ENTROPY NUMBERS OF CERTAIN SUMMATION OPERATORS

Transcription

1 Georgian Mathematical Journal Volume , Number 2, ENTROPY NUMBERS OF CERTAIN SUMMATION OPERATORS J. CREUTZIG AND W. LINDE Abstract. Given nonnegative real sequences a = α k and b = β k we study the generated summation operator [ ] S a,b x := α k β l x l, x = x k, l<k regarded as a mapping from l p Z to l q Z. We give necessary and sufficient conditions for the boundedness of S a,b and prove optimal estimates for its entropy numbers relative to the summation properties of a and b. Our results are applied to the investigation of the behaviour of P α q k W t k q < ε q and P sup α k W t k < ε as ε 0, where t k is some nondecreasing sequence in [0, and W t t 0 denotes the Wiener process Mathematics Subject Classification: Primary: 47B06. Secondary: 47B37, 60G15. Key words and phrases: Summation operator, entropy numbers, small ball probabilities. 1. Introduction Volterra integral operators are known to play an important role within Functional Analysis as well as Probability Theory. A special class of interest is that of so-called weighted Volterra integral operators T ρ,ψ mapping L p 0, into L q 0, and defined by s T ρ,ψ : f ρs 0 ψtft dt, 1.1 where ρ, ψ 0 are suitable functions on 0,. The main task is to describe properties of T ρ,ψ boundedness, degree of compactness, etc. in terms of certain properties of the weight functions ρ and ψ. Although several results have been proved in this direction [12], [5], [6], [10], interesting problems remain open. For example, we do not know of a complete characterization of ρ s and ψ s such that the approximation or entropy numbers of T ρ,ψ behave exactly as those of T 1 on [0, 1]. T 1 = T 1,1 denotes the ordinary integral operator. ISSN X / $8.00 / c Heldermann Verlag

2 246 J. CREUTZIG AND W. LINDE Given sequences a = α k and b = β k, the summation operator S a,b : l p Z l q Z defined by S a,b : x = x k α k β j x j 1.2 can be viewed as a discrete counterpart to weighted Volterra operators. Thus one may expect for S a,b similar compactness properties relative to a and b as for T ρ,ψ relative to ρ and ψ. This is indeed so as long as one considers upper estimates for the entropy numbers e n S a,b. Here the most surprising result Theorem 2.3 asserts that summation operators can behave as badly and irregularly as weighted integral operators. Yet the situation becomes completely different when considering lower estimates for e n S a,b. The deeper reason is as follows: If ρ ψ > 0 on some interval some set of positive measure suffices, then the entropy numbers of T ρ,ψ cannot tend to zero faster than those of T 1 considered on this interval. Since e n T 1 n 1, this order is a natural lower bound for e n T ρ,ψ. In the case of summation operators such a canonical operator as T 1 for T ρ,ψ does not exist. Hence it is not surprising that e n S a,b can tend to zero much faster than n 1. The aim of this paper is to investigate these phenomena more precisely. We state optimal conditions for a and b in order that sup n n e n S a,b <. More precise statements can be formulated under some additional regularity assumptions on a and b, e.g., monotonicity or an exponential decay of the β k s. As is well-known, the ordinary Volterra operator T 1 is closely related to Brownian motion. More precisely, let ξ k k=1 be an i.i.d. sequence of standard normal distributed random variables and let f k k=1 be some orthonormal basis in L 2 0,. Then j<k W t = ξ k T 1 f k t, t 0, 1.3 k=1 is a standard Wiener process over 0,. In this sense, summation operators S a,b : l 2 l q generate Gaussian sequences X = X k l q Z with X k = α k W t k, k Z, 1.4 where t k s are defined via β 2 k = t k+1 t k. Recent results [9], [11] relate the entropy behaviour of an operator with estimates for the probability of small balls of the generated Gaussian random variable. We transform the entropy results proved for S a,b into the small ball estimates for X = X k defined above. Hence we find sufficient conditions for the α k s and t k s such that lim ε 0 ε2 log P X q < ε = These conditions turn out to be in this language the best possible ones. The paper is organized as follows. In Section 2, the main results about the entropy of S a,b are stated. Section 3 provides basic tools and well-known facts,

3 ENTROPY OF SUMMATION OPERATORS 247 which are used in Section 4 to prove our results. Section 5 is devoted to the study of some examples. Finally, in Section 6 we establish some small ball estimates for Gaussian vectors as in 1.4, using our entropy results for S a,b proved before. 2. Notation and Main Results For a given sequence x = x k R and p [1, ] set x p := 1/p x k p if p <, x := sup x k, k= and let l p Z := {x : x p < }. As usual, for p [1, ] the adjoint p is given by 1/p + 1/p = 1. Now, let p, q [1, ] be arbitrary. For nonnegative sequences a = α k and b = β k satisfying 1/q 1/p A k := α q l < and Bk := β p l < 2.1 l k for all k Z with obvious modifications for q = or p = 1, the expression [ ] S a,b x := α k β l x l 2.2 l<k is well-defined for any x = x k l p Z. Our first result is a version of the well-known Maz ja-rosin Theorem see, e.g., [12], pp , characterizing the boundedness of S a,b in terms of A k and B k given in 2.1: Theorem 2.1. Under the above assumptions, the operator S a,b is well-defined and bounded from l p Z to l q Z iff Da, b <, where sup[a k B k ] for p q, Da, b := p p q q [A k B k ] pq p q β p pq 2.3 k 1 for p > q. Moreover, there is a universal C p,q > 0 with 1 C p,q Da, b S a,b : l p Z l q Z C p,q Da, b. 2.4 Our next aim is to describe the compactness of S a,b in terms of entropy numbers. For the introduction of entropy numbers, let E, F be Banach spaces with unit balls B E, B F. If T : E F is a linear bounded operator, we set the dyadic entropy numbers of T to be { e n T := inf ε > 0 : y 1,..., y 2 n 1 F s. t. T B E l<k } yk + εb F, where + means the Minkowski sum. It is well-known that T is compact iff e n T 0, so that it makes sense to consider the speed of decay of e n T as a measure for the compactness of T. 2 n 1 k=1

4 248 J. CREUTZIG AND W. LINDE To avoid some technical and notational problems, from now on it is assumed that p > 1, hence p <. In the following, the number r > 0 defined by 1/r := 1/q + 1/p 2.5 will play a crucial role. Since p > 1, we always have r <. Later on, we will use the monotone rearrangement δk of a sequence δ k. By this we mean that the sequence δk k 0 is the nonincreasing rearrangement of δ k k 0, and that δk k 1 is the nondecreasing rearrangement of δ k k 1 i.e., δ k k 1 is the nonincreasing rearrangement of δ k k 1. For the nonnegative sequences a and b satisfying 2.1, we define { v k := inf m Z : } β p l 2 k 2.6 l m for any k Z, where inf :=, and with these v k s we set v k+1 1/q 2 k/p α q l if v k <, δ k a, b := l=v k +1 0 for v k =. 2.7 Here, as well as in the following, the empty sum has to be read as 0. With r given by 2.5 we define a, b r := 1/r δ k a, b r = δ k a, b r 2.8 and It is easy to see that a, b r, := sup k 1/r δk 1a, b + δ ka, b. 2.9 k 1 α l β l 1 l Z r a, b r and a, b r, c a, b r. On the other hand, there are easy examples such that a, b r, < and α l β l 1 l Z r =, and vice versa, i.e., these two expressions are not comparable. Let us formulate now the main results of this paper. We have the following general upper estimate for the entropy numbers of S a,b : Theorem 2.2. Let S a,b map l p Z into l q Z. 1 There is a numerical constant c > 0 such that sup n e n S a,b c a, b r Whenever the right hand side of 2.10 is finite, we have lim n e ns a,b =

