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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
NEW EXPLICIT-IN-DIMENSION ESTIMATES FOR THE CARDINALITY OF HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS HAVING MIXED SMOOTHNESS ˜ ALEXEY CHERNOV AND DINH DUNG 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 upper and lower bounds for the cardinality of hyperbolic 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 L2 (Ts ) of modified Korobov classes U r,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 (Ts ) as a subset in L2 (Ts ). We give also a classification of tractability for the problem of ε-dimensions of U r,a (Ts ). This theory is extended to high-dimensional approximations of non-periodic functions in the weighted space L2 ([−1, 1]s , w) with the tensor product Jacobi weight w by tensor products of Jacobi polynomials with powers in hyperbolic crosses. 1. Introduction The recent decades have been designated by an increasing interest in numerical approximation of problems in high dimensions, in particular problems involving high-dimensional input and output data depending on a large number s of variables. They naturally appear in a vast number of applications in Mathematical Finance, Chemistry, Physics (e.g. Quantum Mechanics), Meteorology, Machine Learning, etc. Typically, a numerical solution of such problems to the target accuracy ε demands for a high exponentially increasing computational cost ε−δs for some δ > 0, so that numerical computations even for a moderate values of ε will result 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, εdimensions, tractability, exponential tractability. 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 University, Oldenburg, Germany. Dinh Dung’s research work is funded by Vietnam National Foundation for Science and Technology 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. The authors would like to thank Erich Novak for valuable remarks and comments. 1 2 ˜ ALEXEY CHERNOV AND DINH DUNG 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 of variables s 1). This consideration is true in general, but in some cases the curse of dimensionality can be overcome, particularly when the high-dimensional data belong to certain classes of functions having mixed smoothness. Such functions can be optimally represented by means of the hyperbolic cross (HC) approximation. For example, trigonometric polynomials with frequencies in HCs have been widely used for approximating functions with a bounded mixed derivative or difference. These classical trigonometric HC approximations date back to Babenko [2]. For further sources on HC approximations in this classical context we refer to [16, 39] and the references therein. Later on, these terminologies were extended to approximations by wavelets [11, 35], by B-splines [17, 36], and to algebraic polynomial approximations where the power of algebraic polynomials approximants are in HCs [7, 9]. HC approximations 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] 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 smoothness, there are two kinds of HC used as the frequency domain of approximant trigonometric polynomials: continuous HCs s G(s, T ) := k ∈ Zs : max(|ki |, 1) ≤ T i=1 and step HCs formed from the dyadic blocks j s G∗ (s, T ) := j : j ∈ Zs+ , ji ≤ log T , i=1 where j := {k ∈ Zs : 2ji −1 ≤ |ki | < 2ji , i = 1, ..., s} (we refer to [16, 39] for further modifications of these HCs in trigonometric approximations of functions having a mixed smoothness and zero mean value in each variable). These HCs having the asymptotic cardinality T logs−1 T , play an important role in computing asymptotic orders of various characteristics of optimal approximation such as N -widths and ε-dimensions for classes of periodic functions having mixed smoothness. In this work we study approximations by trigonometric polynomials with frequencies from modified HCs s Γ(s, T, a) := k ∈ Ns0 : (ki + a) ≤ T (1.1) i=1 and s Γ± (s, T, a) := k ∈ Zs : (|ki | + a) ≤ T (1.2) i=1 where a > 0 is a fixed parameter. We derive tight and explicit-in-dimension upper and lower bounds for the cardinality of these modified HCs and then apply them in study of ε-dimensions – the inverses of the well known of Kolmogorov N -widths – of a correspondingly modified Korobov function classes U r,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 HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 3 N -width dN (W, X) is defined by dN (W, X) := inf sup inf LN f ∈W g∈LN f −g X, where the outer infimum is taken over all linear subspaces LN in X of dimension ≤ 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 the same object is termed “information complexity” or “ε-cardinality”). It is defined by nε (W, X) := inf N ∈ N : sup inf f ∈W g∈LN f −g X ≤ε , for some a linear subspace LN of X of dimension ≤ N . This approximation characteristic is the inverse of dN (W, X). In other words, the quantity nε (W, X) is the minimal number nε such that the approximation of W by a suitably chosen approximant nε -dimensional subspace L in X gives the approximation error ≤ ε (see [13], [14], [18]). The quantity nε represents a special case of the information complexity which is described by the minimal number n(ε, s) of information needed to solve the corresponding s-variate linear approximation problem of the identity operator within error ε (see [30, 4.1.4, 4.2]). For further information on this topic we refer the interested reader to the surveys in monographs [30, 31] and the references therein. The task of an efficient numerical approximation (for example, numerical solution of a linear, high-dimensional elliptic PDE by the finite element method) raises naturally the question of the optimal selection of the approximation (finite element) subspace with N degrees of freedom. Recalling the above terminology, this reduces to the problem of optimal linear approximation in X of functions from W by linear N -dimensional subspaces, Kolmogorov N -widths dN (W, X) and ε-dimension nε (W, X), where W is a smoothness class of functions having in some sense more regularity 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 suitable for this framework [20] (cf. also [5]). As a model we will consider functions on Rs which are 2π-periodic in each variable, as functions defined on the s-dimensional torus Ts := [−π, π]s for which the end points of the interval [−π, π] are identified for each coordinate component. The space H r1 (Ts ) := H r (T) ⊗ · · · ⊗ H r (T) consists of all periodic functions whose mixed derivatives of order r > 0 are L2 -integrable (i.e. having mixed smoothness of order r). For the unit ball U r1 (Ts ) in the space H r1 (Ts ), the following well-known estimates hold true: A(r, s)N −r (log N )r(s−1) ≤ dN (U r1 , L2 (Ts )) ≤ A (r, s)N −r (log N )r(s−1) , (1.3) or equivalently, B(r, s)ε−1/r | log ε|(s−1)/r ≤ nε (U r1 , L2 (Ts )) ≤ B (r, s)ε−1/r | log ε|(s−1)/r , (1.4) Here, A(r, s), A (r, s) and B(r, s), B (r, s) denote certain constants depending on the smoothness r and the dimension s which are usually not computed explicitly. The inequalities (1.3) were proved by Babenko [2] already in 1960 for the basic ˜ ALEXEY CHERNOV AND DINH DUNG 4 linear approximation by continuous HC spaces of trigonometric polynomials. These estimates are quite satisfactory if s is small and fixed. In the recent work of D˜ ung and Ullrich [20], A(r, s), A (r, s) and B(r, s), B (r, s) have been estimated from above and below explicitly in s when s is large. In their