NEW EXPLICITINDIMENSION ESTIMATES FOR THE CARDINALITY OF HIGHDIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS HAVING MIXED SMOOTHNESS

Abstract. We are aiming at sharp and explicitindimension estimations of the cardinality of sdimensional hyperbolic crosses where s may be large, and applications in highdimensional approximations of functions having mixed smoothness. In particular, we provide new tight and explicitindimension upper and lower bounds for the cardinality of hyperbolic crosses. We apply them to obtain explicit upper and lower bounds for εdimensions – the inverses of the well known Kolmogorov Nwidths – in the space L2(T s ) of modified Korobov classes Ur,a(T s ) on the storus T s := −π, π s . The functions in this class have mixed smoothness of order r and depend on an additional parameter a which is responsible for the shape of the hyperbolic cross and controls the bound of the smoothness component of the unit ball of Kr,a(T s ) as a subset in L2(T s ). We give also a classification of tractability for the problem of εdimensions of Ur,a(T s ). This theory is extended to highdimensional approximations of nonperiodic functions in the weighted space L2(−1, 1s , w) with the tensor product Jacobi weight

CARDINALITY OF HIGH-DIMENSIONAL HYPERBOLICCROSSES AND APPROXIMATION OF FUNCTIONS HAVING

MIXED SMOOTHNESS

ALEXEY CHERNOV AND DINH D ˜ UNG

Abstract We are aiming at sharp and explicit-in-dimension estimations of

the cardinality of s-dimensional hyperbolic crosses where s may be large, and

applications in high-dimensional approximations of functions having mixed

smoothness In particular, we provide new tight and explicit-in-dimension

up-per and lower bounds for the cardinality of hyup-perbolic crosses We apply them

to obtain explicit upper and lower bounds for ε-dimensions – the inverses of the

well known Kolmogorov N -widths – in the space L 2 (Ts) of modified Korobov

classes Ur,a(Ts) on the s-torus Ts:= [−π, π]s The functions in this class have

mixed smoothness of order r and depend on an additional parameter a which

is responsible for the shape of the hyperbolic cross and controls the bound of

the smoothness component of the unit ball of K r,a (T s ) as a subset in L 2 (T s ).

We give also a classification of tractability for the problem of ε-dimensions

of U r,a (T s ) This theory is extended to high-dimensional approximations of

non-periodic functions in the weighted space L 2 ([−1, 1] s , w) with the tensor

product Jacobi weight w by tensor products of Jacobi polynomials with powers

in hyperbolic crosses.

1 IntroductionThe recent decades have been designated by an increasing interest in numericalapproximation of problems in high dimensions, in particular problems involvinghigh-dimensional input and output data depending on a large number s of vari-ables They naturally appear in a vast number of applications in MathematicalFinance, Chemistry, Physics (e.g Quantum Mechanics), Meteorology, MachineLearning, etc Typically, a numerical solution of such problems to the target ac-curacy ε demands for a high exponentially increasing computational cost ε−δs forsome δ > 0, so that numerical computations even for a moderate values of ε willresult in an unacceptably large computation times and memory requirements This

2010 Mathematics Subject Classification Primary 41A25, 41A46, 41A63, 42A10.

Key words and phrases Hyperbolic cross, high-dimensional approximation, N -widths, dimensions, tractability, exponential tractability.

ε-Alexey Chernov acknowledges support by the Hausdorff Center for Mathematics, University

of Bonn, Germany, the University of Reading, United Kingdom and the Carl von Ossietzky versity, Oldenburg, Germany.

Uni-Dinh Dung’s research work is funded by Vietnam National Foundation for Science and nology Development (NAFOSTED) under Grant No 102.01-2014.02 A part of Dinh Dung’s research work was done when he was working as a research professor at the Vietnam Institute for Advanced Study in Mathematics (VIASM) He would like to thank the VIASM for providing a fruitful research environment and working condition.

