2 → H and Qn : C(T ) → `n∞ by Jn (x) :=

n X

xj hj ,

x = (x1 , . . . , xn ) ,

j=1

and

¡ ¢n Qn (f ) := f (tj ) j=1 ,

f ∈ C(T ).

Of course, we have kJn k ≤ 1 as well as kQn k ≤ 1. Define Sn : `n2 → `n∞ via Sn := Qn ◦ u ◦ Jn and set σij := (uhj )(ti ) ,

1 ≤ i, j ≤ n.

Then the operator Sn has the matrix representation (σij )nij=1 . By the choice of hj and tj this is a triangular matrix, thus we are in the situation of Proposition 4 . Using this together with (3.2) it follows that 1/n  n Y · n−1/2 ≥ 2−1 · δ · n−1/2 . |σjj | en (Sn ) ≥ 2−1  j=1

Since δ < τn (u) was arbitrary, this completes the proof of Theorem 1 in view of en (Sn ) ≤ kQn k · en (u) · kJn k ≤ en (u). Before proving Theorem 2 let us introduce the notion of packing numbers of a compact metric space (T, ρ): δn (T ) := sup {ε > 0 : ∃ t1 , . . . , tn+1 ∈ T , ρ(ti , tj ) ≥ ε , i 6= j} .

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Non–determinism of Linear Operators and Lower Entropy Estimates

Note that, as easily seen, we have δn (T )/2 ≤ εn (T ) ≤ δn (T ).

(3.3)

Proof of Proposition 2. Choose n0 ∈ N such that δn (T ) ≤ ε0 whenever n ≥ n0 . Here ε0 is the number such that (1.7) holds for 0 < ε < ε0 . Take now an arbitrary ε < δn (T ). Then there are t1 , . . . , tn+1 ∈ T with pairwise distance bigger or equal to ε. Thus, if Bε (tj ) is the open ε–ball centered at tj , then we derive ti ∈ Bε (tj )c for all i 6= j. Hence, by (2.4) and by the definition of the ψ–SLND property it follows that λu ({tj } | {t1 , . . . , tj−1 }) ≥ λu ({tj } |Bε (tj )c ) ≥ ψ(ε) for j = 1, . . . , n + 1. Consequently, τn+1 (u) ≥ ψ(ε), thus using (3.3), by the properties of ψ (non–decreasing and continuous) we finally derive τn (u) ≥ τn+1 (u) ≥ ψ(δn (T )) ≥ ψ(εn (T )). This proves (1.8) and in view of Theorem 1 we get (1.9) as well. This completes the proof of Theorem 2 .

4. 4.1

Examples and applications

Volterra type operators

Our first objective is to apply Theorem 1 to Volterra type operators defined as follows. Let T ⊆ [0, 1] be some compact subset and let K be a kernel on T × [0, 1] such that Z t (vh)(t) := K(t, x)h(x) dx , t ∈ T , (4.1) 0

is well–defined for h ∈ L2 [0, 1] and bounded as operator into C(T ). Then we set  ÃZ !1/2    tj 2 |K(tj , x)| dx τ˜n (v) := sup min 1≤j≤n  tj−1 where the supremum is taken over all 0 = t0 < t1 < · · · < tn ≤ 1 , tj ∈ T . With this notation the following holds.

Proposition 5. Let v be defined by (4.1) Then for n ≥ 1 it follows that n−1/2 · τ˜n (v) ≤ 2 en (v : L2 [0, 1] → C(T )).

(4.2)

Proof. If there exist t1 , . . . , tn ∈ T with 0 < t1 < · · · < tn ≤ 1 and with min λv ({tj } | {t1 , . . . , tj−1 }) ≥ δ ,

1≤j≤n

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then necessarily τn (v) ≥ δ. Yet, whenever the tj in T are in increasing order, then for each j ≥ 2 this implies λv ({tj } | {t1 , . . . , tj−1 }) ¯ ½¯Z tj ¾ ¯ ¯ = sup ¯¯ K(tj , x)h(x) dx¯¯ : (vh)(ti ) = 0 , i < j , khk ≤ 1 ¯ ) (¯Z0 ¯ tj ¯ ¯ ¯ K(tj , x)h(x) dx¯ : h(x) = 0 , x ∈ [0, tj−1 ] , khk ≤ 1 ≥ sup ¯ ¯ tj−1 ¯ ÃZ !1/2 tj