5 ENTROPY OF SUMMATION OPERATORS 249 The next result shows that estimate 2.10 cannot in general be improved to an estimate neither against a, b r, nor against α l β l 1 l Z r: Theorem 2.3. For any sequence d k > 0, k = 1, 2,... satisfying k d q k < and k d r k = there are sequences a = α k k 1 and b = β k k 1 such that: 1 The operator S a,b : l p l q is bounded. 2 [α k+1 β k ] r <. k=1 v k+1 3 d k = δ k a, b = 2 k/p 4 sup n e n S a,b =. l=v k +1 α q l 1/q for all k 1 On the other hand, under some additional regularity assumptions about a and b assertion 2.11 holds even under weaker conditions. More precisely, the following is valid. Proposition 2.4. For a, b given, assume that α q k /βp k is monotone near ± if q <, or that α k is monotone near ± if q =. Then the condition α l β l 1 l Z r < implies lim n e ns a,b = 0. For sequences b which do not increase too fast e.g., not superexponentially at +, and do not decrease too fast at, we have the following lower estimate for the entropy numbers of S a,b. Theorem 2.5. Assume that there is m 1 such that {k Z : v k = v k } m 2.12 for all k Z. Then, for ρ > 0 with 1/s := ρ 1/p + 1/q > 0 we have sup n ρ e n Sa,b : l p l q c sup n 1/s δn 1a, b + δ na, b 2.13 with c > 0 only depending on p, q, m and ρ. In particular, if ρ = 1, then Moreover, for ρ and s in 2.13 we have sup n e n Sa,b : l p l q c a, b r, lim n ρ e n Sa,b : l p l q c lim n 1/s δn 1a, b + δ na, b Remark. Note that for v k <, k Z, i.e., for l Z β p l =, condition 2.12 is equivalent to for some γ > 0. β k γ l<k 1/p β p l, k Z,

6 250 J. CREUTZIG AND W. LINDE Of course, condition 2.12 is violated provided that a β l = 0 if l l 0 for some l 0 Z or b β p l <. l Z Here we have the following weaker variant of Theorem 2.5. Proposition 2.6. Suppose that there are some k 0, k 1 Z such that in case a condition 2.12 holds for all k k 0 or in case b for all k k 1. Then if 1/s = ρ 1/p + 1/q > 0, from sup n ρ e n S a,b : l p l q < we conclude that sup n 1/s δn 1a, b + δ na, b < and 2.15 holds for S a,b as well. Note that in case a, δ n a, b = 0 for n n 0, and in case b, δ n 1 a, b = 0 for n n 1 with certain n 0, n 1 Z. Examples see Proposition 5.3 show that Theorem 2.5 and Proposition 2.6 become false without an additional regularity assumption. 3. Basic Tools 3.1. Connection to Integral Operators. Let ρ, ψ : 0, [0, be measurable functions with ρ L q x, and ψ L p 0, x 3.1 for any x 0,. Then for f L p 0, the function Tρ,ψ f s s := ρs 0 ψtft dt 3.2 is well-defined. The operator T ρ,ψ is called the Volterra integral operator induced by ρ and ψ. There is a close connection between summation and integral operators. Generally speaking, one can always consider a summation operator as part of a special Volterra integral operator. This allows one to use, for these operators, results established in [10]. Let us denote k := [2 k, 2 k+1 for k Z, and set I k := 2k = [2 2k, 2 2k+1 and J k := 2k+1 = [2 2k+1, 2 2k For any set X 0,, let 1 X be the indicator function of X. We denote by l 0 Z := R Z the space of all real sequences and set L 0 0, to be the space of all equivalence classes of measurable functions. It is easily checked that for any p [1, ] the mappings Φ p I : l 0 Z L 0 0,, x k x k 1 Ik I k 1/p 3.4

7 ENTROPY OF SUMMATION OPERATORS 251 and Φ p J : l 0 Z L 0 0,, x k x k 1 Jk J k 1/p 3.5 induce isometric embeddings of l p Z into L p 0, for p = let, as usual, 1/p = 0. Given now sequences a = α k and b = β k, we set i.e., we have ρ := Φ q Ia and ψ := Φ p J b, 3.6 ρ = α k 1 Ik I k 1/q and ψ = β k 1 Jk J k 1/p. 3.7 The connection between summation and integral operators reads as follows: Proposition 3.1. In the above setting, ρ and ψ defined in 3.6 satisfy 3.1 if and only if a and b satisfy 2.1, and it holds T ρ,ψ Φ p J = Φ q I S a,b. 3.8 Moreover, T ρ,ψ is bounded from L p to L q iff S a,b is so from l p to l q. In this case there is an operator Q : L p 0, l p Z with Q 1 such that for all f L p 0, we have In particular, T ρ,ψ f q = S a,b Qf q. 3.9 T ρ,ψ : L p L q = S a,b : l p l q Proof. The equivalence of 3.1 and 2.1 is clear by the definition of ρ and ψ and of Φ q I and Φ p J, respectively. We easily compute T ρ,ψ Φ p Jx k = [ α k I k 1/q β l x l ]1 Ik = Φ q IS a,b x k. l<k This verifies that 3.8 holds. If T ρ,ψ is bounded, then, due to 3.8, so is the operator S a,b, and S a,b T ρ,ψ. Conversely, if S a,b is bounded and f L p 0,, set Qf := J k 1/p ft dt Due to Hölder s inequality, Qf l p with Qf p f p, and, additionally, we find T ρ,ψ f q = α q q k β l J l 1/p ft dt Sa,b q = Qf q q l<k This proves 3.9, and implies, in particular, that T ρ,ψ S a,b which completes the proof. As a simple corollary we have J k J l