Tech-The authors would like to thank Erich Novak for valuable remarks and comments.

phenomenon is called the curse of dimensionality, a term suggested by Bellmann[3] (in the context of our paper term “high-dimensional” refers to the number ofvariables s  1) This consideration is true in general, but in some cases the curse

of dimensionality can be overcome, particularly when the high-dimensional databelong to certain classes of functions having mixed smoothness Such functions can

be optimally represented by means of the hyperbolic cross (HC) approximation Forexample, trigonometric polynomials with frequencies in HCs have been widely usedfor approximating functions with a bounded mixed derivative or difference Theseclassical trigonometric HC approximations date back to Babenko [2] For furthersources on HC approximations in this classical context we refer to [16, 39] and thereferences therein Later on, these terminologies were extended to approximations

by wavelets [11, 35], by B-splines [17, 36], and to algebraic polynomial tions where the power of algebraic polynomials approximants are in HCs [7, 9] HCapproximations have applications in quantum mechanics and PDEs [44, 25], finance[22], numerical solution of stochastic PDEs [7, 9, 10, 33, 34], and data mining [21]

approxima-to mention just a few (see also the surveys [5] and [23] and the references therein)

In traditional trigonometric approximations of functions having a mixed ness, there are two kinds of HC used as the frequency domain of approximanttrigonometric polynomials: continuous HCs

where j := {k ∈ Zs : b2j i −1c ≤ |ki| < 2j i, i = 1, , s} (we refer to [16, 39] forfurther modifications of these HCs in trigonometric approximations of functionshaving a mixed smoothness and zero mean value in each variable) These HCshaving the asymptotic cardinality  T logs−1T , play an important role in comput-ing asymptotic orders of various characteristics of optimal approximation such as

N -widths and ε-dimensions for classes of periodic functions having mixed ness In this work we study approximations by trigonometric polynomials withfrequencies from modified HCs

of a correspondingly modified Korobov function classes Ur,a(Ts) introduced below.Let us recall the concepts of Kolmogorov N -width [27] and its inverse ε-dimension.Let X be a normed space and W a central symmetric subset in X The Kolmogorov

≤ N There is a vast amount of literature on optimal approximations and these

N -widths, see [41], [32], in particular, for s-variate function classes [39]

In computational mathematics, the so-called ε-dimension nε= nε(W, X) is used

to quantify the computational complexity (in Information-Based Complexity thesame object is termed “information complexity” or “ε-cardinality”) It is definedby

to the problem of optimal linear approximation in X of functions from W by ear N -dimensional subspaces, Kolmogorov N -widths dN(W, X) and ε-dimension

lin-nε(W, X), where W is a smoothness class of functions having in some sense moreregularity than X ⊃ W In the present work, the regularity of the class W will

be measured by L2-boundedness of mixed derivatives sufficiently of higher order.Finite element approximation spaces based on HC frequency domains are suitablefor this framework [20] (cf also [5])

As a model we will consider functions on Rs which are 2π-periodic in each able, as functions defined on the s-dimensional torus Ts := [−π, π]s for which theend points of the interval [−π, π] are identified for each coordinate component.The space Hr1

vari-(Ts) := Hr

(T) ⊗ · · · ⊗ Hr

(T) consists of all periodic functions whosemixed derivatives of order r > 0 are L2-integrable (i.e having mixed smoothness oforder r) For the unit ball Ur1

linear approximation by continuous HC spaces of trigonometric polynomials Theseestimates are quite satisfactory if s is small and fixed.

In the recent work of D˜ung and Ullrich [20], A(r, s), A0(r, s) and B(r, s), B0(r, s)have been estimated from above and below explicitly in s when s is large In theirpaper, the class Ur1

(Ts) is redefined in terms of the traditional dyadic tion of the frequency domain These estimations are based on an approximation bytrigonometric polynomials with frequencies in step HCs G∗(s, T ) and explicit-in-dimension estimations of its cardinality |G∗(s, T )| However, the authors were able

decomposi-to estimate them from above only for very large n ≥ 2δs or very small ε ≤ 2−δs, forsome δ > 0, (see [20, Thms 4.10, 4.11]) This does not give a complete picture ofthe convergence rate in high-dimensional settings The reason is that the step HCapproximations of the class Ur1 involve whole dyadic blocks of frequencies whichhave the cardinality at least 2s

In the present paper, to avoid this fact we suggest to replace Hr1(Ts) by anotherspace Kr,a(Ts) which is defined as a modification of the well-known Korobov space,and construct appropriate continuous HCs for the trigonometric approximations offunctions from this space This will allows to derive very tight and explicit-in-dimension upper and lower estimates for the cardinality of continuous HCs andfurther sharp estimates for ε-dimensions Observing that the asymptotic orders ofthese quantities are similar, we restrict the presentation in this paper to the study

of ε-dimensions (these are directly related to the cost of computational complexity

in IBC) and refer to the extended preprint version of this work [8] for the study of