= tj−1

|K(tj , x)|2 dx

Since also

. µZ

λv ({t1 } |∅) ≥

0

t1

¶1/2 2

|K(t1 , x)| dx

,

we get τn (v) ≥ τ˜n (v) for operators v as in (4.1). Consequently, (4.2) follows by Theorem 1

Example 2. Let K(t, x) := (t − x)α−1 for some α > 1/2, i.e., up to a + constant, v generated by this kernel is the Riemann–Liouville operator Rα introduced in (1.6). Since for that kernel ÃZ

!1/2

tj

2

tj−1

|K(tj , x)| dx

= (2α − 1)−1/2 · (tj − tj−1 )α−1/2 ,

(4.3)

we get the following proposition.

Proposition 6. Let Rα be defined by (1.6). Then it follows that en (Rα : L2 [0, 1] → C(T )) ≥ cα n−1/2 εn (T )α−1/2

(4.4)

with cα = 2−1 (2α − 1)−1/2 Γ(α)−1 .

Proof. Choose any δ < εn (T ). By (3.3) there are 0 ≤ t1 < · · · < tn+1 ≤ 1 in T such that tj+1 − tj ≥ δ for 1 ≤ j ≤ n. In a first case we suppose that t1 ≥ δ. Then (4.3) implies τ˜n+1 (Rα ) ≥ Cα · δ α−1/2 , where Cα = (2α − 1)−1/2 Γ(α)−1 . Hence Proposition 5 applies and leads to en+1 (Rα : L2 [0, 1] → C(T )) ≥ (Cα /2) · n−1/2 · δ α−1/2 . This being true for any δ < εn+1 (T ) proves the proposition in this case. In the case t1 < δ, we set t˜1 := t2 up to t˜n := tn+1 . Doing so, we are in the first case, yet this time with only n elements. Hence, the assertion

Non–determinism of Linear Operators and Lower Entropy Estimates

13

follows as before with τ˜n (Rα ) instead of τ˜n+1 (Rα ). This completes the proof.

Remark 4. Proposition 6 answers Question 2 in [15], p. 223. In the case of the Kolmogorov numbers dn (Rα ) estimate (4.4) was proved in [15] under some additional regularity assumptions about εn (T ). Note that the upper estimate for dn (Rα ) with exactly the same expression as in (4.4) holds as well, cf. Theorem 1.2 in [15]. Corollary 3. Let T ⊆ [0, 1] be a self–similar set satisfying the strong open set condition (cf. [5] for the definition). Let D ∈ (0, 1] be the Hausdorff dimension of T . Then it follows that en (Rα : L2 [0, 1] → C(T )) ≈ n−1/2 n−(α−1/2)/D .

Proof. The assumptions about T imply that εn (T ) ≈ n−1/D (cf. [10]). Hence the assertion is a direct consequence of Proposition 6 and of Theorem 1.3 in [15] . Example 3. Suppose now −1/2

K(t, x) := (t − x)+

· |log(t − x)+ |−β for some β > 1/2 ,

and let vβ be the corresponding operator on L2 [0, 1] with values in C(T ) for some closed set T ⊆ [0, 1]. Then we get ÃZ

!1/2

tj

= (1 − 2β)−1/2 · |log(tj − tj−1 )|1/2 −β .

|K(tj , x)| dx

tj−1

Using exactly the same arguments as in the proof of Proposition 6, implies the following.

Proposition 7. If for β > 1/2 the operator vβ is defined by Z (vβ h)(t) :=

0

t

h(x) (t − x)1/2 |log(t − x)|β

dx ,

then it follows that ¯ ¡ ¢¯1/2 −β en (vβ : L2 [0, 1] → C(T )) ≥ c n−1/2 · ¯log εn+1 (T ) ¯ .

(4.5)

In particular, en (vβ : L2 [0, 1] → C[0, 1]) ≥ c n−1/2 · (log n)1/2 −β .

(4.6)

Problem 1. It seems to be open whether or estimates (4.5) and/or (4.6) are sharp.