paper, the class U r1 (Ts ) is redefined in terms of the traditional dyadic decomposition of the frequency domain. These estimations are based on an approximation by trigonometric polynomials with frequencies in step HCs G∗ (s, T ) and explicit-indimension estimations of its cardinality |G∗ (s, T )|. However, the authors were able to estimate them from above only for very large n ≥ 2δs or very small ε ≤ 2−δs , for some δ > 0, (see [20, Thms. 4.10, 4.11]). This does not give a complete picture of the convergence rate in high-dimensional settings. The reason is that the step HC approximations of the class U r1 involve whole dyadic blocks of frequencies which have the cardinality at least 2s . In the present paper, to avoid this fact we suggest to replace H r1 (Ts ) by another space K r,a (Ts ) which is defined as a modification of the well-known Korobov space, and construct appropriate continuous HCs for the trigonometric approximations of functions from this space. This will allows to derive very tight and explicit-indimension upper and lower estimates for the cardinality of continuous HCs and further sharp estimates for ε-dimensions. Observing that the asymptotic orders of these 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 K r,a (Ts ) will be parametrized by a positive number a > 0 controlling the bound of the smoothness component of the unit ball of K r,a (Ts ) as a subset in L2 (Ts ). The parameter a allows also for simultaneous and sharp derivation of upper and lower bounds for the cardinality of HCs. Let us introduce the spaces K r,a (Ts ). For this we recall that L2 (Ts ) is the Hilbert space of functions on Ts equipped with the inner product (f, g) := (2π)−s f (x)g(x) dx. Ts The norm in L2 (Ts ) is defined as f := (f, f )1/2 . For k ∈ Zs , let fˆ(k) := (f, e−k ) 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 ∈ Zs we define a scalar λa (k) by s λa (k) := λa (kj ) with λa (kj ) := (1 + a−1 |kj |) j=1 and the Korobov function κr,a (in distributional sense) by the relation λa (k)−r ek (x), κr,a (x) := x ∈ Ts . k∈Zs Denote by (f ∗ g)(x) := (f (x − ·), g) the convolution of f and g. Then the Hilbert space K r,a (Ts ) is defined as K r,a (Ts ) := {f : f = κr,a ∗ g, g ∈ L2 (Ts )} with the norm f K r,a (Ts ) := g and the inner product (f, f )K r,a (Ts ) := (g, g ), where f = κr,a ∗ g and f = κr,a ∗ g . HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 5 By the convolution theorem fˆ(k) = λa (k)−r gˆ(k) and thus by Parseval’s identity f 2 K r,a (Ts ) λa (k)2r |fˆ(k)|2 . = (1.5) k∈Zs Thus, the space K r,a (Ts ) can be seen as the set of all functions f ∈ L2 (Ts ) for which the right hand side of (1.5) is finite. Notice also that the norm of the embedding K r,a (Ts ) → L2 (Ts ) is one and does not depend on s, r, a. The notion of space K r,a (Ts ) is a modification of the notion of the classical Korobov space. For r > 1/2, the kernel Ka defined at x and y in Ts as Ka (x, y) := κ2r,a (x−y) is the reproducing kernel for the Hilbert space K r,a (Ts ). For a definitive treatment of reproducing kernel, see, for example, [1]. The linear span of the set of functions {κr,a (· − y) : y ∈ Ts } is dense in K r,a (Ts ). In the language of Machine Learning, this means that the reproducing kernel for the Hilbert space is universal. In the recent paper [19] some upper and lower bounds of multivariate approximation by translates of the Korobov function on sparse Smolyak grids have been 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 K r,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 functions, especially in high-dimensional settings. For further information, see detailed surveys and references in the books [39], [37], [30]. Note that the spaces H r1 (Ts ), K r,a (Ts ) and Hs,r (Ts ) coincide as function spaces equipped with equivalent norms. However, if s is large, the unit balls with respect to the norms of these spaces differ significantly. As will be shown in the present paper, for the space K r,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 K r,a (Ts ) as a subset in L2 (Ts ), and determines crucially different high-dimensional approximation properties of functions from K r,a (Ts ). Let U r,a (Ts ) be the unit ball in the class K r,a (Ts ), i.e. U r,a (Ts ) = {f ∈ K r,a (Ts ) : f K r,a (Ts ) ≤ 1}. In this paper, we derive new upper and lower bounds for nε (U r,a (Ts ), L2 (Ts )) with explicit 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 for the index sets Γ(s, T, a) and Γ± (s, T, a) defined in (1.1) and (1.2) above. The sets Γ(s, T, a) and Γ± (s, T, a) will be referred to as a corner and a symmetric continuous hyperbolic crosses, or shortly HCs. Denote by |G| the cardinality of a finite set G. Notice that the problem of computing |Γ(s, T, a)| and |Γ± (s, T, a)| in our setting is itself interesting as a problem of a classical direction in Number Theory investigating the number of the integer points 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 of the 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 worth mentioning 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. 6 ˜ ALEXEY CHERNOV AND DINH DUNG Motivated by all the above arguments, the main goal of this paper to prove upper and lower bounds for the hyperbolic crosses |Γ(s, T, a)| and |Γ± (s, T, a)| in a new high-dimensional approach as a functions of three variables s, T, a when the dimension s and the real parameter T > 0 may be (but not necessarily is) large and a the real parameter ranging from 0 to ∞. These cardinality estimates are then applied for the estimation of nε (U r,a (Ts ), L2 (Ts )). Although T is the main parameter in the 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 through the 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, a ), where a may 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 simultaneous and sharp derivation of upper and lower bounds for cardinalities of the HCs. We refer to Section 2 for the rigorous construction. We give now a brief overview of the main results of the present paper. As a by-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 be tightly estimated by ts 1 < (−1)s (s − 1)! t + s ∞ k=s (−t)k 1 ts < , k! (s − 1)! t + s − 1 s ≥ 1, t > 0. (1.6) To the knowledge of the authors, so far these estimates have been unknown. From this 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, a ) we derive very tight non-asymptotic bounds for the cardinality of Γ(s, T, a) and Γ± (s, T, a). Let us formulate them. For every s ∈ N and every a > 1/2, there exists T∗ (s, a) > 0 such that there holds the upper bound for |Γ(s, T, a)| for every T ≥ T∗ (s, a) s |Γ(s, T, a)| < T ln T − s ln(a − 1/2) 1 . (s − 1)! ln T − s ln(a − 1/2) + s − 1 This results is proved on the basis of (1.6) and a non-trivial inequality between |Γ(s, T, a)| and the volume of the shifted smooth HC P (s, T, a − 1/2) (see Theorem 2.4 and Theorem 3.5). If the restriction of T being sufficiently large T ≥ T∗ (s, a) 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 lower bound (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 lower bound for |Γ± (s, T, a)| |Γ± (s, T, a)| > T (ln T − s ln(a + 1/2))s 2s , (s − 1)! ln T − s ln(a + 1/2) + s HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 7 and for every s ∈ N, every a > 1/2 and T > (a − 1/2)s , there holds the upper bound for |Γ± (s, T, a)| |Γ± (s, T, a)| < 2s T (ln T − s ln(a − 1/2))s . (s − 1)! ln T − s ln(a − 1/2)) + s − 1 These new results are then utilized in derivation of tight upper and lower bounds for nε (U r,a (Ts ), L2 (Ts )) explicit in ε, s, a. If r > 0, s ∈ N and a > 0, we prove that for every q ∈ [2, ∞) satisfying λ := a − 2/q > 0, and every ε ∈ (0, 1], we have nε (U r,a (Ts ), L2 (Ts )) ≤ qas(1+q) λ−qs ε−(1+q)/r . In the next step of our investigation, these estimates are sharpened by other upper and lower bounds for nε (U r,a (Ts ), L2 (Ts )). In particular, for r > 0, s ≥ 2, a > 1/2, we prove for every ε ∈ (0, 1], nε (U r,a (Ts ), L2 (Ts )) ≤ (2a)s ε−1/r (ln ε−1/r + s ln a)s (s − 1)! ln ε−1/r + s ln a + s − 1 and for every ε ∈ (0, a−rs ), nε (U r,a (Ts ), L2 (Ts )) ≥ (2a)s ε−1/r (ln ε−1/r − s ln a)s − 1, (s − 1)! ln ε−1/r − s ln a + s a where a := a−1/2 and a := a+1/2 a . We also show that the problem of nε (U r,a (Ts ), L2 (Ts )) is weakly tractable but polynomially intractable for every a > 0. The tractability of linear approximation problem 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 functions from the non-periodic modified Korobov space K r,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 to other HC approximations in a Hilbert space. In brief, the paper is organized as follows. In Section 2, we prove preliminary estimates for |Γ(s, T, a)| and |Γ± (s, T, a)| via the volume of corresponding smooth HCs. Section 3 is the core of the present work. There, we prove non-asymptotic tight upper and lower estimates for the volume of smooth HCs P (s, T, a), and derive from them and the results of Section 2 non-asymptotic tight upper and lower 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 (U r,a (Ts ), L2 (Ts )). In Section 4, we also investigate tractabilities of the problem of nε (U r,a (Ts ), L2 (Ts )). In Section 5, we extend the methods and results for periodic approximations to the non-periodic case and approximations by polynomials. The Appendix in Section 6 contains the 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| = dx. Ω ˜ ALEXEY CHERNOV AND DINH DUNG 8 This is a slight abuse of notation since for a (discrete) finite set G we use also the notation |G| for the cardinality of G. However, there is no ambiguity for a given 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, a )| of the corresponding corner smooth HC s P (s, T, a ) := x ∈ Rs+ : (xj + a ) ≤ T . (2.1) j=1 where a and a can in general be nonequal. Consider the set (k + [0, 1]s ). Q(s, T, a) := k∈Γ(s,T,a) Obviously, it holds that |Γ(s, T, a)| = |Q(s, T, a)|. (2.2) Using this equation we will compare |Γ(s, T, a)| with the volume of P (s, T, a ) I(s, T, a ) := |P (s, T, a )|. (2.3) Notice that if T is given and a > a, then Γ(s, T, a ) ⊂ Γ(s, T, a), Γ± (s, T, a ) ⊂ Γ± (s, T, a), P (s, T, a ) ⊂ P (s, T, a). s Define x := ( x1 , ..., xs ) for x ∈ R , where t 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 x ∈ Γ(s, T, a). Therefore, the relation s T ≥ s xj + a + 1 ≥ j=1 s xj + a ≥ j=1 xj + a j=1 implies (2.4) with sharp inclusions and, by (2.2) and (2.3), the inequalities (2.5). The next lemma generalizes and sharpens the left inequality in (2.5). Lemma 2.2. Suppose 0 < δ ≤ 1. Then for every s ∈ N, T ≥ δ s , and a > δ, it holds that |Γ(s, T, a)| < (1/δ)s I(s, T, a − δ), (2.6) and |Γ± (s, T, a)| < (2/δ)s I(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 ˜ ˜ s, T, a) := {x : xj = yj − (1 − δ), y ∈ Q(s, T, a)} Q(δ) := Q(δ, ˜ to simplify the notations) and its subset (we suppress the dependence on s, T, a in Q ˜ + (δ) := {x ∈ Q(δ) ˜ Q : x ≥ 0}. HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 9 For yj = xj + (1 − δ) we observe s T ≥ s ( yj + a) ≥ j=1 ˜ + (δ) implying Q (xj + a − δ), j=1 P (s, T, a − δ) and therefore ˜ + (δ)| ≤ I(s, T, a − δ). |Q Let e ∈ {0, 1}s and consider ˜ e (δ) = {x ∈ Q(δ) ˜ Q : xj < δ if ej = 0 ∧ xj ≥ δ if ej = 1} ˜+ ˜ ˜+ and correspondingly, Q e (δ) = Qe (δ) ∩ Q (δ). This defines disjoint decompositions ˜ Q(δ) = ˜ e (δ), Q ˜ + (δ) = Q e∈{0,1}s ˜+ Q e (δ). e∈{0,1}s ˜ e (δ) and Q ˜+ For the volumes of Q e (δ) we obviously have the relations s ˜ e (δ)| 1 |Q = |e| , + ˜ δ |Qe (δ)| where |e| := ej . j=1 Therefore, ˜ |Q(s, T, a)| = |Q(δ)| = ˜ e (δ)| |Q e∈{0,1}s = δ −|e| −s ˜+ |Q I(s, T, a − δ) e (δ)| ≤ δ e∈{0,1}s and the proof is complete. Due to the specific geometrical structure of the symmetric HC Γ± (s, T, a), the upper bound (2.7) can be improved for δ = 21 as in the following lemma. Lemma 2.3. For every s ∈ N a > 1/2 and T ≥ 1, it holds that 2s I(s, T, a + 21 ) < |Γ± (s, T, a)| < 2s I(s, T, a − 21 ). (2.8) Proof. We set x ∈ Rs : xj ∈ [kj − 12 , kj + 12 ), j = 1, . . . , s . Q± (s, T ) := k∈Γ± (s,T,a) Then we have |Γ± (s, T, a)| = |Q± (s, T )|. The relations s s j=1 s (|xj ± 12 | + a) ≥ (|xj | + a + 21 ) ≥ T ≥ j=1 (|xj | + a − 21 ) j=1 imply |x| ∈ P (s, T, a + 12 ) Q± (s, T, a) |x| ∈ P (s, T, a − 12 ) , 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 − 21 ) for T large enough. Namely, we have the following theorem. 10 ˜ ALEXEY CHERNOV AND DINH DUNG 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 − 21 ), ∀T ≥ T∗ (s, a). (2.9) Observe that by (2.5) one can see that for every s ∈ N, T > 0, and a > 1, the inequality |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 means that Γ(s, T, a) ⊂ P (s, T, a ) for any a > a − 1, in particular, Γ(s, T, a) ⊂ 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 Theorem 2.4 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 HCs The 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, a ) of the smooth HC P (s, T, a), which is, as a matter 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 is related to the s-th remainder of the Taylor series of the exponential exp(−t). In the next important step, we will give tight two-sided non-asymptotic bounds for this remainder. 3.1. Tight non-asymptotic bounds for the volume of smooth HCs and the cardinality of HCs. To formulate the result, for s ∈ N0 , we introduce the function ∞ Fs (t) := (−1)s (−1)n pn (t), t ∈ R, (3.1) n=s where ps (t) := ts /(s!). Observe that exp(−t) = F0 (t) and Fs (t) is the absolute value of the sth remainder of the Taylor series of the exponential exp(−t). The following lemma shows that the problem of estimations of the volume of smooth HC I(s, T, a) can be reduced to estimations of Fs (t). Lemma 3.1. Suppose T > 0, s ∈ N and a > 0. Then there holds the identity 0, T ≤ as , T Fs (ln T − s ln a), T > as . I(s, T, a) = (3.2) Proof. The continuous HC P (s, T, a) is empty if T < as , and contains only the origin of Rs if T = as . This yields the first line in (3.2). Suppose now that T > as . From [12, Lemma 3] we know that s−1 I(s, T, a) = (−1)s+1 (T − as ) + T (ln T − s ln a)n (−1)s−1−n . n! n=1 HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 11 Then for t := ln T − s ln a > 0, there holds s−1 1 tn (−1)s−1−n I(s, T, a) = (−1)s+1 (1 − e−t ) + T n! n=1 s−1 = (−1)s e−t − tn (−1)n n! n=0 ∞ = (−1) s (−1)n n=s tn n! = Fs (t). 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-hand side, whose behaviour is not clearly seen from (3.1). To make it explicit, we prove very tight bounds for the series Fs (t) in terms of the single summand ps−1 . This approximation is somewhat surprisingly good, as we observe from the following non-asymptotic estimates. Theorem 3.2. For any s ∈ N and t > 0, the following estimate holds true t t ps−1 (t) < Fs (t) < ps−1 (t). t+s t+s−1 (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 1 T (ln T − s ln a)s 1 T (ln T − s ln a)s < I(s, T, a) < . (s − 1)! ln T − s ln a + s (s − 1)! (ln T − ln a + s − 1) A combination of Corollary 3.3 and the results in Section 2 implies various lower and 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 T (ln T − s ln a)s , (s − 1)! 