N -widths Along with the smoothness r and dimensionality s, the norms on spaces

Kr,a

(Ts) will be parametrized by a positive number a > 0 controlling the bound ofthe smoothness component of the unit ball of Kr,a

(Ts) as a subset in L2(Ts) Theparameter a allows also for simultaneous and sharp derivation of upper and lowerbounds for the cardinality of HCs

Let us introduce the spaces Kr,a

(Ts) For this we recall that L2(Ts) is theHilbert space of functions on Tsequipped with the inner product

be the k-th Fourier coefficient of f , where ek(x) := ei(k,x) Then for a given r ≥ 0,

a > 0 and a vector k ∈ Zswe define a scalar λa(k) by

κr,a(x) := X

k∈Z s

λa(k)−rek(x), x ∈ Ts.Denote by (f ∗ g)(x) := (f (x − ·), g) the convolution of f and g Then the Hilbertspace Kr,a

(Ts) is defined as

Kr,a(Ts) := {f : f = κr,a∗ g, g ∈ L2(Ts)}

with the norm kf kKr,a (T s ) := kgk and the inner product

(f, f0)Kr,a (T s ):= (g, g0), where f = κr,a∗ g and f0= κr,a∗ g0

By the convolution theorem ˆf (k) = λa(k)−rg(k) and thus by Parseval’s identityˆ

kf k2

K r,a (T s )= X

k∈Z s

λa(k)2r| ˆf (k)|2 (1.5)Thus, the space Kr,a

(Ts) can be seen as the set of all functions f ∈ L2(Ts) for whichthe right hand side of (1.5) is finite Notice also that the norm of the embedding

Kr,a(Ts) ,→ L2(Ts) is one and does not depend on s, r, a

The notion of space Kr,a(Ts) is a modification of the notion of the classicalKorobov space For r > 1/2, the kernel Ka defined at x and y in Tsas Ka(x, y) :=

κ2r,a(x−y) is the reproducing kernel for the Hilbert space Kr,a(Ts) For a definitivetreatment of reproducing kernel, see, for example, [1] The linear span of theset of functions {κr,a(· − y) : y ∈ Ts} is dense in Kr,a

(Ts) In the language ofMachine Learning, this means that the reproducing kernel for the Hilbert space isuniversal In the recent paper [19] some upper and lower bounds of multivariateapproximation by translates of the Korobov function on sparse Smolyak grids havebeen established

A similar notion of generalized Korobov space Hs,r(Ts) was introduced in [30,A.1, Appendix] This space is defined in the same way as the definition of Kr,a

(Ts)

by replacing the scalar λa(k)r by the scalar %s,r(k) depending on two parameters

β and β1 Korobov spaces and their modifications are important for the study

of approximation and computation problems of smooth multivariate periodic tions, especially in high-dimensional settings For further information, see detailedsurveys and references in the books [39], [37], [30]

func-Note that the spaces Hr1(Ts), Kr,a(Ts) and Hs,r(Ts) coincide as function spacesequipped with equivalent norms However, if s is large, the unit balls with respect tothe norms of these spaces differ significantly As will be shown in the present paper,for the space Kr,a

(Ts), the scaling parameter a defining different equivalent norms,

as noticed above controls the bound of the smoothness component of the unit ball of

Ur,a(Ts) = {f ∈ Kr,a(Ts) : kf kKr,a (T s )≤ 1}

In this paper, we derive new upper and lower bounds for nε(Ur,a

(Ts), L2(Ts)) withexplicit dependence on the parameters ε, s and a The core of our theory in both,periodic and non-periodic settings, is based upon sharp cardinality estimates forthe index sets Γ(s, T, a) and Γ±(s, T, a) defined in (1.1) and (1.2) above The setsΓ(s, T, a) and Γ±(s, T, a) will be referred to as a corner and a symmetric continuoushyperbolic crosses, or shortly HCs