14

4.2

Werner Linde

Multi–dimensional fractional integration operators

Given α = (α1 , . . . , αN ) with αj > 1/2 the kernel Kα on [0, 1]N × [0, 1]N is defined by Kα (t, x) :=

N Y

Kj (tj , xj ) ,

t = (t1 , . . . , tN ) , x = (x1 , . . . , xN ) ,

j=1

with Kj (s, y) :=

1 α −1 (s − y)+j , Γ(αj )

0 ≤ s, y ≤ 1.

The corresponding multi–dimensional fractional integration operator RαN from L2 [0, 1]N to C[0, 1]N is then defined by Z N (Rα h)(t) := Kα (t, x) h(x) dx. (4.7) [0,1]N

Proposition 8. Suppose that α1 = · · · = αν < αν+1 ≤ · · · ≤ αN . Then it follows that τn (RαN ) ≥ c n−α1 +1/2 (1 + log n)α1 (ν−1) . (4.8) Proof. This is a direct consequence of (2.9) and of Proposition 5.8 in [9] . Remark 5. Let us rephrase the preceding proposition. For each n ≥ 1 there are points t1 , . . . , tn in [0, 1]N such that for all j ≤ n ¡ © ª¢ dist Kα (tj , · ), span Kα (ti , · ) , i < j ≥ c n−α1 +1/2 (1 + log n)α1 (ν−1) . Here as before the distance is taken in L2 [0, 1]N . Surprisingly, even in the easiest case α1 = · · · = αN = 1, N ≥ 2, i.e., in the case of the N –dimensional integration operator, we were not able to construct those points t1 , . . . , tn directly.

Problem 2. We do not know whether or not the estimate (4.8) is sharp. This question is tightly related to the classical open problem of the behavior of en (RαN ) for ν ≥ 3. Recall that the case ν = 2 was treated in [23] and [2]. We conjecture that the right power of the log–term in (4.8) is α1 (ν −1)+1/2 provided that ν ≥ 2. A positive answer to that question would also answer the problem of entropy numbers. Next we turn to an operator tightly related to RαN . Given H = (H1 , . . . , HN ) with 0 < Hj < 1 (the vectors H and α are related via Hj = αj − 1/2) we define a kernel K H on [0, 1]N × RN by K H (t, x) :=

N Y j=1

K j (tj , xj ) ,

t = (t1 , . . . , tN ) , x = (x1 , . . . , xN ) ,

Non–determinism of Linear Operators and Lower Entropy Estimates

15

where H −1/2

K j (s, y) := (s − y)+ j

H −1/2

− (−y)+ j

,

0 ≤ s ≤ 1, y ∈ R .

The corresponding operator from L2 (RN ) to C[0, 1]N is defined as in (4.7) N . Observe that S 1 = S as defined in (1.10). and denoted by SH H H

Proposition 9. Suppose 0 < H1 = · · · = Hν < Hν+1 ≤ · · · ≤ HN < 1. Then it follows that N τn (SH ) ≥ c n−H1 (1 + log n)(H1 +1/2)(ν−1) .

Proof. Use (2.9) as well as Corollary 7.6 in [8]. 4.3

Integral operators related to fractional motions

N Let LC 2 (R ) be the (complex) Hilbert space of square integrable complex N valued functions on RN . Let E ⊆ LC 2 (R ) be the (real) Hilbert space consisting of functions h satisfying h(x) = h(−x) for x ∈ RN . Given a number H ∈ (0, 1) and a compact subset T ⊆ RN we define now an operator N from E into the (real) space C(T ) as follows: BH Z ³ ´ dx N (BH h)(t) := eiht,xi − 1 h(x) H+N/2 , t ∈ T. N |x| R

Here | · | denotes the Euclidean distance in RN . It is easy to see that indeed N is a bounded operator from E into the space of real–valued continuous BH functions over T .