8 252 J. CREUTZIG AND W. LINDE Corollary 3.2. For a and b satisfying 2.1, and ρ and ψ defined as in 3.7 we have 1/2 e n T ρ,ψ e n S a,b 2 e n T ρ,ψ. Proof. First note that e n T 2 e n J T for any operator T : E F and any isometric embedding J : F F 0. Hence, by 3.8 we conclude e n S a,b 2 e n Φ q I S a,b = 2 e n T ρ,ψ Φ p J 2 e n T ρ,ψ Φ p J 2 e n T ρ,ψ. On the other hand, by 3.9 we have use Lemma 4.2 in [10], yet observe that there the entropy numbers were defined slightly different so that additional factor 2 had to be added e n T ρ,ψ 2 e n S a,b Q 2 e n S a,b Q 2 e n S a,b, which completes the proof. For later use we cite now some of the main results in [10] about weighted Volterra integral operators. Let ρ and ψ now be arbitrary nonnegative measurable functions on 0, satisfying 3.1. For s > 0, set Rs := ρ Lq s, and Ψs := ψ Lp 0,s The following version of the Maz ja Rosin Theorem can be found in [10], Theorem 6.1: Theorem 3.3. Let 1 p, q and ρ, ψ 0 on 0,. Then T ρ,ψ is bounded from L p 0, into L q 0, iff Dρ, ψ <, where sup RsΨs if p q, s>0 Dρ, ψ := [ Rs Ψs p /q ] p q pq pq 3.14 p q ψs p ds for p > q. 0 Moreover, there are universal constants c p,q, C p,q > 0 such that c p,q Dρ, ψ T ρ,ψ : L p L q C p,q Dρ, ψ Now we turn to upper estimates for the entropy numbers of T ρ,ψ. Therefore set { s u k := inf s > 0 : ψt p dt 2 }, k k Z, 3.16 and define uk+1 δ k ρ, ψ := 2 k/p 0 u k 1/q. ψt q dt 3.17

9 ENTROPY OF SUMMATION OPERATORS 253 Similarly to the setting for a and b, with r given by 2.5 define and ρ, ψ r := δ k ρ, ψ 1/r r = δ k a, b r 3.18 ρ, ψ r, := sup k 1/r δka, b + δ k+1a, b k 1 With this notation we have [10], Theorem 4.6: Theorem 3.4. Let p > 1 and 1 q. Then for all ρ, ψ as above, the following statements are valid. 1 For T ρ,ψ as in 3.2, we have sup n e n T ρ,ψ C ρ, ψ r. n 2 Whenever ρ, ψ r <, we even have lim n e nt ρ,ψ C ρ ψ r Remarks. 1 There exist functions ρ and ψ cf. [10], Theorem 2.3 with ρ ψ r < such that lim n e n T ρ,ψ =. But note that since these ρ s and ψ s are not of form 3.7 with suitable sequences a and b, hence they cannot be used to construct similar examples for summation operators. However to prove Theorem 2.3 the general ideas may be taken over in order. 2 In [10] lower estimates for e n T ρ,ψ were also proved. But unfortunately all these lower bounds turn out to be zero in the case of disjointly supported functions ρ and ψ. So they do not provide any information about lower bounds for e n S a,b Entropy Numbers of Diagonal Operators. For a given sequence σ = σ k k 1 of nonnegative numbers, one defines the diagonal operator D σ x := σ k x k k N, x = x k k N Considered as a mapping from l p N to l q N, the entropy of these operators can be estimated by means of σ k s. As an example, we have the following special case of general results in [1] or [13]. Theorem 3.5. Let p, q [1, ] be arbitrary, and let σ = σ k k 0 be a nonnegative bounded sequence of real numbers. Let ρ > 0 satisfy 1/s := ρ 1/p + 1/q > 0. Then there are constants c 1, c 2 > 0 depending on p, q and ρ only such that c 1 sup n 1/s σ n sup For the lower bound, one even has Let us draw an easy conclusion out of this: n ρ e n Dσ : l p N l q N c 2 sup n 1/s σn n 1/q 1/p σ n 6 e n D σ. 3.23

10 254 J. CREUTZIG AND W. LINDE Corollary 3.6. In the above setting, as well as holds. c 1 lim n 1/s σn lim n ρ e n D σ 3.24 lim nρ e n D σ 2 c 2 lim n 1/s σn 3.25 Proof. Without loss of generality, assume that σ n s are already nonincreasing, e.g., σ n = σ n. Then estimate 3.24 is an immediate consequence of To prove 3.25, fix a number N N and set D N x i i=1 := σ 1 x 1,..., σ N x N, 0,..., D N 0 := D σ D N. Because of e 2n 1 D σ e n D N + e n D N 0 and the exponential decay of e n D N as n cf. [2], , we get lim nρ e n D σ 2 lim n ρ e n D N 0 2 sup n ρ e n D0 N Now D0 N is not a diagonal operator with nonincreasing entries, but this difficulty can be easily overcome. For x = x i i N let S Nx := x i+n i N and S + Nx := 0,..., 0, x 1,... be the operators of shifting N times to the right or to the left in l p N or l q N, respectively. If D N 1 x i i=1 := x 1 σ N+1, x 2 σ N+2,... is the diagonal operator defined by σ N+i i 1, we clearly have D N 0 = S + N D N 1 S N and D N 1 = S N D N 0 S + N, and therefore note that both shift operators have norm one it holds by the multiplicity of the entropy numbers But for D N 1 e n D N 0 = e n D N 1. Theorem 3.5 applies and leads to sup n ρ e n D1 N c 2 sup n 1/s σ n+n with s > 0 defined in this theorem. Since n 1/s n + N 1/s, this yields sup n ρ e n D0 N c 2 supn + N 1/s σ n+n = c 2 sup m 1/s σ m m>n Now we combine 3.26 and 3.27 to find that lim nρ e n D σ 2c 2 sup n 1/s σ n 3.28 n>n

11 ENTROPY OF SUMMATION OPERATORS 255 holds for all N N. Taking the infimum over the right side of 3.28, we end up with lim nρ e n D σ 2c 2 inf sup n 1/s σ n = 2c 2 lim n 1/s σ n, N N n>n which proves It is of more use for us to consider diagonal operators mapping l p Z into l q Z, generated, say, by a sequence σ k. However these operators are isomorphic to the product of two diagonal operators generated by σ k 1 k 1 and σ k k 1. It is then quite easy to establish Corollary 3.7. Let p, q [1, ] be arbitrary, and let σ = σ k be a nonnegative bounded sequence of real numbers. For given ρ > 0 assume 1/s := ρ 1/p + 1/q > 0. Then there are constants c 1, c 2 > 0 depending only on p, q and ρ such that c 1 sup and, moreover, as well as n 1/s σ n 1 + σ n sup lim n ρ e n D σ c 1 lim nρ e n D σ c 2 n ρ e n D σ c 2 sup n 1/s σn 1 + σ n 3.29 lim n 1/s σn 1 + σ n lim n 1/s σn 1 + σ n. 4. Proof of the Results Let us start with the proof of Theorem 2.1. Proof of Theorem 2.1. Let a = α k and b = β k be given, satisfying 2.1, and define ρ and ψ by means of 3.7. Then Rs and Ψs are given by Because of Proposition 3.1 and Theorem 3.3 we have the estimate 1 C p,q Dρ, ψ S a,b : l p Z l q Z C pq, Dρ, ψ with Dρ, ψ given in So it remains to show that with Da, b from 2.3 we have c 1 Dρ, ψ Da, b c 2 Dρ, ψ. 4.1 Let us treat the case p q first. Take any s I k for some k Z. Then on the one hand we have Rs A k, while, on the other hand, Ψs = B k. Now assume s J k for some k Z. Then Rs = A k+1, while Ψs B k+1. Combining both cases, we find sup s>0 Rs Ψs sup [A k B k ]. 4.2 Since A k B k = R2 2k Ψ2 2k, we even have equality in 4.2, which proves 4.1 for p q.