Denote by |G| the cardinality of a finite set G Notice that the problem of puting |Γ(s, T, a)| and |Γ±(s, T, a)| in our setting is itself interesting as a problem

com-of a classical direction in Number Theory investigating the number com-of the integerpoints in a domain such as a ball and a sphere [43], [6], [26], a hyperbolic domain[29], [12], [24], [15], etc Specially, in [15] the convergence rate of cardinality ofthe intersection of HCs was computed and applied to estimations of dN(W, L2(Ts))where W is a class of several L2(Ts)-bounded mixed derivatives It is also worthmentioning that the problem of estimation of the cardinality of the hyperbolic cross(1.1) is related to the classical Dirichlet divisor problem in Number Theory, see [42,Chapter XII] for further details

Motivated by all the above arguments, the main goal of this paper to prove upperand lower bounds for the hyperbolic crosses |Γ(s, T, a)| and |Γ±(s, T, a)| in a newhigh-dimensional approach as a functions of three variables s, T, a when the dimen-sion s and the real parameter T > 0 may be (but not necessarily is) large and a thereal parameter ranging from 0 to ∞ These cardinality estimates are then appliedfor the estimation of nε(Ur,a

(Ts), L2(Ts)) Although T is the main parameter inthe study of cardinality of HCs, the parameters s and a may have a serious effect

on the estimates when s is large, or when the positive parameter a ranges throughthe critical value 1 Our method of estimation is based on comparison |Γ(s, T, a)|and |Γ±(s, T, a)| with the volume of smooth HCs P (s, T, a) (cf their definition in(2.1)) and tight non-asymptotic estimates for the latter A new crucial element

in our volume-based estimation approach is that the cardinalities of Γ(s, T, a) and

Γ±(s, T, a) are compared with volumes of shifted smooth HCs P (s, T, a0), where a0may be not equal to a This shift is made possible by introduction of the parameter

a which is a new and essential ingredient of the present work allowing for neous and sharp derivation of upper and lower bounds for cardinalities of the HCs

simulta-We refer to Section 2 for the rigorous construction

We give now a brief overview of the main results of the present paper As aby-product of our analysis, we prove that the volume of smooth HCs P (s, T, a)can be reduced to the sth remainder of the Taylor series of exp(−t), which can betightly estimated by

1(s − 1)!

ts

t + s − 1, s ≥ 1, t > 0. (1.6)

To the knowledge of the authors, so far these estimates have been unknown Fromthis basic result and new tight two-sided estimates of the cardinalities of Γ(s, T, a)and Γ±(s, T, a) by the volume of shifted smooth HCs P (s, T, a0) we derive verytight non-asymptotic bounds for the cardinality of Γ(s, T, a) and Γ±(s, T, a) Let

|Γ(s, T, a)| and the volume of the shifted smooth HC P (s, T, a − 1/2) (see rem 2.4 and Theorem 3.5) If the restriction of T being sufficiently large T ≥ T∗(s, a)

Theo-is omitted, a slightly relaxed upper bound (3.5) will be proved for any (even small)value of T satisfying T > (a − 1)s, a > 1 This estimate comes along with the lowerbound (3.4) which is proved for any (even small) value of T satisfying T > as,

a > 0, see Theorem 3.4 for the details

For every s ∈ N, every a > 0 and every T > (a + 1/2)s, there holds the lowerbound for |Γ±(s, T, a)|

and for every s ∈ N, every a > 1/2 and T > (a − 1/2)s, there holds the upperbound for |Γ±(s, T, a)|

We also show that the problem of nε(Ur,a

(Ts), L2(Ts)) is weakly tractable butpolynomially intractable for every a > 0 The tractability of linear approximationproblem for the generalized Korobov space Hs,r(Ts) was studied in [30, Pages 184–185]

All of these methods and results are extended to HC approximations of functionsfrom the non-periodic modified Korobov space Kr,a

(Is, w) in the weighted space

L2(Is, w) with the Jacobi weight w by Jacobi polynomials with powers in the corner

HC Γ(s, T, a), where Is:= [−1, 1]s We believe that they can be also extended toother HC approximations in a Hilbert space

In brief, the paper is organized as follows In Section 2, we prove preliminaryestimates for |Γ(s, T, a)| and |Γ±(s, T, a)| via the volume of corresponding smoothHCs Section 3 is the core of the present work There, we prove non-asymptotictight upper and lower estimates for the volume of smooth HCs P (s, T, a), andderive from them and the results of Section 2 non-asymptotic tight upper andlower estimates for |Γ(s, T, a)| and |Γ±(s, T, a)| Utilizing these results, we prove

in Section 4 lower and upper estimates for nε(Ur,a

(Ts), L2(Ts)) In Section 4, wealso investigate tractabilities of the problem of nε(Ur,a