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)| < T (ln T − s ln(a − 1))s 1 . (s − 1)! 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 s T ln T − s ln(a − 1/2) 1 |Γ(s, T, a)| < , (s − 1)! ln T − s ln(a − 1/2)s − 1 ∀T ≥ T∗ (s, a). 12 ˜ ALEXEY CHERNOV AND DINH DUNG Analogously, Lemma 2.3 and Corollary 3.3 provide sharpened upper and lower bounds 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)| > 2s T (ln T − s ln(a + 1/2))s , (s − 1)! ln T − s ln(a + 1/2) + s (3.6) and for every s ∈ N, every a > 1/2 and T > (a − 1/2)s , |Γ± (s, T, a)| < 2s T (ln T − s ln(a − 1/2))s . (s − 1)! ln T − s ln(a − 1/2)) + s − 1 (3.7) Theorem 3.7. For every s ∈ N, a > 0, 0 < δ ≤ 1, δ < a and T > 0, it holds that |Γ(s, T, a)| < δ −1 T 1+1/δ (a − δ)−s/δ , (3.8) and |Γ± (s, T, a)| < 2δ −1 T 1+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 the case where T ≥ as > (a − δ)s . By Lemmas 2.2, 3.1 and Theorem 3.2 we have that |Γ(s, T, a)| ≤ δ −s I(s, T, a − δ) T [ln T − s ln(a − δ)]s (s − 1)!(ln T + s[1 − ln(a − δ)] − 1) [δ −1 (ln T − s ln(a − δ))]s−1 < δ −1 T (s − 1)! < δ −s < δ −1 T exp[δ −1 (ln T − s ln(a − δ))] = δ −1 T 1+1/δ (a − δ)−s/δ . 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 the dimension s when s → ∞. It will be used in the study of HC approximations and the problem of ε-dimensions in Sections 4–5. Remark 3.8. The recent work [28] contains an independent investigation of approximation 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 dimension s. It contains somewhat related estimates of the volume of the smooth HC I(s, T, a) for the special case a = 1. We mention in particular [28, Lemma 3.3] showing T (ln T )s−2 T (ln T )s−1 T (ln T )s−1 − ≤ I(s, T, 1) ≤ . (3.9) (s − 1)! (s − 2)! (s − 1)! Our result in Corollary 3.3 above yields in the special case a = 1 the estimate T (ln T )s−1 ln T T (ln T )s−1 ln T ≤ I(s, T, 1) ≤ (3.10) (s − 1)! ln T + s (s − 1)! ln T + s − 1 which holds for any s ∈ N and T > 1 and, moreover, proves the corresponding result for the case a = 1. The upper bound in our result (3.10) is always sharper than the upper bound in (3.9), whereas the lower bound in (3.9) is sharper for large HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 13 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 parameter a > 0 (taking fractional values as well) allows for more flexibility and, as a consequence, better upper and lower estimates for the cardinality of the hyperbolic crosses in Lemma 2.1 – 2.3, Theorem 2.4 and Corollary 3.5 – 3.7. 3.2. Proof of Theorem 3.2. We define Fs+1 (t) . (3.11) ps (t) Observe that Theorem 3.2 is equivalent to the following statement. For any s ∈ N and t > 0, we have t t < hs−1 (t) < . (3.12) t+s t+s−1 Therefore, to prove Theorem 3.2 we will verify (3.12). The proof of (3.12) requires some auxiliary lemmas. Note that (3.3) and (3.12) are claimed for a fixed couple (s, t) and are therefore, non-asymptotic error estimates. Thus, we may assume that t is a fixed positive real number and write ps , Fs , hs instead of ps (t), Fs (t), hs (t) if possible, in order to simplify and shorten the notations. hs (t) := Lemma 3.9. For every t > 0 and s ∈ N0 , we have 0 < Fs (t) < ∞. (3.13) Proof. Suppose s ≥ t. Then pn s+1 n+1 ≥ > 1, = pn+1 t t ∀n ≥ s, i.e. the sequence {pn }n≥s converges monotonously to zero. Thus Fs = (ps − ps+1 ) + (ps+2 − ps+3 ) + · · · > 0 >0 >0 and (3.13) follows. In order to analyse the case t > s we observe the identity s−1 (−1)s Fs = exp(−t) − (−1)n n=0 tn n! and the relation pn−2 n−1 s−1 = ≤ < 1, pn−1 t t 2 ≤ n ≤ s, s−1 i.e. the finite sequence {pn }n=0 is monotonously increasing. If s is even, we have Fs (t) = exp(−t) + (−p0 + p1 ) + · · · + (−ps−2 + ps−1 ) > 0. >0 >0 >0 If s is odd we regroup the terms and obtain −Fs (t) = (exp(−t) − p0 ) + (p1 − p2 ) + · · · + (ps−2 − ps−1 ) < 0. 0 is equivalent to Fs+1 (t) > 0. Similarly, hs (t) < 1 is equivalent to Fs (t) > 0, see (3.14). The rest follows from Lemma 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 − s Fs s Fs =1− = 1 − hs−1 , ps t ps−1 t hence the sequence {hs }∞ s=0 is defined via the recurrence relation hs (t) = 1 − s hs−1 (t), s ∈ N, t h (t) = 1 − e−t . 0 Lemma 3.11. The two-sided bound t t < hs (t) < t+s+1 s+t holds for every t > 0 and s ∈ N0 if s ≤ t − 1. (3.15) (3.16) Proof. The proof is by induction on s. By simple calculations we have t < h0 (t) = 1 − e−t < 1 t+1 and 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 s s t s t hs = 1 − hs−1 < 1 − =1− = . t t t+s t+s t+s For the lower bound we have s s t s t−1 t hs = 1 − hs−1 > 1 − =1− = ≥ t t t+s−1 t+s−1 t+s−1 t+s+1 where the last estimate holds if and only if s ≤ t − 1. Indeed, it is equivalent to (t − 1)(t + s + 1) ≥ t(t + s − 1) ⇔ −(t + s) + t − 1 ≥ −t ⇔ t − 1 ≥ s. Lemma 3.12. For every t > 0 and s ∈ N0 , we have t < hs (t). t+s+1 (3.17) HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 15 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 t+s+1 ∆s (t) := hs+1 (t) − hs (t) = 1 − hs (t), s ∈ N0 , (3.18) t where the last relation follows from (3.15). Then (3.17) is equivalent to ∀s ∈ N0 . ∆s (t) < 0, (3.19) Lemma 3.11 implies (3.17) and hence (3.19) for s ≤ t−1. Suppose now that s > t−1 and show (3.19) by contradiction. For this, we utilize (3.15) to obtain the following recurrence relation for ∆s : ∆s = s(s − 1) t + s + 1 s−t+1 + ∆s−2 . t(s + t − 1) t2 t+s−1 (3.20) Indeed, we have t t t+s ∆s−1 = hs − hs−1 = hs − (1 − hs ) = − + hs , s s s and therefore, hs = s t+s ∆s−1 + t s . Hence, by (3.18) t+s+1 s t t + s + 1 s(t + s + 1) ∆s−1 + =1− − ∆s−1 t t+s s t+s t(t + s) s(t + s + 1) 1 − ∆s−1 =− t+s t(t + s) 1 s(t + s + 1) s−1 t+s 1 =− + + ∆s−2 t+s t(t + s) t+s−1 t t+s−1 1 s(t + s + 1) s(s − 1) t + s + 1 = −1 + + ∆s−2 t+s t(t + s − 1) t2 t+s−1 s(s − 1) t + s + 1 s2 + s − t2 + t + ∆s−2 = (t + s)t(t + s − 1) t2 t+s−1 s−t+1 s(s − 1) t + s + 1 = + ∆s−2 . t(s + t − 1) t2 t+s−1 ∆s = 1 − Suppose now that (3.19) is wrong, i.e. ∆s−2 (t) ≥ 0 for some s > t + 1. Then from (3.20) it follows that ∆s (t) is positive: ∆s (t) > 1 1 min 1, t t + 1+ 1 t 2 ∆s−2 > 1 1 min 1, t t + ∆s−2 > 0. Hence, for every k ∈ N0 the increments ∆s+2k (t) are positive and admit the lower bounds 1 1 k 1 ∆s+2k (t) > min 1, + ∆s+2k−2 (t) > min 1, → ∞, k → ∞. t t t t This is a contradiction to Corollary 3.10 which yields ∆s+2k (t) = hs+2k+1 (t) − hs+2k (t) < 1. The proof is complete. ˜ ALEXEY CHERNOV AND DINH DUNG 16 Lemma 3.13. For every t > 0 and s ∈ N0 , we have t . t+s hs (t) < (3.21) Proof. By Lemma 3.11 it suffices to prove (3.21) for s > t − 1. From Lemma 3.12 we have t > 0. s (t) := hs (t) − t+s+1 On the other hand, (3.15) implies s (t) + t = hs (t) = 1 − t+s+1 s − =1− t+s s hs−1 (t) t s t s − s−1 (t) = t t+s t s−1 (t). This means that s (t) = t s − (t + s)(t + s + 1) t s−1 (t) > 0. Hence, t2 t t < + t+s s(t + s)(t + s + 1) t + s t(s + 1) t t(t + st + s2 + s) = < , = s(t + s)(t + s + 1) s(t + s + 1) t+s−1 hs−1 (t) = s−1 (t) + 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 lower bounds for the ε-dimension nε (U r,a (Ts ), L2 (Ts )). As auxiliary results we derive an upper bound for the L2 (Ts )-error in the HC approximation by trigonometric polynomials with frequencies from Γ± (s, T, a) (Jackson inequality) as well as the corresponding inverse estimate (Bernstein inequality). We give also a classification of tractability for the problem of ε-dimensions of U r,a (Ts ). 