12 256 J. CREUTZIG AND W. LINDE Let us now assume p > q. First note that the integral in the definition of Dρ, ψ has only to be taken over J k. But if s J k 1, then Rs = A k while Ψs B k, hence recall ψs = β k 1 J k 1 1/p Dρ, ψ Da, b. Conversely, whenever s [2 2k+3/2, 2 2k+2 ] J k, we have Rs = A k+1, while Ψs Consequently, we obtain J k yielding p q [ Rs Ψs p pq pq /q ] p q ψs p ds which proves 4.2 in this case as well. 1/p B p k β p k 2 1/p B k+1. Da, b 2 1/q Dρ, ψ, 2 1/q [ A k+1 B p /q k+1 pq p q β p k ] p q pq, Proof of Theorem 2.2. Let a, b be given, and ρ, ψ be defined via 3.7. In view of Proposition 3.1 and Theorem 3.4 we have to show only that δ k a, b = δ k ρ, ψ, 4.3 where δ k s are defined in 2.7 and 3.17, respectively. But for v k as in 2.6 and u k as in 3.16, we have u k J vk = [2 2k+1, 2 2k+2 ]. Inserting this into the definition of δ k s yields 4.3, and hence Theorem 2.2. Proof of Theorem 2.3. We treat the case q < first. Let d k be given with d q k < and d r k =. 4.4 k N We assume 0 < d k 1. For any k N, we can find s k N such that k N s k 1 d q k 2s k. 4.5 According to our assumptions, it is possible to find a partition K m m N of N with inf K m+1 = sup K m + 1, and k K m d r k m r. 4.6 Set ν m := max d 1 k k K m Further we define. Then for any k K m there are n k N such that ν m d k r n k 2ν m d k r. 4.7 γ k,j := l<k1 + 2s l n k + j s k 4.8

13 ENTROPY OF SUMMATION OPERATORS 257 for k N and j = 1,..., n k + 1. In particular, γ 1,1 = 0 and γ k,nk +1 = γ k+1,1. Now we are ready to define a partition of N suitable for our purposes: For any k N and j = 1,..., n k set as well as and Clearly, A + k,j = A k,j = s k, and A + k,j := {γ k,j + 1,..., γ k,j + s k }, 4.9 I k,j := {γ k,j + s k + 1} 4.10 A k,j := {γ k,j + s k + 2,..., γ k,j+1 } A + k,1 I k,1 A k,1 A+ k,2 A k,n k A + k+1,1, 4.12 where denotes the natural ordering of intervals in N. Denoting n k U k := A + k,j I k,j A k,j = {γk,1 + 1,..., γ k+1,1 }, 4.13 j=1 it is clear that N = k 1 U k. Let us define β 0 := 1, and 2 k 1/p β l := for l U k n 1/p k 1 + 2s k 1/p This way we ensure note that U k = n k 1 + 2s k that for v k s defined in 2.6 we have v k = γ k,1 for k 1 and v k = 0 if k 0 so that U k = {v k + 1,..., v k+1 }, k Now, let us set 2 k/p d k if l I α l := n 1/q k,j for some k N, j = 1,..., n k, k 0 otherwise. We can easily compute that v k+1 l=v k +1 a q l = n k l= kq/p d q k n k = 2 kq/p d q k In particular, we know a = α l l Z l q Z, and δ k a, b = d k and hence 3 holds. If l I k,j, then l 1 U k, which implies r n k 2 k/p d k αl β l 1 = l N k N j=1 n 1/q k n 1/p k 2 k 1/p 1 + 2s k 1/p r d r k n k k N n k 1 + 2s k r/p k N dk /2s k 1/p r.

14 258 J. CREUTZIG AND W. LINDE Because of the definition of s k we assured 2s k d q k, and hence we can estimate further r αl β l 1 d 1+q/p r k = d q k <, 4.18 k N l N k N i.e., 2 is valid. Using Theorem 2.1 one easily checks that S a,b is bounded, i.e.,1 is satisfied as well. So it remains to verify 4. To do so, let us fix k N and j {1,..., n k }. Then consider the vector x k,j = x k,j l l Z given by s 1/p x k,j k if l A + k,j, l := s 1/p k for l A k,j, otherwise. Recall that A + k,j = A k,j = s k, so x k,j p 2. Setting y k,j := S a,b x k,j, we have d s 1/p k y k,j k l = n 1/r k 1 + 2s if l I k,j k 1/p otherwise. For m N, set N m := k K m n k. Let us denote by e k,j the standard basis of l Nm p or l Nm q, respectively, where k runs through K m and j through 1,..., n k. Then we define two mappings, an injection and a quasi-projection X m : l N m p l p Z, e k,j x k,j, 4.21 Y m : l q Z l N m q, e l We have X m 2 and Y m 1. Additionally, e k,j if l I k,j, 0 otherwise where D σ m Y m S a,b X m = D σ m, 4.23 is the diagonal operator generated by the sequence σ m k,j = s 1/p k n 1/r d k n 1/r k 1 + 2s k 1/p k 3 1/p d k 1/6 ν 1 m Here we have used the definition of n k. It is well-known cf. [13] or 3.23 that Ne N id : l N p l N q 1/6 N 1/r, 4.25 so, putting things together and using 4.23, 4.24 and 4.25 we get N m e Nm S a,b 1/6 νm 1 N m e Nm id : l N m p l N m q c ν 1 m N 1/r m. 4.26

15 ENTROPY OF SUMMATION OPERATORS 259 But due to the definition of n k we have which, inserted in 4.26, yields, with regard to 4.6, N m νm r d r k, 4.27 k K m 1/r N m e Nm S a,b c d r k c m, 4.28 k K m so that lim n e n S a,b =. Let now q =, so r = p. This means we have sup n d n < while n d p n =. We adapt the construction above, changing only the definition of variables where q is involved. So define s k := 2 k, and set α l := 2 k/p d k for l I k,j, while all other variables are defined as above they have different values now, though. It is trivial to see that α l l and that δ k a, b = d k. Further, we estimate l Z[α l β l 1 ] p = k N n k j=1 d p k n k 1 + 2s k p sup d p n 2 kp <. k N As above, Theorem 2.1 shows the boundedness of S a,b. The construction of X m and Y m remains exactly the same, and since 4.25 remains valid for q =, the arguments given above apply also here, so 4.28 holds in this case, too. Next we want to prove Proposition 2.4. For this it clearly suffices to show the following: Lemma 4.1. Assume α q l /βp l 1 to be monotone near ± if q <, or α k to be monotone near ± for q =. Then α l β l 1 l Z l r always implies a, b r <. Remark. It is notable that in the setting of Lemma 4.1, one cannot establish the estimate a, b r c α l β l 1 l r, as simple examples show. Proof of Lemma 4.1. Let us first assume q <. Since all possible situations can be treated similarly, we treat, e.g., the case where α q l /βp l 1 is monotonically increasing near, say, for k > k 0. So let k > k 0 be given with v k+2 < and v k v k+1. Then, α q l β p l 1 αq v k+1 β vk+1 1 p holds for any l {v k +1,..., v k+1 } and hence v δ k a, b r k+1 = 2 kr/p α q l l=v k +1 r/q 2 kr/p v k+1 1 β p l l=v k r/q α r v k+1 β rp /q v k k+1 αr v k β rp /q v k+1 1

