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Let P(D) be the differential operator generated by a polynomial P, and let U P 2 be the class of multivariate periodic functions f such that kP(D)(f)k2 6 1. The problem of computing the asymptotic order of the Kolmogorov nwidths dn(U P 2 , L2) in the general case when U P 2 is compactly embedded into L2 has been open for a long time. In the present paper, we solve it in the case when P(D) is nondegenerate
Kolmogorov n-Widths of Function Classes Defined by a Non-Degenerate Differential Operator Patrick L. Combettes1 and Dinh D˜ ung2∗ 1 Sorbonne Universit´es – UPMC Univ. Paris 06 UMR 7598, Laboratoire Jacques-Louis Lions F-75005 Paris, France plc@math.jussieu.fr 2 Information Technology Institute Vietnam National University 144 Xuan Thuy, Cau Giay Hanoi, Vietnam dinhzung@gmail.com August 20, 2014 -- version 3.0 Abstract [P ] Let P (D) be the differential operator generated by a polynomial P , and let U2 be the class of multivariate periodic functions f such that P (D)(f ) 2 1. The problem of computing the [P ] [P ] asymptotic order of the Kolmogorov n-widths dn (U2 , L2 ) in the general case when U2 is compactly embedded into L2 has been open for a long time. In the present paper, we solve it in the case when P (D) is non-degenerate. Keywords. Kolmogorov n-widths · Non-degenerate differential operator · Mathematics Subject Classifications (2010) 41A10; 41A50; 41A63 ∗ Corresponding author. Email: dinhzung@gmail.com. 1 1 Introduction The aim of the present paper is to study Kolmogorov n-widths of classes of multivariate periodic functions given by a differential operator. In order to describe the exact setting of the problem let us introduce some notation. We first recall the notion of Kolmogorov n-widths [12, 19]. Let X be a normed space, let F be a nonempty subset of X such that F = −F , and let Gn be the class of all vector subspaces of X of dimension at most n. The Kolmogorov n-width, dn (F, X ), of F in X is given by dn (F, X ) = inf sup inf G∈Gn f ∈F g∈G f −g (1.1) X. This notion quantifies the error of the best approximation to the elements of F by elements in a vector subspace in X of dimension at most n [19, 26, 27]. Recently, there has been strong interest in applications of Kolmogorov n-width and its dual Gelfand n-widths to compressive sensing [3, 9, 10, 20], and in Kolmogorov n-width and its inverse ε-dimension of classes mixed smoothness in high-dimensional approximations [4, 8]. ε-dimension and more general information complexity is a tool for study of tractability of high-dimensional approximation problems; see [4, 8, 16, 17, 18] for details and references. We consider functions on Rd which are 2π-periodic in each variable as functions defined on the d-dimensional torus Td = [−π, π]d . Denote by L2 (Td ) the Hilbert space of functions on Td equipped with the standard scalar product, i.e., (∀f ∈ L2 (Td ))(∀g ∈ L2 (Td )) f |g = 1 (2π)d (1.2) f (x)g(x)dx, Td and by S (Td ) the space of distributions on Td . The norm of f ∈ L2 (Td ) is f 2 = f | f and, d d i k|· ˆ given k ∈ Z , the kth Fourier coefficient of f ∈ L2 (T ) is f (k) = f | e . Every f ∈ S (Td ) can