(Ts), L2(Ts)) In Section 5,

we extend the methods and results for periodic approximations to the non-periodiccase and approximations by polynomials The Appendix in Section 6 containsthe detailed proof of Theorem 2.4 stating a sharpened upper bound for |Γ(s, T, a)|required in Section 3

2 Preliminary estimates via the volume of smooth HCs

For a domain Ω ⊂ Rs

, let us denote by |Ω| the volume of Ω ⊂ Rs, that is,

|Ω| =Zdx

This is a slight abuse of notation since for a (discrete) finite set G we use alsothe notation |G| for the cardinality of G However, there is no ambiguity for agiven set A natural way to estimate the cardinalities |Γ(s, T, a)| and |Γ±(s, T, a)|from above and from below is to compare them with the volume |P (s, T, a0)| of thecorresponding corner smooth HC

Γ(s, T, a0) ⊂ Γ(s, T, a), Γ±(s, T, a0) ⊂ Γ±(s, T, a), P (s, T, a0) ⊂ P (s, T, a).Define bxc := (bx1c, , bxsc) for x ∈ Rs, where btc denotes the integer part of

t ∈ R The following lemma gives preliminary upper and lower bounds of |Γ(s, T, a)|via the volume I(s, T, a) based on direct set inclusions

Lemma 2.1 For every s ∈ N, T > 0, and a > 0, there hold the inclusions

Q(s, T, a + 1) ( P (s, T, a) ( Q(s, T, a) (2.4)and consequently,

|Γ(s, T, a + 1)| < I(s, T, a) < |Γ(s, T, a)| (2.5)Proof We observe that x ∈ Q(s, T, a) if and only if bxc ∈ Γ(s, T, a) Therefore, therelation

Lemma 2.2 Suppose 0 < δ ≤ 1 Then for every s ∈ N, T ≥ δs, and a > δ, itholds that

|Γ(s, T, a)| < (1/δ)sI(s, T, a − δ), (2.6)and

|Γ±(s, T, a)| < (2/δ)sI(s, T, a − δ) (2.7)Proof Since |Γ±(s, T, a)| < 2s|Γ(s, T, a)|, it is sufficient to prove (2.6) For δ = 1,estimate (2.6) is equivalent to the left inequality in (2.5) if changing a to a + 1 For

0 < δ < 1 we introduce a (1 − δ)-shifted set

For the volumes of ˜Qe(δ) and ˜Q+

e(δ) we obviously have the relations

and the proof is complete

Due to the specific geometrical structure of the symmetric HC Γ±(s, T, a), theupper bound (2.7) can be improved for δ = 12 as in the following lemma

Lemma 2.3 For every s ∈ N a > 1/2 and T ≥ 1, it holds that

|x| ∈ P (s, T, a +1

2) ( Q±(s, T, a) ( |x| ∈ P (s, T, a −1

2) ,and consequently, by symmetry (2.8)

Unfortunately, we have no analogue of Lemma 2.3 for the corner HC Γ(s, T, a)

We able to establish only the upper bound |Γ(s, T, a)| < I(s, T, a −1

2) for T largeenough Namely, we have the following theorem

Theorem 2.4 For any s ∈ N, and a > 12, there exists T∗= T∗(s, a) > 0 such that

|Γ(s, T, a)| ≤ I(s, T, a −1

2), ∀T ≥ T∗(s, a) (2.9)Observe that by (2.5) one can see that for every s ∈ N, T > 0, and a > 1, theinequality

|Γ(s, T, a)| < I(s, T, a − 1)directly follows from the sharp set inclusion Γ(s, T, a) ( P (s, T, a − 1) This meansthat Γ(s, T, a) 6⊂ P (s, T, a0) for any a0> a − 1, in particular, Γ(s, T, a) 6⊂ P (s, T, a −1/2) Therefore, the proof of (2.9) requires a completely new idea and technique

To focus the reader’s attention on the main path in our theory, the proof of Theorem2.4 which is technically complicate is given in Appendix in Section 6