Proposition 10. Let 0 < a < 1 be any fixed number and suppose that N as operator from E into C(T ) satisfies the |t| ≥ a for all t ∈ T . Then BH ψ–SLND property with ψ(ε) = c εH for a certain c > 0 only depending on H and N and for 0 < ε < a. Proof. The proof follows the ideas developed in [7], Ch. © 18, and in [24]. ª Let ϕ be a fixed function in C0∞ (RN ) with supp(ϕ) ⊆ x ∈ RN : |x| ≤ 1 and with ϕ(0) = 1. Take t ∈ T and ε ∈ (0, a) and define a function ht,ε by ht,ε (x) := Cϕ εH+N ϕ(εx) ˆ e−i ht,xi |x|H+N/2 , where

µZ Cϕ :=

Here

2

2H+N

|ϕ(x)| ˆ |x|

¶−1/2 dx

RN

1 ϕ(x) ˆ = (2π)N/2

Z e−i hx,si ϕ(s) ds RN

x ∈ RN

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Werner Linde

denotes the usual Fourier transform of ϕ. Note that Cϕ > 0 because of ϕˆ ∈ S(RN ). First observe that by the choice of Cϕ we get µZ ¶1/2 2 kht,ε k2 ≤ Cϕ |ϕ(x)| ˆ |x|2H+N dx =1 (4.9) RN

for t ∈ T and ε > 0. Easy calculations lead to ½ µ ¶ µ ¶¾ s−t t = Cϕ (2π) ε ϕ −ϕ − ε ε ¶ µ s−t = Cϕ (2π)N/2 εH ϕ ε © ª where we used ε < a ≤ |t| and supp(ϕ) ⊆ x ∈ RN : |x| ≤ 1 . Consequently, N h )(s) = 0 and, whenever |t − s| ≥ ε, by the same argument we derive (BH t,ε moreover, because of ϕ(0) = 1 it follows N (BH ht,ε )(s)

N/2 H

N (BH ht,ε )(t) = Cϕ (2π)N/2 εH .

(4.10)

Summing up, whenever 0 < ε < a and t ∈ T , hence |t| ≥ a, then by (4.9), N h )(s) = 0 whenever |t − s| ≥ ε we get (4.10) and by (BH t,ε ¯ N ¯ λB N ({t} |Bε (t)c ) ≥ ¯(BH ht,ε (t)¯ ≥ c εH H

where c := Cϕ (2π)N/2 . This completes the proof. N . First, if Let us introduce two other operators tightly related to BH N N 0 < H < 1 with H 6= 1/2 for N = 1 define VH from L2 (R ) to C(T ) (as above T is a compact subset of RN ) by Z h i N (VH h)(t) := |t − x|H−N/2 − |x|H−N/2 h(x) dx , t ∈ T. RN

Proposition 11. Let 0 < a < 1 be any fixed number and suppose that |t| ≥ a for all t ∈ T . Then VHN as operator from L2 (RN ) into C(T ) satisfies the ψ–SLND property with ψ(ε) = c εH for a certain c > 0. Proof. By the results in [16] we have with some universal cH > 0 ­ N ∗ ® ­ ® N ∗ (BH ) (δt ), (BH ) (δs ) = cH (VHN )∗ (δt ), (VHN )∗ (δs ) for all t, s ∈ T . Here the scalar product on the left hand side is taken in E defined above while on the right hand side is the one in L2 (RN ). Thus the assertion follows by Corollary 1 and by Proposition 10 . In the one–dimensional case another operator is of interest. Namely, let SH be defined as in (1.10). Then it is well–known (cf. [6] or [21]) that ­ 1 ∗ ® 1 ∗ (BH ) (δt ), (BH ) (δs ) = cH h(SH )∗ (δt ), (SH )∗ (δs )i

Non–determinism of Linear Operators and Lower Entropy Estimates

17

for all t, s ≥ 0. Consequently, by the same arguments as before we get the following.

Proposition 12. Let 0 < a < b be two fixed numbers. Then SH as operator from L2 (R) into C([a, b]) satisfies the ψ–SLND property with ψ(ε) = c εH for a certain c > 0. N from E into C(T ) for some Our next aim is to apply Theorem 2 to BH N compact subset T in RN . Recall that h ∈ LC 2 (R ) belongs to E if and only if h(x) = h(−x). Some problem arises because Proposition 10 only holds for those t ∈ T with |t| ≥ a for a certain a > 0. Thus if 0 ∈ T some extra considerations are necessary. To overcome this difficulty we have to add a rank one operator. More precisely, the following is valid.

Proposition 13. Let T ⊆ RN be a compact set. Then there is an operator F from L2 (RN ) into C(T ) of rank at most one such that for all n ≥ 1 N en (BH + F : L2 (RN ) → C(T )) ≥ c n−1/2 εn (T )H .