16 260 J. CREUTZIG AND W. LINDE On the other hand, for any l {v k+1 + 1,..., v k+2 + 1} we have of course α q l β p l 1 αq v k+1 +1β vk+1 p, which implies v k+2 +1 l=v k+1 +1 α r l β r l 1 v k+2 β p l l=v k+1 We combine 4.29 and 4.30 to see that δ k a, b r αv r k+1 +1 αr k+1 v 2 k β rp /q v k+1 β rp /q v k+1 1 v k+2 +1 l=v k+1 +1 α r l β r l holds for all k > k 0 with v k+2 < and v k < v k+1. In order to prove that k>k 0 δ k a, b r <, in view of 4.31 it suffices to show that with K := {k Z : k > k 0 and v k < v k+1 v k+2 < } we have k K v k+2 +1 αl r βl 1 r 2 αl r βl 1. r 4.32 l=v k+1 +1 l Z This takes place if every summand appearing in the right-hand sum appears at most two times in the sums on the left-hand side. Thus we have to show only that for any l Z the set M l := { k Z : v k v k+1, v k l v k } consists of at most two elements. So assume that k 1, k 2, k 3 M l with k 1 < k 2 < k 3. Since v k3 < v k3 +1, we get v k1 +2 v k2 +1 v k3 < v k But by the definition of M l we must have v k3 +1 l 1 v k1 +2, which is a contradiction. So we know M l 2, and 4.32 is shown. Let us now consider the case q =, so r = p. Again, we demonstrate only the summability in +, provided that α k is monotonically increasing for k > k 0. The arguments above have only to be modified slightly: For any k > k 0 with v k < v k+1 we have while v k+2 +1 l=v k+1 +1 δ k a, b p = 2 k sup v k <l v k+1 α p l = 2 k α p v k+1, v α p l βp l 1 k+2 αp v k+1 +1 β p l αv p k+1 2 k+1. l=v k+1 So we conclude that it suffices to verify 4.32 for the case q =, which is proved completely analogously as above for the case q <. Next we prove Theorem 2.5. To do so we construct suitable embeddings of diagonal operators. Let m N be a number for which the regularity condition 2.12 holds. For l Z we set l + 2m + 4 Z := {l + 2m + 4k : k Z} 4.34

17 ENTROPY OF SUMMATION OPERATORS 261 as usual. If l {0,..., 2m + 3}, then we define π l : Z l + 2m + 4 Z, π l k := l + 2m + 4k Now let D l be a diagonal operator generated by the sequence With these operators, we have v πlk+1 σk l := δ πl ka, b = 2 π lk/p j=v πl k+1 α q j 1/q Proposition 4.2. For any l {0,..., 2m + 3}, there is a bounded linear operator X l : l p Z l p Z with X l C m and for all x l p Z. D l x 2 S a,b X l x 4.37 Before we prove Proposition 4.2, let us mention a useful lemma: Lemma 4.3. Assume v k v k+1. Then 2 k 1 v k+1 1 j=v k 1 β p l 2 k Proof. The upper estimate is trivial. For the lower estimate, combine l 2 k with l < 2 k 1. β p l l<v k+1 l v k β p l<v k 1 β p Proof of Proposition 4.2. We prove the proposition for the case l = 0, the other cases can be treated analogously. Let e k be the standard base of l p Z. We define the mapping X 0 by setting X 0 e k := x k, where x k s are defined below. The construction guarantees that the vectors y k := S a,b x k are disjointly supported, that x k p c, and moreover y k 1/2σk 0 with σk 0 defined in 4.36 for l = 0. From this, one can directly conclude the assertion. So let k Z be given, and set k 0 := 2m + 4k = π 0 k. If v k0 = v k0 +1, then σk 0 = δ k0 a, b = 0, and we set x k := 0. Otherwise, we distinguish two cases: Case 1: v k0 1 v k0. Then we fix k 1 {k 0 + m + 2, k 0 + 2m + 2} such that v k1 v k1 +1 this is possible due to our assumption Since k 1 1 > k m, 2.12 also shows that v k1 1 > v k0 +1. Now we define x k = x k j j Z via β p 1 j if j {v k0 2,..., v k0 1}, x k j := 2 k 0/p C k β p 1 j for j {v k1 1,..., v k1 +1 1}, otherwise, where C k := v k0 1 j=v k0 2 v k β p l β p l 4.40 j=v k1 1

18 262 J. CREUTZIG AND W. LINDE Lemma 4.3 tells us that C k [2 2m 4, 2 m+1 ]. It is easy to obtain x k p p = 2 k 0 [ v k0 1 j=v k0 2 β p j + C p k v k j=v k1 1 ] β p j k0+1 2 k0+2m+1 2 2m+2. Let y k = yl k l Z := S a,b x k. We will not calculate all values of yl k s, since it is only important to note that yl k 0, and 0 if l < v k0 2, v k0 yl k = 2 k0/p 1 α l β p j when l {v k0 + 1,..., v k0 +1}, 4.41 j=v k0 2 0 for l v k1 +1. Case 2: v k0 = v k0 1. In this case, because of v k0 < v k0 +1, we know that 2 k 0 1 β p v k0 2 k Let k 1 be fixed like in case 1, and set β p 1 v k0 for j = v k0, x k j := 2 k 0/p C k β p 1 j for j {v k1 1,..., v k1 +1 1}, 0 otherwise, where this time C k := β p vk1+1 1 v k0 j=v k β p l 4.44 Because of 4.42 and Lemma 4.3, we know that C k [2 2m 4, 2 m+1 ]. Again, it is easy to obtain x k p 2 2m+2/p. Setting y k := S a,b x k, we conclude that 0 if l < v k0, yl k = 2 k0/p α l β vk0 when l {v k0 + 1,..., v k0 +1}, for l v k1 +1. So in both cases we have defined our vectors x k. Since k 1 k 0 + 2m + 2 π 0 k + 1 2, we easily see that the y k s are as disjointly supported as the x k s are with v k0 +1 y k q 2 k 0/p 1 α q j j=v k0 +1 1/q = 1/2σ 0 k Taking into account y k = S a,b X 0 e k, we conclude that 4.37 holds. Before we can finally prove Theorem 2.5, a technical difficulty has to be fixed. Note that for a subsequence σ πn of a sequence σ n, the sequence σ πn n cannot, in general, be estimated from below by σ n, e.g. σ πn n could be constantly 0