3 Non-asymptotic bounds for the volume of smooth HCs and the

cardinality of HCsThe inequalities (2.5)–(2.7), (2.8) and (2.9) allow us to estimate |Γ(s, T, a)| and

|Γ±(s, T, a)| by the volume I(s, T, a0) of the smooth HC P (s, T, a), which is, as amatter of fact, a simpler task Indeed, below we will show that the integral I(s, T, a)for any a > 0, can be represented as a sum of an infinite series This series isrelated to the s-th remainder of the Taylor series of the exponential exp(−t) Inthe next important step, we will give tight two-sided non-asymptotic bounds forthis remainder

3.1 Tight non-asymptotic bounds for the volume of smooth HCs andthe cardinality of HCs

To formulate the result, for s ∈ N0, we introduce the function

I(s, T, a) = (−1)s+1(T − as) + T

s−1

X(ln T − s ln a)n(−1)s−1−n

Then for t := ln T − s ln a > 0, there holds

This yields the second line in (3.2)

Note that for s = 0 it formally holds

I(0, T, a) = T F0(ln T ) = 1

Relation (3.2) is exact but involves an infinite summation on the right-handside, whose behaviour is not clearly seen from (3.1) To make it explicit, we provevery tight bounds for the series Fs(t) in terms of the single summand ps−1 Thisapproximation is somewhat surprisingly good, as we observe from the followingnon-asymptotic estimates

Theorem 3.2 For any s ∈ N and t > 0, the following estimate holds true

t

t + sps−1(t) < Fs(t) <

t

t + s − 1ps−1(t). (3.3)The proof of Theorem 3.2 will be given in Subsection 3.2

This theorem and Lemma 3.1 imply directly

Corollary 3.3 For s ∈ N, a > 0 and T > as, it holds that

T (ln T − s ln a)s

(ln T − ln a + s − 1).

A combination of Corollary 3.3 and the results in Section 2 implies various lowerand upper bounds for |Γ(s, T, a)| and |Γ±(s, T, a)| in the following theorems.Lemma 2.1 and Corollary 3.3 provide an lower and upper bounds for |Γ(s, T, a)|.Theorem 3.4 We have for every s ∈ N, every a > 0 and every T > as,

|Γ(s, T, a)| > 1

(s − 1)!

T (ln T − s ln a)s

ln T − s ln a + s, (3.4)and for every s ∈ N, every a > 1 and T > (a − 1)s,

|Γ(s, T, a)| < 1

(s − 1)!

T (ln T − s ln(a − 1))s

ln T − s ln(a − 1) + s − 1. (3.5)Theorem 2.4 and Corollary 3.3 give the following sharpened upper bound for

|Γ(s, T, a)| which becomes valid when T is sufficiently large

Theorem 3.5 For any s ∈ N and a > 1/2, there exists T∗(s, a) > 0 such that

|Γ(s, T, a)| < 1

(s − 1)!

T ln T − s ln(a − 1/2)
s

ln T − s ln(a − 1/2)s − 1, ∀T ≥ T∗(s, a)

Analogously, Lemma 2.3 and Corollary 3.3 provide sharpened upper and lowerbounds for |Γ±(s, T, a)|.

Theorem 3.6 We have for every s ∈ N, every a > 0 and every T > (a + 1/2)s,

|Γ(s, T, a)| < δ−1T1+1/δ(a − δ)−s/δ, (3.8)and

|Γ±(s, T, a)| < 2δ−1T1+2/δ(a − δ)−2s/δ.Proof Let us prove the inequality (3.8) in the lemma The other one can be proven

in a similar way Since |Γ(s, T, a)| = 0 for 0 < T < as, it is enough to consider thecase where T ≥ as> (a − δ)s By Lemmas 2.2, 3.1 and Theorem 3.2 we have that

Theorem 3.7 shows that if a > 1 and 0 < δ ≤ 1 are any fixed numbers such that

λ := a − δ > 1, then the number of integer points in the hyperbolic cross Γ(s, T, a)and Γ±(s, T, a) is decreasing exponentially as λ−s/δ and λ−2s/δ with respect to thedimension s when s → ∞ It will be used in the study of HC approximations andthe problem of ε-dimensions in Sections 4–5