be identified with the formal Fourier series fˆ(k)ei f= k|· (1.3) , k∈Zd where the sequence (fˆ(k))k∈Zd forms a tempered sequence [23, 27]. By Parseval’s identity, L2 (Td ) is the subset of S (Td ) of all distributions f for which |fˆ(k)|2 < +∞. (1.4) k∈Zd Let α = (α1 , . . . , αd ) ∈ Nd and let f ∈ S (Td ). We set Zd0 (α) = (k1 , . . . , kd ) ∈ Zd (∀j ∈ {1, . . . , d}) αj = 0 ⇒ kj = 0 . (1.5) α As usual, we set |α| = dj=1 αj and, given z = (z1 , . . . , zd ) ∈ Cd , z α = dj=1 zj j . The αth derivative of f ∈ S (Td ) is the distribution f (α) ∈ S (Td ) given through the identification (ik)α fˆ(k)ei f (α) = k|· (1.6) . k∈Zd0 (α) 2 The differential operator Dα on S (Td ) is defined by Dα : f → (−i)|α| f (α) . Now let A ⊂ Nd be a nonempty finite set, let (cα )α∈A be nonzero real numbers, and define a polynomial by cα xα . P: x→ (1.7) α∈A The differential operator P (D) on S (Td ) generated by P is cα Dα . P (D) = (1.8) α∈A Set [P ] W2 = f ∈ S (Td ) P (D)(f ) ∈ L2 (Td ) , [P ] denote the seminorm of f ∈ W2 f [P ] W2 (1.9) by (1.10) = P (D)(f ) 2 , and let [P ] U2 [P ] = f ∈ W2 f [P ] W2 (1.11) 1 . [P ] [P ] The problem of computing asymptotic orders of dn (U2 , L2 (Td )) in the general case when W2 is compactly embedded into L2 (Td ) has been open for a long time; see, e.g., [25, Chapter III] for details. Our main contribution is to solve it for a non-degenerate differential operator P (D) by establishing the asymptotic order [P ] dn U2 , L2 (Td ) n−r logνr n, (1.12) where r and ν depend only on P . The first exact values of n-widths of univariate Sobolev classes were obtained by Kolmogorov [12] (see also [13, pp. 186–189]). The problem of computing the asymptotic order [P ] of dn (U2 , L2 (Td )) is directly related to hyperbolic crosses trigonometric approximations and to n-widths of classes multivariate periodic functions with a bounded mixed smoothness. This line of work was initiated by Babenko in [1, 2]; in particular, the asymptotic orders of n-widths in L2 (Td ) of these classes were established in [1]. Further work on asymptotic orders and hyperbolic cross approximation can be found in [6, 7, 25] and recent developments in [14, 22, 24, 28]. In [5], the strong asymptotic order of dn (U2A , L2 (Td )) was computed in the case when U2A is the closed unit ball of the space W2A of functions with several bounded mixed derivatives (see Subsection 4.4 for a precise definition). Recently, Kolmogorov n-widths in the classical isotropic Sobolev space H s of classes of multivariate periodic functions with anisotropic smoothness have been investigated in high-dimensional settings [4, 8], where although the dimension n of the approximating subspace is the main parameter in the study of convergence rates with respect to n going to infinity, the parameter d may seriously affect this rate when d is large. 3 The paper is organized as follows. In Section 2, we provide as auxiliary results Jackson-type [P ] and Bernstein-type inequalities for trigonometric approximations of functions from W2 . We also [P ] characterize the compactness of U2 in L2 (Td ) and of non-degenerateness of P (D). In Section 3, [P ] we present the main result of the paper, namely the asymptotic order of dn (U2 , L2 (Td )) in the case when P (D) is non-degenerate. In Section 4, we derive norm equivalences relative to · W [P ] and, 2 [P ] dn (U2 , L2 (Td )) based on them, we provide examples of n-widths operators. 