4.1. HC approximations. For a finite set M ⊂ Zs , we denote by the T (M ) subspace of trigonometric polynomials with frequencies in M , i.e., trigonometric polynomials g of the form g= gˆ(k)ek . k∈M We abbreviate T (s, T, a) := T (Γ± (s, T, a)). For a function f ∈ L2 (Ts ), we define the Fourier operator ST as fˆ(k)ek . ST (f ) := k∈Γ± (s,T,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 the error of the orthogonal projection. HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 17 Lemma 4.1. For arbitrary T ≥ 1, s ∈ N, a > 0 and r ≥ 0, we have f − ST (f ) ≤ (T /as )−r f K r,a (Ts ) ∀f ∈ K r,a (Ts ). , Proof. Indeed, from (1.5) we have for every f ∈ K r,a (Ts ), f − ST (f ) 2 fˆ(k) = 2 k∈Γ± (s,T,a) ≤ λa (k)2r fˆ(k) λa (k)−2r sup k∈Γ± (s,T,a) 2 k∈Γ± (s,T,a) −2r = inf k∈Γ± (s,T,a) λa (k) ≤ (T /as )−2r f f 2 K r,a (Ts ) 2 K r,a (Ts ) . Corollary 4.2. For arbitrary T ≥ 1, we have f −g sup inf f ∈U r,a (Ts ) g∈T (s,T,a) = sup f − ST (f ) ≤ (T /as )−r . f ∈U r,a (Ts ) Next, we prove a Bernstein type inequality. Lemma 4.3. For arbitrary T ≥ 1, we have f ≤ (T /as )r f , f ∈ T (s, T, a). K r,a (Ts ) Proof. Indeed, by (1.5) we have for every f ∈ T (s, T, a), f 2 K r,a (Ts ) λa (k)2r |fˆ(k)|2 = k∈T (s,T,a) ≤ |fˆ(k)|2 ≤ (T /as )2r f λa (k)2r sup k∈T (s,T,a) 2 . k∈T (s,T,a) Corollary 4.4. Let ε ∈ (0, 1]. Then we have |Γ± (s, as ε−1/r , a)| − 1 ≤ nε (U r,a (Ts ), L2 (Ts )) ≤ |Γ± (s, as ε−1/r , a)|. (4.1) Proof. Let T := as ε−1/r and N := |Γ± (s, T, a)|, then (4.1) is equivalent to dN (U r,a (Ts ), L2 (Ts )) ≤ (T /as )−r ≤ dN −1 (U r,a (Ts ), L2 (Ts )). (4.2) The first inequality in (4.2) follows from Corollary 4.2 and the relation dim T (s, T, a) = |Γ± (s, T, a)| = N. (4.3) To establish the second one, we need a result proven by Tikhomirov in [40, Theorem 1], stating that for an n-dimensional subspace Ln in a Banach space X, and Bn (δ) := {f ∈ Ln : f X ≤ δ} it holds that dn−1 (Bn (δ), X) = δ. (4.4) Consider the subset B(T ) := {f ∈ T (s, T, a) : f ≤ (T /as )−r } in L2 (Ts ). By Lemma 4.3 we have B(T ) ⊂ U r,a (Ts ). Applying (4.4), by (4.3) we get dN −1 (U r,a (Ts ), L2 (Ts )) ≥ dN −1 (B(T ), L2 (Ts )) = (T /as )−r . ˜ ALEXEY CHERNOV AND DINH DUNG 18 4.2. Estimates of ε-dimensions. Corollary 4.4 combined with results of Section 3 allows to derive a quantitative upper bounds for ε-dimensions. Theorem 4.5. Let r > 0, s ∈ N, a > 0. Then for every q ∈ [2, ∞) satisfying the inequality λ := a − 2/q > 0, and every ε ∈ (0, 1], we have nε (U r,a (Ts ), L2 (Ts )) ≤ qas(1+q) λ−qs ε−(1+q)/r . Proof. For fixed q ∈ [2, ∞) and ε ∈ (0, 1] we define δ := 2/q and λ := a − 2/q > 0, and T := as ε−1/r . By Corollary 4.4 and Theorem 3.7 we have nε (U r,a (Ts ), L2 (Ts )) ≤ |Γ± (s, T, a)| < 2δ −1 T 1+2/δ (a − δ)−2s/δ = qas(1+q) λ−qs ε−(1+q)/r . Next, we sharpen the results in Theorem 4.5 by establishing tight upper and lower bounds on nε (U r,a (Ts ), L2 (Ts )) as a function of three variables ε, s, a, respectively. Theorem 4.6. Let r > 0, s ≥ 2, a > 1/2. Then we have for every ε ∈ (0, 1], nε (U r,a (Ts ), L2 (Ts )) ≤ (2a)s ε−1/r (ln ε−1/r + s ln a)s (s − 1)! ln ε−1/r + s ln a + s − 1 (4.5) and for every ε ∈ (0, a−rs ) nε (U r,a (Ts ), L2 (Ts )) ≥ where a := a a−1/2 and a := (2a)s ε−1/r (ln ε−1/r − s ln a)s − 1, (s − 1)! ln ε−1/r − s ln a + s (4.6) a+1/2 a . Proof. Both upper and lower bounds follow by sequential application of Corollary 4.4 and Theorem 3.6. The upper bound (4.5) holds true for any ε ∈ (0, 1], since the restriction T > (a − 12 )s is satisfied for T := as ε−1/r and this range of ε. On the other hand, the lower bound (4.6) holds in the range T > (a + 12 )s which is equivalent to ε ∈ (0, a−rs ). An important case of Theorem 4.6 where a = 1 reads as follows. Corollary 4.7. Let r > 0, s ≥ 2. Then we have for every ε ∈ (0, 1], 2s nε (U r,1 , L2 (Ts )) ≤ ε−1/r (ln ε−1/r + s ln 2)s−1 (s − 1)! and for every ε ∈ (0, (2/3)rs ) nε (U r,1 , L2 (Ts )) ≥ 2s ε−1/r [ln ε−1/r − s ln(3/2)]s − 1. (s − 1)! ln ε−1/r − s ln(3/2) + s Remark 4.8. It is worth to emphasize that in high-dimensional approximation, the form of the upper and lower bounds for nε (U r,a (Ts ), L2 (Ts )) as in Theorem 4.6 is more natural and suitable than the form of those as in (1.4). Indeed, for the latter one, the traditional terms ε−1/r | log ε|(s−1)/r are a priori split from constants which are actually a function of dimension parameter s (and smoothness parameter r), and therefore, any high-dimensional estimate based on them leads to a rougher bound. The situation is analogous for Kolmogorov N -widths of classes of functions having a mixed smoothness if the terms N −r (log N )r(s−1) are a priori split from constants depending on s as in (1.3) (cf. [20, 28]). HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 19 4.3. Tractabilities of the problem of ε-dimensions. Let us give notions of tractability for a problem of ε-dimensions. Let W be a subset in L2 (Ts ). We say that the problem of ε-dimensions of W is weakly tractable if lim ε−1 +s → ln nε (W, L2 (Ts )) = 0, ε−1 + s ∞ and intractable otherwise. The problem of ε-dimensions of W is polynomially tractable if there are nonnegative numbers C, p and q such that nε (W, L2 (Ts )) ≤ Csq ε−p for all ε ∈ (0, 1] and s ∈ N. For details about notions of tractability see [30, Section 4.4, Chapter 4]. Usually, in Information-Based Complexity, it is assumed that the information is linear (like Fourier coefficients) and this leads to the linear N -widths λN (W, X). In a Hilbert space X, the Kolmogorov N -widths dN (W, X) and linear N -widths λN (W, X) are the same. This allows us to study the tractability of the problem of nε (U r,a (Ts ), L2 (Ts )) in the sense information complexity for the corresponding traditional linear problem in the worst case setting, see [30, Section 4.2, Chapter 4] for details. Let us show that the tractability of the problem of nε (U r,a (Ts ), L2 (Ts )) can be seen as the tractability of a linear tensor product problem over the Hilbert spaces K r,a (Ts ) and L2 (Ts ) (see [30, Section 5.1, Chapter 5]) for details). First of all, observe that K r,a (Ts ) = K r,a (T) ⊗ · · · ⊗ K r,a (T) is the s-fold tensor product Hilbert space of K r,a (T), and L2 (Ts ) = L2 (T) ⊗ · · · ⊗ L2 (T) the s-fold tensor product Hilbert space of L2 (T). By the definitions one can see that the univariate identity operator I : K r,a (T) → L2 (T) is a compact linear operator. Then the operator I ∗ I : K r,a (T) → K r,a (T) is a positive definite self-adjoint operator which is also compact. Clearly, I ∗ I is the identity operator in K r,a (T). ∞ Let (µk , ϕk ) k=1 be the eigenpairs of the operator I ∗ I in an non-increasing order of the eingenvalues µk , so that I ∗ Iϕk = µk ϕk ; From the definition of K r,a µk ≥ µk+1 ; (ϕk , ϕs )K r,a (T) = δk,s . (T) it follows that µk = 1 + a−1 k/2 −2r , ∀k ∈ N, and consequently, µ1 = 1, and µk = O(k −2r µ2 = (1 + a−1 )−2r < 1 (4.7) ) implying in particular that µk = o (ln k)−2 (ln ln k)−2 , k → ∞. (4.8) Observe that the problem of nε (U r,a (Ts ), L2 (Ts )) can be considered as the approximation problem Is : K r,a (Ts ) → L2 (Ts ) where Is (f ) = f , i.e., Is is the s-variate identity operator. On the other hand, Is can be formally defined as the tensor product operator: Is = I ⊗ · · · ⊗ I : K r,a (Ts ) → L2 (Ts ). ˜ ALEXEY CHERNOV AND DINH DUNG 20 Moreover, the norm of the embedding K r,a (Ts ) → L2 (Ts ) is one. This means that the approximation problem Is : K r,a (Ts ) → L2 (Ts ) is a linear tensor product problem in the worst case setting for both the absolute error criterion and normalized error criterion. Therefore, applying Theorem [30, Theorem 5.5] on the tractability of a linear tensor product problem, by (4.7) and (4.8) we obtain Theorem 4.9. Let r > 0, s ∈ N, a > 0. Then the problem of nε (U r,a (Ts ), L2 (Ts )) is weakly tractable but polynomially intractable. 