Proof. Suppose first that dist(0, T ) > diam(T ) (here the distance is the Euclidean one in RN ). In view of Proposition 10 it follows that λB N ({t} |Bε (t)c ) ≥ c εH H

provided that 0 < ε < diam(T ). By assumption δ1 (T ) = diam(T ), thus by the proof of Theorem 2 we derive N τn (BH ) ≥ c εn (T )H

for all n ≥ 1 and the assertion follows by Theorem 1 choosing F = 0. Let now T be an arbitrary compact subset of RN . Then we choose a t0 ∈ RN such that for T0 := t0 + T the above relation between the distance N to zero and its diameter holds. Let E ⊆ LC 2 (R ) be as above and let I : E → E be the isometry given by (Ih)(x) := e−iht0 ,xi h(x). To make the N mapping E following calculations precise, denote by B 0 the operator BH into C(T0 ). Finally, we define an isometry J : C(T0 ) → C(T ) by setting (Jf )(t) := f (t + t0 ) if t ∈ T . Then, if h ∈ E and t ∈ T we obtain Z eiht+t0 ,xi − 1 −iht0 ,xi 0 (J B Ih)(t) = ·e h(x) dx |x|H+N/2 RN Z Z e−iht0 ,xi − 1 eiht,xi − 1 h(x) dx − h(x) dx = H+N/2 H+N/2 RN |x| RN |x| N = (BH h)(t) + (F h)(t)

where the rank one operator F is defined by N (F h)(t) := −(BH h)(−t0 ) · 1.

18

Werner Linde

Since I and J are isometries, the first step leads to N en (BH + F ) = en (B 0 ) ≥ c n−1/2 εn (T0 )H = c n−1/2 εn (T )H

which completes the proof.

Corollary 4. Suppose that a compact set T ⊆ RN satisfies εn (T ) ≥ f (n) for some regularly varying positive function f on R. Then this implies N en (BH ) ≥ c n−1/2 f (n)H .

The same assertion holds for VHN and if N = 1 also for SH .

Proof. This follows easily from Proposition 13 using that the entropy numbers of finite rank operators tend to zero exponentially (cf. [4]).

5.

Concluding Remarks

Let X = (X(t))t∈T be an arbitrary centered Gaussian process over an index set T . We assume that σ 2 := sup |X(t)|2 < ∞.

(5.1)

t∈T

Then similarly as in (1.4) we may define quantities τn (X) measuring the degree of determinism of the paths of X. We set ½ ¾ 1/2 τn (X) := sup min [V (X(tj )|X(t1 ), . . . , X(tj−1 ))] : t1 , . . . , tn ∈ T 1≤j≤n

where, as usual, the conditional variance of a random variable X with respect to a random vector Z is defined by V(X|Z) := E [X − E(X|Z)]2 . Of course, with σ defined by (5.1) we have σ = τ1 (X) ≥ τ2 (X) ≥ · · · ≥ 0. Moreover, it is not difficult to see that limn→∞ τn (X) = 0 for a.s. bounded X. Recall that a Gaussian process X is a.s. bounded (cf. [11]) if and only if ½ ¾ sup E sup |X(t)|2 : T0 ⊆ T , card(T0 ) < ∞ < ∞. t∈T0

Suppose now that T is a compact metric space, let H be a separable Hilbert space and let u be an operator from H to C(T ) such that for some (or, equivalently, all) orthonormal basis (fk )k≥1 in H the sum ∞ X k=1

ξk u(fk )

Non–determinism of Linear Operators and Lower Entropy Estimates

19

converges a.s. in C(T ). Here the ξk are independent standard normal random variables. Then this operator u generates a centered Gaussian process X on T by ∞ X X(t) := ξk (ufk )(t) , t ∈ T. (5.2) k=1

Note that each centered Gaussian process X with a.s. continuous paths can be represented as in (5.2) with suitable H and u. It is easy to see that in this case we have τn (u) = τn (X). Thus, for example, Proposition 9 is equivalent to the following: N Proposition 14. Let BN H = (BH (t))t∈[0,1]N be the (N, 1)–fractional Brownian sheet with Hurst index H = (H1 , . . . , HN ) as defined in [24] or in [25] . Then it follows that −H1 τn (BN (1 + log n)(H1 +1/2)(ν−1) H) ≥ c n

where as before ν is the multiplicity of H1 and the Hj are ordered in increasing order. Let us state another consequence of τn (u) = τn (X). If we combine Theorem 1 with Theorem 5.1 in [12] we get the following.