19 ENTROPY OF SUMMATION OPERATORS 263 and σ n be 1. In particular, for the sequences σ l defined in 4.36, σ l n cannot be estimated from below by δ na, b, but fortunately this is possible for the maximum: Lemma 4.4. Let σ n n 0 be a nonnegative sequence tending to zero, and let σ π1 n,..., σ πn n be partial sequences of σ n, where the π j s satisfy N j=1 π j N = N. Then for any s > 0 we have and for all k N sup n 1/s σn 2N 1/s max sup n 1/s σ πj n n, 4.47 j=1,...,n holds. Nk 1/s σkn 2 N 1/s k 1/s max σ π j n k 4.48 j=1,...,n Proof. For x 0 denote x := sup{n N : n x}. Given n N, there are k 1,..., k n N with σ ki σn/2. For l := n/n + 1 we conclude that there is j N such that π j N {k 1,..., k n } l. Consequently, for this j we have σ πj n l σn/2, and since n/l N, we find n 1/s σ n 2 n/l 1/s l 1/s σ πj n l 2N 1/s l 1/s σ πj n n, which implies The second assertion is analogous. Proof of Theorem 2.5. Using Lemma 4.2 from [10], it is easy to deduce from Proposition 4.2 and Corollary 3.7 that sup n ρ e n S a,b 1/4 max sup l 2m+3 1/4 max sup l 2m+3 n ρ e n D l n 1/s σ l n + σ l n with D l and σ l defined in On the other hand, by Lemma 4.4 with δ n := δ n a, b we get sup n 1/s δ n 1 + δ n sup n 1/s δn 1 + sup n 1/s δ n C max sup n 1/s σ l n 1 l=0,...,2m+3 + C max sup n 1/s σ l n l=0,...,2m+3 2 C max n 1/s σ l n 1 + σ l n. l=0,...,2m+3 sup Combined with 4.49, this shows the first assertion. The last estimate is based on the fact that for a n, b n 0 we have sup n a n + b n sup n a n + sup n b n 2 supa n + b n. n

20 264 J. CREUTZIG AND W. LINDE Such estimates cannot be established for the infimum. So for estimate 2.15 we have to be a little more careful. First we define two subspaces of l p Z, namely l + p := { } x k l p Z : x k = 0 for k < 0, 4.50 as well as l p := { x k l p Z : x k = 0 if k 0, } β j x j = j<0 The corresponding restrictions of S a,b are denoted analogously, e.g., S + a,b := S a,b l + p, S a,b := S a,b l p We observe that Im S a,b + Im S a,b = {0}. It is easy to deduce that e n S + a,b + e ns a,b C p,q e n S a,b A careful inspection of the proof of Proposition 4.2 reveals that X l e k l + p k 0, and that X l e k l p if k < 1. As above, this yields e n S a,b + C max e n D σ 0 l 2m+3 lkk 0 if and e n S a,b C max e n D σ. 0 l 2m+3 l kk 2 From this we easily derive with the help of 3.23 and 4.48 that n ρ e n S + a,b C n1/s δ 2m+4n and n ρ e n Sa,b C n 1/s δ k k 2m m+4n But it is not hard to see that to continue the estimate 4.55 into δ k k 2m+4 δ 22m+4n, which allows 2m+4n n ρ e n S a,b C n 1/s δ 22m+4n Combined with 4.54 and 4.53, this reveals n ρ e n S a,b C n δ 1/s 22m+4n 1 + δ 22m+4n. From this we are directly led to 2.15.

21 ENTROPY OF SUMMATION OPERATORS Examples and Special Cases At first we will study polynomial growth of the α k s and β k s: Let us assume that for α, β R we have α k = k α and β k = k β if k 1, and α k = 0 = β k otherwise. First of all, 2.1 is satisfied iff α > 1/q for q <, and iff α 0 when q =. For A k and B k given by 2.1 we find that then A k k α+1/q while k β+1/p, β < 1/p, B k log k, β = 1/p, c, β > 1/p. 5.1 Assume first p q. Then Theorem 2.1 tells us that S a,b is bounded iff A k B k is so; by treating the three cases given in 5.1 separately and regarding 1/p 1/r α one can easily check that S a,b is bounded from l p to l q iff α + β 1/r. For p > q, the condition in Theorem 2.1 seems to be more complicated; however, using 5.1 one can verify that in this case we can estimate [A k B p /q k ] pq p q β p k 1 k pq p q 1 α β, β < 1/p, k pq k pq p q 1/q α 1 log k q p 1, β = 1/p, p q 1/q α βp, β > 1/p. pq 5.2 For β 1/p this is always summable, while for β < 1/p we infer that S a,b is bounded from l p to l q iff pq 1 α β < 1, which is equivalent to α+β > 1/r. p q Since 1/p > 1/r α we have q < in this case, we find that for p > q the operator S a,b is bounded iff α + β > 1/r. Next we wish to calculate δ k a, b. For this purpose we assume from now on that β < 1/p. Then it is easy to see that for v k as in 2.6 we have v k 2 k/1 βp, and consequently, δ k a, b 2 k/p 2 k+1/1 βp j=2 k/1 βp j αq 1/q 2 k1/q α/1 βp +1/p. Since it is easily verified that 1/q α/1 βp < 1/p if and only if β > 1/r α, we derive that S a,b is compact iff β > 1/r α, and in this case, we have an exponential decrease of δ k a, b. Through easy direct calculations one verifies that δ k a, b decrease even more rapidly for β 1/p. Summing up our results and using Theorem 2.2, we verify the following proposition: Then Proposition 5.1. For α, β R, define S α,β : l p N l q N by S α,β : x = x k k N k α k 1 l=1 l β x l k N.

22 266 J. CREUTZIG AND W. LINDE i S α,β is bounded from l p to l q iff α 0 and α + β 1/r if q =, α > 1/q and α + β 1/r if p q <, α > 1/q and α + β > 1/r for p > q; and ii S α,β is compact iff α > 1/q α 0, if q =, and α + β > 1/r. In this case, we have lim ne n S α,β = 0. As a second class of interest we investigate the case that β k s grow exponentially: Proposition 5.2. Let a = α k be arbitrary, and let β k = 2 k/p, k Z. Then we have and c 1 a, b r, sup n e n S a,b c 2 a, b r, 5.3 c 1 lim n 1/r τ n lim n e n S a,b lim n e n S a,b c 2 lim n 1/r τ n, 5.4 with τ n := δ n 1a, b + δ na, b. Proof. Note first that δ k a, b = 2 k/p α k for k Z. Let V : l p Z l p Z denote an auxiliary summation operator defined by V x k := 2 k/p x j 2 j/p. Using Theorem 2.1 one easily checks that V is bounded. We set now D : l p Z l q Z to be the diagonal operator generated by the sequence δ k a, b. Then it is trivial to obtain S a,b = D V, yielding e n S a,b Ce n D. On the other hand, let R : l p Z l p Z be given by Re k := 2e k 1 e k. We find that D = S a,b R, and thus e n D C e n S a,b. Now we have to estimate e n D only, which has already been done in Corollary 3.7. Next we study the case of a rapid i.e., superexponential increase of β k : Proposition 5.3. Set β k = 2 2k for k 0 and β k = 0 for k < 0. Then for any sequence a = α k we have while sup l N j<k a, b r, c sup l 1/r α l β l l, 5.5 l N l e l S a,b C sup l 1/r α l β l 1 l. 5.6 l N In particular, if we set α l := 2 l β 1 l 1 whenever l 0, and α l = 0 otherwise, then a, b r, = while sup n n e n S a,b <.