5. Non-periodic HC approximations The above theory admits an extension to non-periodic functions with slight modifications. In this section we outline the main results. Suppose Is = [−1, 1]s is the reference s-dimensional cube and denote by L2 (Is , w) the Hilbert space of functions equipped with the weighted inner product (f, g)w = f (x)g(x)w(x)dx, Is where the Jacobi weight is give by s (1 − xj )α (1 + xj )β w(x) = j=1 for some fixed parameters α, β > −1. By f w = (f, f )w we denote the induced (α,β) ∞ norm. Further, let {Pk }k=0 be the family of orthonormal Jacobi polynomials, i.e. (α,β) Pk (x)P (α,β) (x)(1 − x)α (1 + x)β dx = δk . I Then s (α,β) Pk (α,β) (x) = Pkj (xj ) j=1 is an orthonormal basis on L2 (Is , w) and for any f ∈ L2 (Is , w) it holds that f 2 w |fk |2 , = where (α,β) fk = (f, Pk )w . k∈Ns0 Define α+β+1 . 2 Assuming a > 0 we consider the subspaces a := K r,a (Is , w) := {f ∈ L2 (Is , w) : f K r,a (Is ,w) < ∞} of L2 (Is , w) endowed with the norm f 2 K r,a (Is ,w) |λa (k)|2r |fk |2 . = k∈Ns0 To obtain an expression for the above norms in the differential form we recall that (α,β) Jacobi polynomials Pk can be characterized as unique solution of the following differential equation, see e.g. [38, Sect. 4.2] (α,β) LPk (α,β) (x) = k(k + α + β + 1)Pk (x) HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 21 where the differential operator L admits the representation L = (1 − x)−α (1 + x)−β d dx (1 − x)1+α (1 + x)1+β d dx . This yields for F := a2 I + L (α,β) FPk (α,β) (x) = (k + a)2 Pk (x) implying for F (s) = F ⊗ · · · ⊗ F (s times) (α,β) F (s) Pk (α,β) (x) = λa (k)2 Pk (x). The operator F is self-adjoint with respect to (·, ·)w (F (s) f, g)w = (f, F (s) g)w and it holds that f 2 K r,a (Is ,w) f (x)((F (s) )r f )(x)w(x)dx. = (5.1) Is Notice that (5.1) is indeed a symmetric expression, cf. [4]. For a function f ∈ L2 (Is , w) we define projection (α,β) ΠT (f ) := f k Pk k∈Γ(s,T,a) and (α,β) P(s, T, a) := span Pk : k ∈ Γ(s, T, a) . Similarly to Lemma 4.1 and Lemma 4.3 we obtain the Jackson and Bernstein inequalities Lemma 5.1. For arbitrary T ≥ 1, we have f − ΠT (f ) L2 (Is ,w) ≤ (T /as )−r f K r,a (Is ,w) , ∀f ∈ K r,a (Is , w). Lemma 5.2. For arbitrary T ≥ 1, we have f K r,a (Is ,w) ≥ (T /as )r f L2 (Is ,w) , ∀f ∈ P(s, T, a). Let U r,a (Is , w) be the unit ball in K r,a (Is , w). Similarly to the periodic case, Jackson and Bernstein inequalities imply. Corollary 5.3. Let ε ∈ (0, 1]. Then we have |Γ(s, as ε−1/r , a)| − 1 ≤ nε (U r,a (Is , w), L2 (I, w)) ≤ |Γ(s, as ε−1/r , a)|. From these corollaries and the corresponding explicit-in-dimension estimates of |Γ(s, T, a)| in Sections 2 and 3 we prove the following results. Theorem 5.4. Let r > 0, s ∈ N, a > 0. Then for any q ∈ [1, ∞) satisfying the inequality λ := a − 1/q > 0, and any ε ∈ (0, 1] we have nε (U r,a (Is , w), L2 (Is , w)) ≤ qas(1+q) λ−qs ε−(1+q)/r . Theorem 5.5. Let r > 0, s ≥ 2. Then for any a > 1 and ε ∈ (0, 1] it holds that nε (U r,a (Is , w), L2 (Is , w)) ≤ ε−1/r (ln ε−1/r + s ln a ˜)s as , (s − 1)! ln ε−1/r + s ln a ˜+s−1 ˜ ALEXEY CHERNOV AND DINH DUNG 22 where a ˜ := a a−1 . Moreover, for any a > 0 and ε ∈ (0, 1] it holds that as ε−1/r (ln ε−1/r )s − 1. (s − 1)! ln ε−1/r + s nε (U r,a (Is , w), L2 (Is , w)) ≥ Furthermore, for any a > 1/2 there exists ε∗ = ε∗ (s, a) ∈ (0, 1) such that for any ε ∈ (0, ε∗ ] it holds that nε (U r,a (Is , w), L2 (Is , w)) ≤ where a := as ε−1/r (ln ε−1/r + s ln a)s , (s − 1)! ln ε−1/r + s ln a + s − 1 a a−1/2 . Corollary 5.6. Let r > 0, s ∈ N. Then there is an ε∗ = ε∗ (s, a) ∈ (0, 1) such that for every ε ∈ (0, ε∗ ], nε (U r,1 (Is , w), L2 (Is , w))) ≤ 1 ε−1/r (ln ε−1/r + s ln 2)s−1 (s − 1)! and for every ε ∈ (0, 1] nε (U r,1 (Is , w), L2 (Is , w)) ≥ ε−1/r (ln ε−1/r )s 1 − 1. (s − 1)! ln ε−1/r + s Theorem 5.7. Let r > 0, s ∈ N, a > 0. Then the problem of the ε-dimension nε (U r,a (Is , w), L2 (Is , w)) is weakly tractable but polynomially intractable. 6. Appendix: Proof of Theorem 2.4 6.1. Auxiliary results. The prove of Theorem 2.4 relies on a number of auxiliary results summarized in this section. We start with basic notational agreements. We denote |Γ(0, T, a)| := 1 (6.1) for further convenience. For a function f : R → R and a, b ∈ R we abbreviate the sum b f (m) := m=a f (m), m∈M (a,b) where M (a, b) := a + n : n ∈ N0 , a + n ≤ b . For definiteness, we agree that the sum over an empty set equals zero. We have the following dimension-by-dimension decomposition s |Γ(s, T, a)| = # k ∈ Γ(s, T, a) : | supp k| = j j=0 s = j=0 s = j=0 j s # k ∈ Nj : (ki + a) ≤ T aj−s j i=1 s |Γ(j, T aj−s , a + 1)|. j Note that |Γ(j, T aj−s , a + 1)| = 0 ⇔ (a + 1)j ≤ T aj−s . (6.2) HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 23 This implies in particular that all summands in (6.2) are nonzero if and only if the term with j = s is nonzero. Extracting this last term we apply the decomposition (6.2) again and obtain for M = a, a + 1 . . . M s−1 s |Γ(j, T mj−s , m + 1)|. j |Γ(s, T, a)| = |Γ(s, T, M + 1)| + m=a j=0 The first term on the right-hand side vanishes if M satisfies (M + 1)s > T . Suppose as ≤ T and let n0 be the largest nonnegative integer such that n0 ≤ T 1/s − a. Then for M0 := a + n0 it holds that M0 ≤ T 1/s and M0 + 1 > T 1/s implying |Γ(s, T, M + 1)| = 0 and therefore T 1/s s−1 s |Γ(j, T mj−s , m + 1)|. j |Γ(s, T, a)| = m=a j=0 (6.3) Notice that the right-hand side of (6.3) might contain zero summands. Moreover, decomposition (6.3) is formally true also when |Γ(s, T, a)| = 0. Indeed, in this case as > T and the summation over m in (6.3) runs over an empty set, which we formally interpret as zero. We have a similar decomposition for the integral. Denote I := [0, 1] and set for the notational convenience I(0, T, a)) := 1. (6.4) 1 Then for any a > 2 s−1 I(s, T, a − 1 2) = I(s, T, a + 1 2) + j=0 s−j s j (t + a − 21 )−1 , a + 21 ) dt I(j, T t∈Is−j =1 and consequently, for m ˜ = a, a + 1 . . . I(s, T, a− 12 ) = m ˜ s−1 I(s, T, m+ ˜ 12 )+ m=a j=0 s j s−j (t +m− 12 )−1 , m+ 21 ) dt. I(j, T t∈Is−j =1 The first term on the right-hand side first is zero if (m ˜ + 12 )s ≥ T . Thus, setting 1/s m ˜ := T T 1/s s−1 I(s, T, a − 1 2) = m=a j=0 s j s−j (t + m − 12 )−1 , m + 12 ) dt. (6.5) I(j, T t∈Is−j =1 Note that the above sum may contain some zero summands. The proof of Theorem 2.4 will be based on the comparison of |Γ(s, T, a)| and I(s, T, a − 12 ) involving their similar series representations (6.3) and (6.5). The fundamental idea of the proof relies on the structure of (6.3) and (6.5) allowing to represent the volume of the continuous and smooth s-dimensional HCs as a sum of volumes of HCs of lower dimensions: j-dimensional crosses for j = 0, . . . , s − 1. To simplify this quite technical comparison, we introduce several auxiliary sequences depending on the parameter T and the vectors [j]n := (j0 , . . . , jn ) ∈ Nn+1 and 0 s [m]n := (m0 , . . . , mn ) ∈ Rn+1 for j = (j , ..., j ) ∈ N , m = (m , ..., m ) ∈ Rs+ 0 s−1 0 s−1 0 + and 0 ≤ n ≤ s − 1. In what follows the first components in [j]n and [m]n will be always associated with the parameters s and a respectively: j0 := s, m0 := a. ˜ ALEXEY CHERNOV AND DINH DUNG 24 This redundant notation will greatly simplify the forthcoming expressions. For fixed [j]n , [m]n and T we define T, Tn [j]n , T, [m]n := n = 0, Tn−1 [j]n , T, [m]n (mn )jn −jn−1 , 1 ≤ n ≤ s − 2. (6.6) In what follows we will skip the arguments and write Tn ≡ Tn ([j]n , T, [m]n ) to simplify the notations. For fixed [j]n , m0 , T and any 1 ≤ n ≤ s − 1 we introduce the partial sums 1/j Tn−1n−1 1/j0 T0 j0 jn−1 ... j1 jn Xn ([j]n , T, m0 ) := ··· m1 =m0 +1 |Γ(jn , Tn , mn + 1)|. mn =mn−1 +1 (6.7) We observe that (6.3) and (6.7) imply the relation jn −1 Xn ([j]n , T, m0 ) = Xn+1 ([j]n+1 , T, m0 ). (6.8) jn+1 =0 Finally, we introduce the sums j0 −1 jn−2 −1 jn−1 −1 j1 −1 ··· Cn (j0 , T, m0 ) := j1 =n+1 j=n Xn ([j]n , T, m0 ), jn−1 =3 jn =2 jn−2 −1 j0 −1 j1 −1 1 ··· Bn (j0 , T, m0 ) := j1 =n j=n−1 Xn ([j]n , T, m0 ), jn−2 −1 j0 −1 j1 −1 0 ··· An (j0 , T, m0 ) := (6.9) jn−1 =2 jn =1 j1 =n j=n−1 Xn ([j]n , T, m0 ). jn−1 =2 jn =0 Note that the sums over jn in An , Bn are trivial and consist only of one term with jn = 0 and 1 respectively. With this notations, we obtain the following representation for the cardinality of the continuous HC Γ(s, T, a + 1). Lemma 6.1. Assume that T > 0, a + 1 > 0 and s ≥ 2. Then s−1 |Γ(s, T, a + 1)| = An (s, T, a) + Bn (s, T, a) . (6.10) n=1 Proof. The relations (6.3) and (6.9) imply 1/j0 T0 j0 −1 |Γ(j0 , T0 , m0 + 1)| = m1 =m0 +1 j1 =0 j0 |Γ(j1 , T1 , m1 + 1)| = A1 + B1 + C1 . (6.11) j1 We claim Cn = An+1 + Bn+1 + Cn+1 , 1 ≤ n ≤ s − 3, An+1 + Bn+1 , n = s − 2. (6.12) HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 25 Indeed, by (6.8) and (6.9) we have for 1 ≤ n ≤ s − 3 j0 −1 jn−2 −1 jn−1 −1 j1 −1 ··· Cn (j0 , T, m0 ) = j1 =n+1 j=n j0 −1 Xn ([j]n , T, m0 ) jn−1 =3 jn =2 jn−2 −1 jn−1 −1 jn −1 j1 −1 ··· = j1 =n+1 j=n Xn+1 ([j]n+1 , T, m0 ) jn−1 =3 jn =2 jn+1 =0 = An+1 + Bn+1 + Cn+1 and js−4 −1 js−3 −1 j1 −1 s−1 ··· Cs−2 = j1 =s−1 j=s−2 s−1 Xs−2 ([j]s−2 , T, m0 ) js−3 =3 js−2 =2 2 3 s−2 1 ··· = j1 =s−1 j=s−2 Xs−1 ([j]s−1 , T, m0 ) js−3 =3 js−2 =2 js−1 =0 = As−1 + Bs−1 . Therefore, using (6.11) and (6.12) repeatedly for n = 1, . . . , s − 2 we obtain the assertion. Next, we obtain lower bounds for the volume of the smooth HC P (s, T, a). For this, we need more auxiliary sequences. For fixed [j]n , [m]n , T and t ∈ Rjn−1 −jn we define T, n = 0, jn−1 −jn Rn ([j]n , T, [m]n , t) := (t + mn − 21 )−1 , 1 ≤ n ≤ s − 2. Tn ([j]n , T, [m]n ) =1 (6.13) To shorten the notations we write Rn (t) ≡ Rn ([j]n , T, [m]n , t) and for 1 ≤ n ≤ s−1 the partial sums 1/j Tn−1n−1 1/j0 Yn ([j]n , T, m0 ) := j0 jn−1 ... j1 jn T0 × ··· mn =mn−1 +1 m1 =m0 +1 × I(jn , Rn (t), mn + (6.14) 1 2 ), t∈Ijn−1 −jn We claim that for 2 ≤ jn < jn−1 < · · · < j0 and m0 + 1 2 > 0 it holds that jn −1 Yn ([j]n , T, m0 ) > Yn+1 ([j]n+1 , T, m0 ). (6.15) jn+1 =0 This estimate follows from the strict convexity of the volume of the jth dimensional smooth HC P (j, t, a) (2 ≤ j ≤ s − 1) and relies on the following lemma. Lemma 6.2. For any natural numbers j, s satisfying 2 ≤ j ≤ s − 1 and T > 0, a − 12 > 0 it holds that s−j (t + a − 21 )−1 , a + 12 ) dt > I(j, T aj−s , a + 12 ). I(j, T t∈Is−j =1 (6.16) ˜ ALEXEY CHERNOV AND DINH DUNG 26 Proof. By Lemma 3.1, we can verify that for fixed j and a, the first and the second derivatives of the function Φ(T ) := I(j, T, a + 21 ) are positive for T > (a + 12 )j . Hence, Φ is increasing and strictly convex on the interval (a + 12 )j , ∞ . Consider s−j (t + a − 21 )−1 , u(t) := T G(t) := Φ(u(t)). =1 Since the function u(t) is strictly convex on Is−j in all variables t = (t1 , . . . , ts−j ), so is G(t). For a strictly convex function g on [0, 1] we have g(1/2) < 1 2 1 1 [g(x) + g(1 − x)] dx = 0 g(x) dx. 0 Applying this inequality to the (s − j)-variate function G as a univariate function in each variable ti consecutively with the other variables held fixed, we get s−j (t + a − 12 )−1 , a + 21 ) dt = I(j, T Is−j G(t) dt Is−j =1 > Is−j−1 G( 21 , t, ..., ts−j ) dt...dts−j ········· > G( 12 , ..., 12 ) = I(j, T aj−s , a + 12 ). The proof is complete. Now, it is easy to verify that the combination of (6.5) and (6.16) implies the relation (6.15). We leave the details to the interested reader. Similarly to (6.9) we introduce the sums j0 −1 jn−2 −1 jn−1 −1 j1 −1 ··· Fn (j0 , T, m0 ) := Yn ([j]n , T, m0 ), jn−1 =3 jn =2 j1 =n+1 j=n jn−2 −1 j0 −1 j1 −1 1 ··· En (j0 , T, m0 ) := j1 =n j=n−1 Yn ([j]n , T, m0 ), jn−2 −1 j0 −1 j1 −1 0 ··· Dn (j0 , T, m0 ) := (6.17) jn−1 =2 jn =1 j1 =n j=n−1 Yn ([j]n , T, m0 ). jn−1 =2 jn =0 Lemma 6.3. Assume that T > (a + 12 )s , a + 1 2 > 0 and s ≥ 2. Then s−1 I(s, T, a + 12 ) > Dn (s, T, a) + En (s, T, a) . (6.18) n=1 Proof. Observe that relation (6.5) and definitions (6.13), (6.14), (6.17) imply I(j0 , R0 , m0 + 21 ) 1/j0 T0 j0 −1 = m1 =m0 +1 j1 =0 = D1 + E1 + F1 . j0 −j1 j0 j1 (t + m1 − 12 )−1 , m1 + 12 ) dt (6.19) I(j1 , T0 t∈Ij0 −j1 =1 HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 27 Furthermore, we claim that Dn+1 + En+1 + Fn+1 , 1 ≤ n ≤ s − 3, Fn > n = s − 2. Dn+1 + En+1 , (6.20) Indeed, by (6.17) and (6.15) j0 −1 jn−2 −1 jn−1 −1 j1 −1 ··· Fn (j0 , T, m0 ) = j1 =n+1 j=n j0 −1 Yn ([j]n , T, m0 ) jn−1 =3 jn =2 jn−2 −1 jn−1 −1 jn −1 j1 −1 ··· > j1 =n+1 j=n Yn+1 ([j]n+1 , T, m0 ) jn−1 =3 jn =2 jn+1 =0 and (6.20) follows. Using (6.19) and (6.20) repeatedly for n = 1, . . . , s − 2 implies the assertion. To prove the main result in Theorem 2.4 we will compare |Γ(s, T, a + 1)| with I(s, T, a + 12 ) by using the representation in (6.10) and the inequality in (6.18). More precisely, we will compare Bn (s, T, a) with the corresponding term En (s, T, a) and An (s, T, a) with the corresponding term Dn (s, T, a) due to the fact that the summation ranges in the right side of in (6.10) and (6.18) are the same. It is easy to see that (6.1) and (6.4) imply An (s, T, a) = Dn (s, T, a). (6.21) The relation between Bn (s, T, a) and En (s, T, a) is less obvious and requires the following auxiliary lemma. Lemma 6.4. For 1 ≤ n ≤ s − 1 and jn = 1 it holds that |Γ 1, Tn , mn + 1 | ≤ Tn−1 (mn )1−jn−1 − (mn + 1), I 1, Rn , mn + t∈Is−1 1 2 dt = Tn−1 ln 1 2 1 2 mn − mn + (6.22) 1−jn−1 − (mn + 21 ), (6.23) where Tn = Tn ([j]n , T, [m]n ) and Rn (t) = Rn ([j]n , T, [m]n , t) are defined in (6.6) and (6.13) respectively. Proof. For 1 ≤ n ≤ s − 1 we have |Γ(1, Tn , mn + 1)| = Tn − (mn + 1) ≤ Tn − (mn + 1). The bound (6.22) from the definition (6.6) yielding Tn = Tn−1 (mn )1−jn−1 . Recalling (6.13) and jn = 1 we obtain for the integral jn−1 −1 I(1, Rn , mn + t∈Is−1 1 2 ) dt = = Tn−1 and hence the assertion (6.23). (t + mn − 12 )−1 dt − (mn + 12 ) Tn−1 t∈Is−1 =1 mn + ln mn − 1 2 1 2 jn−1 −1 − (mn + 12 ), ˜ ALEXEY CHERNOV AND DINH DUNG 28 6.2. Proof of Theorem 2.4. We are now in a position to prove Theorem 2.4. Proof of Theorem 2.4. It is convenient to us to prove the theorem in the following equivalent form. For any s ∈ N, and a + 12 > 0 there exists T∗ = T∗ (s, a) > 0 such that |Γ(s, T, a + 1)| ≤ I(s, T, a + 12 ), ∀T ≥ T∗ (s, a). (6.24) Define for 1 ≤ n ≤ s − 1 and positive integers s = j0 > j1 > · · · > jn−1 > jn = 1, a = m0 < m1 < · · · < mn−1 < mn (6.25) an auxiliary family n−1 j −j Ln ([j]n , [m]n ) := m −1 ln =1 mn − mn + 1−jn−1 1 2 1 2 − mn1−jn−1 . Using Lemma 6.1 and Lemma 6.3 and observing (6.21) we get s−1 1 2) I(s, T, a + − |Γ(s, T, a + 1)| > (En − Bn ). (6.26) n=1 The summands on the right-hand side of (6.26) are estimated by Lemma 6.4 as follows jn−2 −1 j0 −1 j1 −1 E n − Bn = 1 ··· Yn ([j]n , T, m0 ) − Xn ([j]n , T, m0 ) j1 =n j=n−1 jn−1 =2 jn =1 j0 −1 j1 −1 jn−2 −1 1 ··· = j1 =n j=n−1 jn−1 =2 jn =1 × (6.27) 1/j Tn−1n−1 1/j T0 0 × j0 jn−1 ... j1 jn m1 =m0 +1 1 2 T Ln ([j]n , [m]n ) − ··· mn =mn−1 +1 1 = T αn (s, a) − βn (s, T, a), 2 where αn (s, a) and βn (s, T, a) are defined as jn−2 −1 j0 −1 ··· αn (s, a) := j1 =n jn−1 =2 1/j Tn−1n−1 1/j0 j0 jn−2 ... j1 