Remark 3.8 The recent work [28] contains an independent investigation of proximation numbers of the embedding spaces of mixed smoothness r into L2(Ts)focusing on the dependence of constants in lower and upper bounds on the dimen-sion s It contains somewhat related estimates of the volume of the smooth HCI(s, T, a) for the special case a = 1 We mention in particular [28, Lemma 3.3]showing

ap-T (ln ap-T )s−1(s − 1)! −T (ln T )

s−2

(s − 2)! ≤ I(s, T, 1) ≤ T (ln T )

s−1

(s − 1)! . (3.9)Our result in Corollary 3.3 above yields in the special case a = 1 the estimate

values of T satisfying ln T > s(s − 1) Our lower bound (3.10) is sharper whenever

ln T ≤ s(s − 1) holds true

It is important to mention here that the newly introduced in our paper eter a > 0 (taking fractional values as well) allows for more flexibility and, as aconsequence, better upper and lower estimates for the cardinality of the hyperboliccrosses in Lemma 2.1 – 2.3, Theorem 2.4 and Corollary 3.5 – 3.7

Lemma 3.9 For every t > 0 and s ∈ N0, we have

For every t > 0 and s ∈ N, definition (3.11) implies the identities

ps(t) = Fs(t) + Fs+1(t), hs(t) = 1 −Fs(t)

ps(t), ps(t) =

t

sps−1(t). (3.14)The first two of them imply that {hs(t)}s≥0 is a bounded sequence:

Corollary 3.10 For any s ∈ N0 and t > 0, we have

0 < hs(t) < 1

Proof Obviously, ps(t) is positive, therefore hs(t) > 0 is equivalent to Fs+1(t) > 0.Similarly, hs(t) < 1 is equivalent to Fs(t) > 0, see (3.14) The rest follows fromLemma 3.9

The following three lemmas deal with the proof of (3.12) for any real t > 0 and

s ∈ N From (3.14) we observe

hs= 1 −Fs

ps

= 1 − st

Fs

ps−1

= 1 − s

ths−1,hence the sequence {hs}∞

s=0 is defined via the recurrence relation

holds for every t > 0 and s ∈ N0 if s ≤ t − 1

Proof The proof is by induction on s By simple calculations we have

t

t + 1 < h0(t) = 1 − e

−t< 1and the basis is true To prove the inductive step, we show that (3.12) implies(3.16) as long as s ≤ t − 1 The upper bound follows directly from the lower bound

in (3.12) and (3.15), precisely

hs= 1 −s

ths−1< 1 −

st

hs= 1 −s

ths−1> 1 −

st

Proof First we observe that (3.17) holds true if and only if the sequence {hs}s≥0

is strictly decreasing To prove this statement we define the increments

∆s(t) := hs+1(t) − hs(t) = 1 − t + s + 1

t hs(t), s ∈ N0, (3.18)where the last relation follows from (3.15) Then (3.17) is equivalent to

Lemma 3.11 implies (3.17) and hence (3.19) for s ≤ t−1 Suppose now that s > t−1and show (3.19) by contradiction For this, we utilize (3.15) to obtain the followingrecurrence relation for ∆s:

∆s= s − t + 1t(s + t − 1)+

s(s − 1)

t2

t + s + 1

t + s − 1∆s−2. (3.20)Indeed, we have

1

t + s − 1 +

s − 1t

+s(s − 1)

∆s(t) >1

t min



1,1t

+



1 + 1t

+ ∆s−2> 0.Hence, for every k ∈ N0 the increments ∆s+2k(t) are positive and admit the lowerbounds

∆s+2k(t) > 1

t min



1,1t

+ ∆s+2k−2(t) > k

t min



1,1t

Lemma 3.13 For every t > 0 and s ∈ N0, we have

t

t + s

= t(t + st + s

2+ s)s(t + s)(t + s + 1) =

t(s + 1)s(t + s + 1) <

t

t + s − 1,where the last inequality holds if and only if s > t − 1 Changing s − 1 → s yields(3.21) for s > t − 2 The proof of (3.12) is complete

4 Upper and lower bounds for the ε-dimensions

In this section, we utilize the results from Section 3 and establish upper and lowerbounds for the ε-dimension nε(Ur,a(Ts), L2(Ts)) As auxiliary results we derive

an upper bound for the L2(Ts)-error in the HC approximation by trigonometricpolynomials with frequencies from Γ±(s, T, a) (Jackson inequality) as well as thecorresponding inverse estimate (Bernstein inequality) We give also a classification

of tractability for the problem of ε-dimensions of Ur,a

Obviously, ST is the orthogonal projection onto T (s, T, a)

The following lemma and corollary give upper bounds with respect to T for theerror of the orthogonal projection