23 ENTROPY OF SUMMATION OPERATORS 267 Proof. It is easy to see that v k = log 2 k for k k 0, where as usual x := inf{n N : n x}. In particular, v k v k+1 iff k = 2 l 1 for some l 1. So for k = 2 l 1 we derive δ k a, b = 2 k/p α vk+1 = 2 2l 1/p α l c p β l α l. 5.7 Since k 1/r l 1/r, this implies 5.5. On the other hand, let V : l p Z l p Z be defined by V x l l Z := β 1 l 1 β j x j l Z. 5.8 Then we have by Theorem 2.1 that V C p and, additionally, j<l S a,b = D αl β l 1 V. 5.9 Thus e n S a,b C p e n D αl β l 1 and so 5.6 follows by virtue of Theorem 3.5. Finally, we will estimate the first entropy numbers of finite-dimensional standard summation operators. Estimates for the approximation, Gelfand and Kolmogorov numbers of such operators can be found in [8]. Let N 2 be arbitrary, and set λ := min{1, 1/r}. Then for the standard summation operator S1 N : l N p l N q with S1 N x i N i=1 := N x j i=1 we have Proposition 5.4. There are universal constants c 0, c 1, c 2 > 0 such that the estimate holds for all n c 0 N λ. j<i N 1/r c 1 n e ns1 N : l N p l N N 1/r q c 2 n Proof. We start with the upper estimate. Therefore choose m N such that v k s are given now by 2 m N < 2 m+1. 1 for k 0 v k = 2 k if 1 k m when k > m Thus δ k a, b = 2 k/p +k/q = 2 k/r for k = 0,..., m 1, and for k = m we have δ m a, b = 2 m/p N 2 m 1/q 2 m/r. By Theorem 2.2 we obtain sup n e n S1 N m 1/r c δk r c 2 m/r c N 1/r. k=0

24 268 J. CREUTZIG AND W. LINDE For the lower estimate, set [ j 1 I j := N, j N ], 1 j N, and denote by P N the conditional expectation operator mapping from L p [0, 1] to L p [0, 1] generated by the partition I 1,... I N of [0, 1], i.e., N P N f := 1 Ij I j 1 ft dt. j=1 It is easily checked that P N = 1. Further we define ϕ p to be an embedding of l N p into L p [0, 1] with N ϕ p x j := x j 1 Ij I j 1/p. j=1 I j Recalling I j = N 1, we derive easily ϕ q N 1/r S N 1 ϕ 1 p P N f = N k=1 1 Ik j<k I j f dt Next define R N : L p [0, 1] L q [0, 1] via Using 5.12, we find R N fs := T 1 = R N + N k=1 s k 1 N ft dt 1 Ik s. ϕ q N 1/r S N 1 ϕ 1 p P N We derive an upper bound for R N by estimating we assume q < ; the case q = may be treated in the same way R N f q = N s q 1/q ft dt ds k=1 I k k 1 N N s k 1 N k=1i k N N 1/r k=1 I k q/p ds I k q/p 1/q ft p dt q/p 1/q ft p dt The last sum can either be estimated by f p when p q, or by N 1/q 1/p f p if p q. This shows that R N N λ.

25 ENTROPY OF SUMMATION OPERATORS 269 From now on, d 1, d 2,... shall denote fixed constants that is, the value of d i remains the same at each occurrence. Because of the additivity of the entropy numbers see, e.g., [2] and equality 5.13 we come to e n T 1 R + e n ϕ q N 1/r S N 1 ϕ 1 p P N d 1 N λ + d 2 N 1/r e n S N 1. Since it is well-known that e n T 1 d 3 /n cf. [7], this reveals d 2 N 1/r e n S N 1 d 3 n d 1 N λ 5.14 for all n, N N. Let c 0 := d 3 2d 1 1, then 5.14 provides that for n c 0 N λ we have N 1/r e n S N 1 d 3 2d 2 n. For p < q, Proposition 5.4 is somewhat unsatisfying, because we can estimate e n S N 1 only for n c 0 N 1/r. Recall that in this case, 1/r < 1. We leave this as an open problem. Problem. 1 Find optimal estimates for e n S N 1 : l N p l N q when p < q and c 0 N 1/r n N. 6. Some Probabilistic Applications Let T be an operator from a separable Hilbert space H into a Banach space E, and let ξ k k=1 be an i.i.d. sequence of N 0, 1-distributed random variables. Then the behaviour of e n T as n is known to be closely related with the properties of the E-valued Gaussian random variable X := ξ k T f k, 6.1 k=1 where f k k=1 denotes some orthonormal basis in H. For example, in [4] it was proved that n=1 n 1/2 e n T < implies the a.s. convergence of 6.1 in E, while, conversely, the a.s. convergence of 6.1 yields sup n n 1/2 e n T < cf. [14]. A faster decay of e n T as n is reflected by a better small ball behaviour of X. More precisely the following holds cf. [9], [11] and [3]: Proposition 6.1. Let 0 < α < 2 and T : H E generating X via 6.1. Then the following statements are equivalent: i lim n 1/α e n T := c α T <. ii lim ε 2α 2 α log P X E < ε := d α X >. ε 0 1 Recently M. Lifshits kindly showed us a proof of the following fact: For any h > 0 and any pair p, q 1, there is cp, q, h > 0 such that e n S N cp, q, hn 1 N 1/r, n 1 hn. Since the upper estimate is valid for all n N, this almost answers the problem.

26 270 J. CREUTZIG AND W. LINDE Moreover, for some universal c 0 > 0. Conversely, if T satisfies then and d α X c 0 c α T 2α 2 α 6.2 lim n 1/α e n T := κ α T > 0, lim ε 2α 2 α log P X E < ε := λ α X < 0 ε 0 λ α X c 0 κ α T 2α 2 α. We want to apply these relations between the entropy and the small ball behaviour to the summation operator S a,b : l 2 Z l q Z, 1 q. To do so, let W = W t t 0 be a standard Wiener process over [0,. Taking the natural unit vector basis e k in l 2 Z leads to X = k= ξ k S a,b e k, 6.3 where the l q Z-valued Gaussian random variable X = X k is defined by X k := α k W t k 6.4 with t k := j<k βj 2 being an increasing sequence of nonnegative numbers such that lim t k = 0. In our situation we have r = 2q and v k 2+q k s given in 2.6 can be calculated by With this notation the following is valid: v k = sup{m Z : t m < 2 k }. 6.5 Theorem 6.2. Let α k be a sequence of nonnegative numbers satisfying 2.1 and let t k be a nondecreasing sequence with lim t k = 0. For X = k X k defined by 6.4 and v k s as in 6.5 we have the following. i If then k= v 2 kq k q α q 2+q l l=v k +1 <, 6.6 lim ε 0 ε2 log P X q < ε = ii Assume that [t k+1 t k ] 1 α q k is monotone near ±. Then 6.7 holds provided that k= [t k t k 1 ] q 2q 2+q α 2+q k <. 6.8