jn−1 jn−1 1 T0 ··· m1 =m0 +1 × mn =mn−1 +1 ×Ln ([j]n , [m]n ), jn−2 −1 j0 −1 ··· βn (s, T, a) := j1 =n jn−1 =2 1/j Tn−1n−1 1/j0 j0 jn−2 ... j1 jn−1 jn−1 1 T0 ··· m1 =m0 +1 1. mn =mn−1 +1 (6.28) The right-hand side of (6.27) is positive for a sufficiently large T ≥ T∗ (s, a) if α(s, a) > 0 and βn (s, T, a) = o(T ) for a fixed s and T → ∞, which we will prove next. HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 29 Positivity of α(s, a) follows from positivity of Ln ([j]n , [m]n ) for the asserted range 1 of indices (6.25). Indeed, it follows by convexity of g(x) = x+m−1/2 : For any m > 12 1 1 1 = g(1/2) < m 2 1 [g(x) + g(1 − x)] dx = 0 g(x) dx = ln 0 m + 21 . m − 21 This implies ln mn + mn − 1 2 1 2 jn−1 −1 1 mn − jn−1 −1 > 0, since jn−1 − 1 ≥ 1. Therefore αn (s, a) is positive for any 1 ≤ n ≤ s − 1. For a given 1 ≤ n ≤ s − 1, in order to prove upper bounds for βn (s, T, a), we fix a multiindex [j]n−1 = (j0 , ..., jn−1 ) in the outer summation of βn (s, T, a) satisfying (6.25), and consider the inner summation admitting the simple upper bound 1/j0 1/j0 Tn−1 ([j]n−1 ,[m]n−1 )1/jn−1 T0 ··· m1 =m0 +1 Tn−1 ([j]n−1 ,[m]n−1 )1/jn−1 T0 1≤ mn =mn−1 +1 ··· 1 m1 =1 mn =1 =: Pn ([j]n−1 , T ). We claim that for all 1 ≤ n ≤ s − 1, Pn ([j]n−1 , T ) ≤ Cn (s, [j]n−1 )T (s−1)/s , (6.29) where n−1 C1 (s, [j]0 ) := 1, λi ([j]i ), Cn (s, [j]n−1 ) := i=1 λi ([j]i ) := 1+ (ji −1)(ji −ji−1 ) ji +(ji −1)(ji −ji−1 ) (ji − 1)(ji − ji−1 )/ji = −1, , ji−1 − 1, (ji − 1)(ji − ji−1 )/ji = −1. The proof is by induction over n. Obviously, 1/j0 P1 ([j]0 , T ) = T0 ≤ T 1/s ≤ T (s−1)/s . Assume now that (6.29) holds true and observe 1/j0 T0 1/j1 Tn ([j]n ,[m]n )1/jn T1 ··· Pn+1 ([j]n , T ) = 1 m1 =1 m=1 mn+1 =1 (6.30) 1/j0 T0 T1 ([j]1 , [m]1 )(j1 −1)/j1 ≤ Cn (s, [j]n−1 ) m1 =1 by inductive assumption, where n−1 C1 (s, [j]0 ) := 1, Cn (s, [j]n−1 ) := λi ([j]i ), i=2 ˜ ALEXEY CHERNOV AND DINH DUNG 30 It holds that T1 ([j]1 , [m]1 ) = T0 m1j1 −j0 and the function x(j1 −1)(j1 −j0 )/j1 is decreasing. Thus 1/j0 1/j0 T0 T0 T1 ([j]1 , [m]1 ) (j1 −1)/j1 (j1 −1)/j1 = m1 =1 T0 (j −1)(j1 −j0 )/j1 m1 1 m1 =1 1/j0 ≤ T0 (j −1)/j1 T0 1 x(j1 −1)(j1 −j0 )/j1 dx 1+ 1 If (j1 − 1)(j1 − j0 )/j1 = −1, we have 1/j0 T0 (j1 −1)/j1 T1 ([j]1 , [m]1 )(j1 −1)/j1 ≤ T0 m1 =1 1 + x 1+ j /j0 ≤ 1+ (j1 −1)/j1 T0 (j1 −1)(j1 −j0 ) 1 j1 T0 + T0 1/j0 (j1 −1)(j1 −j0 ) j1 j1 +(j1 −1)(j1 −j0 ) j0 j1 (j1 −1)/j1 = T0 = T0 1 1+ + (j1 − 1)(j1 − j0 ) j1 + (j1 − 1)(j1 − j0 ) (j1 − 1)(j1 − j0 ) j1 + (j1 − 1)(j1 − j0 ) (j1 − 1)(j1 − j0 ) T (s−1)/s . j1 + (j1 − 1)(j1 − j0 ) (6.31) If (j1 − 1)(j1 − j0 )/j1 = −1, we have 1/j0 T0 T1 ([j]1 , [m]1 )(j1 −1)/j1 = (j −1)/j1 T0 1 1/j0 T0 1 + ln x 1 m1 =1 (j1 −1)/j1 = T0 1 ln T0 j0 1 1 + ln T j0 1+ ≤ T (s−2)/(s−1) (6.32) ≤ (j0 − 1)T (s−1)/s . In (6.31) and (6.32) we used the relation j1 < j0 = s and the fact that the function (x − 1)/x is monotonously increasing for x > 1. Combining (6.30)–(6.32) we prove (6.29). 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Institute for Mathematics, Carl von Ossietzky University, 26111 Oldenburg, Germany E-mail address: alexey.chernov@uni-oldenburg.de Vietnam National University, Hanoi, Information Technology Institute, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam E-mail address: dinhzung@gmail.com [...]... spaces with bounded mixed derivatives and applications to elliptic problems with random loading, Appl Numer Math., 62(2012), 360-377 8 A Chernov and D D˜ ung, New explicit-in-dimension estimates for the cardinality of highdimensional hyperbolic crosses and approximation of functions having mixed smoothness, preprint, arXiv:1309.5170 9 A Chernov and C Schwab, Sparse p-version BEM for first kind boundary... jn−1 jn−1 1 HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 31 which together with (6.27) implies 1 En − Bn ≥ T αn (s) − C(s)γn (s)T (s−1)/s 2 Obviously, the first term in the right-hand side of the estimate dominates The coefficients c, αn and γn are independent of T Therefore, for a fixed s there exists T∗ = T∗ (s, a) such that for all T > T∗ (s, a) and all 1 ≤ n ≤ s − 1 it holds... situation is analogous for Kolmogorov N -widths of classes of functions having a mixed smoothness if the terms N −r (log N )r(s−1) are a priori split from constants depending on s as in (1.3) (cf [20, 28]) HIGH-DIMENSIONAL HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 19 4.3 Tractabilities of the problem of ε-dimensions Let us give notions of tractability for a problem of ε-dimensions Let W be... HYPERBOLIC CROSSES AND APPROXIMATION OF FUNCTIONS 13 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 parameter a > 0 (taking fractional values as well) allows for more flexibility and, as a consequence, better upper and lower estimates for the cardinality of the hyperbolic crosses. .. (6.3) and (6.5) The fundamental idea of the proof relies on the structure of (6.3) and (6.5) allowing to represent the volume of the continuous and smooth s-dimensional HCs as a sum of volumes of HCs of lower dimensions: j-dimensional crosses for j = 0, , s − 1 To simplify this quite technical comparison, we introduce several auxiliary sequences depending on the parameter T and the vectors [j]n... 2.3, Theorem 2.4 and Corollary 3.5 – 3.7 3.2 Proof of Theorem 3.2 We define Fs+1 (t) (3.11) ps (t) Observe that Theorem 3.2 is equivalent to the following statement For any s ∈ N and t > 0, we have t t < hs−1 (t) < (3.12) t+s t+s−1 Therefore, to prove Theorem 3.2 we will verify (3.12) The proof of (3.12) requires some auxiliary lemmas Note that (3.3) and (3.12) are claimed for a fixed couple (s, t) and. .. 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 lower bounds for the ε-dimension nε (U r,a (Ts ), L2 (Ts )) As auxiliary results we derive an upper bound for the L2 (Ts )-error in the HC approximation. .. 4–5 Remark 3.8 The recent work [28] contains an independent investigation of approximation 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 dimension s It contains somewhat related estimates of the volume of the smooth HC I(s, T, a) for the special case a = 1 We mention in particular [28, Lemma 3.3] showing T... in the Hyperbolic Cross, Mathematical Notes, 63 (1998), no 3, 319–324 13 D D˜ ung, Some approximative characteristics of the classes of smooth functions of several variables in the metric of L2 , Uspekhi Mat Nauk, 34(1979), 189–190 14 D D˜ ung, Mean ε-dimension of the functional class BG,p , Mat Zametki, 28(1980), 727–736 15 D D˜ ung The number of integral points in some sets and approximation of functions. .. bounds for nε (U r,a (Ts ), L2 (Ts )) as in Theorem 4.6 is more natural and suitable than the form of those as in (1.4) Indeed, for the latter one, the traditional terms ε−1/r | log ε|(s−1)/r are a priori split from constants which are actually a function of dimension parameter s (and smoothness parameter r), and therefore, any high-dimensional estimate based on them leads to a rougher bound The situation ... [16, 39] for further modifications of these HCs in trigonometric approximations of functions having a mixed smoothness and zero mean value in each variable) These HCs having the asymptotic cardinality. .. allows for more flexibility and, as a consequence, better upper and lower estimates for the cardinality of the hyperbolic crosses in Lemma 2.1 – 2.3, Theorem 2.4 and Corollary 3.5 – 3.7 3.2 Proof of. .. the bound of the smoothness component of the unit ball of K r,a (Ts ) as a subset in L2 (Ts ) The parameter a allows also for simultaneous and sharp derivation of upper and lower bounds for the