Proposition 15. Suppose that τn (X) ≥ c n−a (1 + log n)β for some a > 0 and β ∈ R. Then this implies µ ¶ − log P sup |X(t)| < ε ≥ c0 ε−1/a log(1/ε)β/a . t∈T

More applications to Gaussian processes are possible and will be subject of a separate paper.

Acknowledgments The author is very grateful to his student Helga Schack for her help to precise the proof of Proposition 10. Furthermore we thank Thomas K¨ uhn for the discussion about Remark 2. Finally our thank goes to Yimin Xiao for sending us some of his recent papers about the property of local non– determinism.

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[3] Carl, B., Heinrich, S., and K¨ uhn, Th. (1988). s-numbers of integral operators with H¨ older continuous kernels over metric compacta, J. Funct. Anal., 81, 54-73. [4] Carl, B., and Stephani, I. (1990). Entropy, Compactness and Approximation of Operators, Cambridge Univ. Press, Cambridge. [5] Falconer, K. J. (1985). The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge. [6] Herbin, E. (2006). From N parameter fractional Brownian motions to N parameter multifractional Brownian motions, Rocky Mountain J. Math., 36, 1249–1284. [7] Kahane, J. P. (1985). Some Random Series of Functions, Cambridge Univ. Press, Cambridge. [8] Kuelbs J., and Li, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal., 116, 133-157. [9] K¨ uhn, Th., and Linde, W. (2002). Optimal series representation of fractional Brownian sheets, Bernoulli, 8, 669-696. [10] Lalley, S. P. (1988). The packing and covering functions of some self–similar fractals, Indiana Univ. Math. J., 37, 699–709. [11] Ledoux, M., and Talagrand, M. (1991). Probability in Banach Spaces, Springer, Berlin. [12] Li, W. V., and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab., 27, 1556–1578. [13] Lifshits, M. A., Linde, W., and Shi, Z. (2006). Small deviations for Riemann-Liouville processes in Lq -spaces with respect to fractal measures, Proc. London Math. Soc., 92, 224–250. [14] Lifshits, M. A., Linde, W., and Shi, Z. (2006). Small deviations of Gaussian random fields in Lq –spaces, Electron. J. Probab., 11, 1204–1233. [15] Linde, W. (2004). Kolmogorov numbers of Riemann–Liouville operators over small sets and applications to Gaussian processes, J. Appr. Theory, 128, 207–233. [16] Lindstrøm, T. (1993). Fractional Brownian fields as integrals of white noise, Bull. London Math. Soc., 25, 83–88. [17] Pietsch, A. (1978). Operator Ideals, Verlag der Wissenschaften, Berlin. [18] Pietsch, A. (1987) Eigenvalues and s–Numbers, Cambridge Univ. Press, Cambridge. [19] Pisier, G. (1989). The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge. [20] Pitt, L. D. (1978). Local times for Gaussian vector fields, Indiana Univ. Math. J., 27, 309–330. [21] Samorodnitsky, G., and M. S. Taqqu, M. S. (1994). Stable non–Gaussian Random Processes, Chapman & Hall, New York. [22] Steinwart, I. (2000). Entropy of C(K)–valued operators, J. Approx. Theory, 103, 302–328. [23] Talagrand, M. (1994). The small ball problem for the Brownian sheet, Ann. Probab., 22, 1331–1354. [24] Wu, D., and Xiao, Y. (2007). Geometric properties of fractional Brownian sheets, J. Fourier Anal. Appl., 13, 1–37. [25] Xiao, Y. (2006). Properties of local non–determinism of Gaussian and stable random fields and their applications, Ann. Fac. Sci. Toulouse Math., 15, 157–193.

Received , Revision received Fakult¨ at f¨ ur Mathematik und Informatik, FSU Jena, Ernst–Abbe–Platz 2 07743 Jena, Germany e-mail: [email protected]