27 ENTROPY OF SUMMATION OPERATORS 271 Proof. Combining Proposition 6.1 for α = 1 with Theorem 2.2, one easily sees that 6.6 implies lim ε 2 log P X q < ε = d 1 X >. 6.9 ε 0 Yet by the second part of Theorem 2.2 and estimate 6.2 we even have d 1 X= 0 which completes the proof of i. Assertion ii is a direct consequence of Proposition 2.4. Recall that β 2 k = t k+1 t k. Remarks. 1 Theorem 2.3 shows that, in general, condition 6.6 cannot be replaced by the weaker assumption 6.8. More precisely, for each q [1, ] there are α k s and t k s satisfying 6.8, yet lim ε 2 log P X q < ε =. ε 0 2 In the case q = Theorem 6.2 reads as follows. Suppose that Then k= 2 k sup v k <l v k+1 α 2 l < lim ε 0 ε2 log Psup α k W t k < ε = Again, 6.11 becomes wrong under a weaker condition k= [t k t k 1 ] α 2 k < Recall that 6.12 suffices for α k s monotone near ±. Observe that Theorem 6.2 provides us with sufficient conditions for log P X q < ε ε α 6.13 in the case α = 2. Of course, similar conditions would also be of interest for α s different of 2, yet we could not handle this problem by our methods. Recall that we have derived upper bounds for the entropy of summation operators by the corresponding results about integration operators. And there only the case α = 2 has been treated. But for t k s not increasing too fast we can prove necessary conditions for 6.13 and all α > 0. To be more precise, suppose that t k s satisfy Then the following holds. { } tk+1 sup : t k > 0 < t k

28 272 J. CREUTZIG AND W. LINDE Proposition 6.3. Let X = X k be a Gaussian sequence defined by 6.4 with t k s satisfying If lim ε α log P X q < ε > 6.15 ε 0 for some α > 0, then necessarily with and v k s defined by 6.5. sup n 1/α+1/q δn 1 + δ n < 6.16 n 1 δ n = 2 n/2 v k+1 1/q α q l l=v k +1 Proof. Using Proposition 6.1, we see that 6.15 implies, n Z, 6.17 sup n ρ e n S a,b <, 6.18 n 1 where, as before, β 2 k = t k+1 t k and ρ = 1/α + 1/2. Let us first treat the case t k > 0 for all k Z and sup k t k =. Then condition 6.14 implies sup {k : v k = v k } < 6.19 in fact, here 6.14 and 6.19 are even equivalent, i.e., condition 2.12 is satisfied. Thus Theorem 2.5 applies and by 6.16 we obtain sup n 1/s δn 1 + δ n <, n 1 where 1/s = ρ 1/2 + 1/q = 1/α + 1/q. This completes the proof in this case. Next assume t k > 0 and sup k t k <. Then by 6.14 there is k 0 N such that sup {k : v k = v k } <. k k 0 Here Proposition 2.6 applies and completes the proof as before. If we have sup k t k < and t k = 0 for k k, {n Z : δ n 0} <, so there is nothing to prove. Finally, if t k = 0 for k k and sup k t k =, this time 6.14 yields sup {k : v k = v k } < k k 0 for a certain k 0 N, and we may proceed as in the preceding case. Next we ask for conditions which ensure log P X q < ε ε α for some α > 0.

29 ENTROPY OF SUMMATION OPERATORS 273 Proposition 6.4. Let t k and X be as in Proposition 6.3. If for some α > 0 and δ n s defined by 6.17 then necessarily lim n 1/α+1/q δn 1 + δ n > 0, lim ε α log P X q < ε < 0. ε 0 Proof. This can be proved exactly as Proposition 6.3 by using Theorem 2.5, Proposition 2.6 and Proposition 6.1. Let us illustrate the preceding results by an example. We define random variables Y k k 1 by Y k := 1 2 k+1 1 W l q 2 k l 1/2 1/q, q <, and Y k := l=2 k sup 2 k l<2 k+1 W l l 1/2, q =. Proposition 6.5. Let γ k k 1 be a sequence of nonnegative real numbers, and for q [1, ], let 1/r = 1/2 + 1/q. i If γ k k N r <, then lim ε 2 log P γ q ε 0 k Y q k < εq = 0. k=1 ii For γ k decreasing and log P γ q k Y q k < εq ε α we necessarily k=1 have sup k 1/α+1/q γ k <. k 1 iii Suppose that γ k s are decreasing with γ k k 1/α 1/q for a certain α > 0. Then it follows log P k=1 γ q k Y q k < εq ε α. In all three cases, for q =, one has to read P sup γ k Y k < ε everywhere. k 1 Proof. We set α l = 2 k/q γ k l 1/2, 2 k l < 2 k+1 for any k 0, and α l = 0 if l < 0. Then it is easy to obtain γ k Y k q = X q 6.20 with X = X l l Z = α l W l l Z. Let b be given by β k = 1 if k 0, β k = 0 otherwise. Then t l = l for l 0, and t l = 0 otherwise. For v k s defined in 2.6 we have v k = 0 for k 0 and v k = 2 k 1 for k 1. Thus 2 k+1 1/q δ k a, b = 2 k/2 k/q 1 γ k l=2 k l q/2

30 274 J. CREUTZIG AND W. LINDE if q < and δ k a, b = 2 k/2 γ k sup l 1/2 = γk 2 k l<2 k+1 for q =. In any case we obtain 2 1/2 γ k δ k a, b γ k, 6.21 so that γ k k N l r N iff δ k k N l r N, and i follows from Theorem 6.2. Conversely, if γ k s are decreasing, by 6.21 we have sup k 1 k 1/α+1/q γ k < iff the same holds for δ k s. So ii is a consequence of Proposition 6.3 and Similarly, assertion iii is proved. References 1. B. Carl, Entropy numbers of diagonal operators with application to eigenvalue problems. J. Approx. Theor , B. Carl and I. Stephani, Entropy, compactness and approximation of operators. Cambridge Univ. Press, Cambridge, J. Creutzig, Gaußmaße kleiner Kugeln und metrische Entropie. Diplomarbeit, FSU Jena, R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal , D. E. Edmunds, W. D. Evans, and D. J. Harris, Approximation numbers of certain Volterra integral operators. J. London Math. Soc , D. E. Edmunds, W. D. Evans, and D. J. Harris, Two-sided estimates of the approximation numbers of certain Volterra integral operators. Studia Math , D. E. Edmunds and H. Triebel, Function spaces, entropy numbers and differential operators. Cambridge Univ. Press, Cambridge, A. Hinrichs, Summationsoperatoren in l p -Räumen. Diplomarbeit, FSU Jena, J. Kuelbs and W. V. Li, Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal , M. Lifshits and W. Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion. Preprint, 1999; to appear in Mem. Amer. Math. Soc. 11. W. V. Li and W. Linde, Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab , V. G. Maz ja, Sobolev spaces. Springer Verlag, Berlin, A. Pietsch, Operator Ideals. Verlag der Wissenschaften, Berlin, V. N. Sudakov, Gaussian measures, Cauchy measures and ɛ-entropy. Soviet Math. Dokl , Received Authors address: Friedrich Schiller Universität Jena Ernst Abbe Platz Jena, Germany jakob@creutzig.de, lindew@minet.uni-jena.de