2 for non-degenerate differential Preliminaries 2.1 Notation, standing assumption, and definitions Let Θ be an abstract set, and let Φ and Ψ be functions from Θ to R. Then we write (∀θ ∈ Θ) Φ(θ) (2.1) Ψ(θ) if there exist constants C1 and C2 in ]0, +∞[ such that (∀θ ∈ Θ) C1 Φ(θ) unit vectors of Rd are denoted by (uj )1 j d . C2 Φ(θ). The Ψ(θ) Definition 2.1 Let A be a nonempty finite subset of Rd+ . The polyhedron spanned by A is the convex hull conv(A) of A, ∆(A) = α ∈ A λα λ ∈ [1, +∞[ ∩ conv(A) = {α} , (2.2) and E(A) the set of vertices of ∆(A). In addition, ∀x ∈ Rd+ mA (x) = max xα (2.3) α∈A and (∀t ∈ [0, +∞[) ΩA (t) = k ∈ Nd mA (k) (2.4) t . Throughout the paper, we make the following standing assumption. Assumption 2.2 A is a nonempty finite subset of Nd and (cα )α∈A are nonzero real numbers. We set cα xα , P: x→ M : x → max |xα |, α∈E(A) α∈A and τ = inf |P (k)|. (2.5) k∈Zd Moreover, for every t ∈ [0, +∞[, K(t) = k ∈ Zd |P (k)| t and V (t) = fˆ(k)ei f ∈ S (Td ) f = k∈K(t) 4 k|· . (2.6) Remark 2.3 Suppose that t ∈ ]τ, +∞[. Then K(t) = ∅ and dimV (t) = |K(t)|, where |K(t)| denotes the cardinality of K(t). In addition, if |K(t)| < +∞, then V (t) is the space of trigonometric polynomials with frequencies in K(t). Definition 2.4 The Newton diagram of P is ∆(A) and the Newton polyhedron of P is Γ(P ) = conv(A). The intersection of Γ(P ) with a supporting hyperplane of Γ(P ) is called a face of Γ(P ). The dimension of a face ranges from 0 to d − 1. A vertex is a 0-dimensional face. The set of vertices of Γ(P ) is ϑ(P ) and the set of faces of Γ(P ) is Σ(P ). The differential operator P (D) is non-degenerate if P and, for every σ ∈ Σ(P ), Pσ : Rd → R : x → α∈σ cα xα do not vanish outside the coordinate planes of Rd , i.e., d ∀x ∈ R d ⇒ xj = 0 ∀σ ∈ Σ(P ) P (x)Pσ (x) = 0. (2.7) j=1 2.2 Trigonometric approximations We first prove a Jackson-type inequality. Lemma 2.5 Let t ∈ ]0, +∞[ and define the linear operator St : S (Td ) → S (Td ) by ∀f ∈ S (Td ) fˆ(k)ei St (f ) = k|· (2.8) . k∈K(t) [P ] Let f ∈ W2 and and suppose that t > τ . Then the distribution f − St (f ) represents a function in L2 (Td ) f − St (f ) t−1 f 2 [P ] W2 (2.9) . Proof. Set g = f − St (f ). Then g ∈ S (Td ). On the other hand, Parseval’s identity yields f 2 [P ] W2 |P (k)|2 |fˆ(k)|2 . = (2.10) k∈Zd Hence, |fˆ(k)|2 |ˆ g (k)|2 = k∈Zd k∈Zd \K(t) |P (k)|−2 sup k∈Zd \K(t) t−2 f |P (k)|2 |fˆ(k)|2 k∈Zd \K(t) 2 [P ] , W2 (2.11) which means that f − St (f ) represents a function in L2 (Td ) for which (2.9) holds. 5 Corollary 2.6 Let t ∈ ]τ, +∞[. Then f −g sup inf [P ] f ∈U2 g∈V (t) f −g∈L2 (Td ) 2 t−1 . (2.12) Next, we prove a Bernstein-type inequality. Lemma 2.7 Let f ∈ V (t) ∩ L2 (Td ) and let t ∈ ]τ, +∞[. Then f t f [P ] W2 (2.13) 2. Proof. By (2.10), we have f 2 [P ] W2 |P (k)|2 |fˆ(k)|2 = k∈K(t) |fˆ(k)|2 sup |P (k)|2 k∈K(t) t2 f 2 2, (2.14) k∈K(t) which provides the announced inequality. 2.3 Compactness and non-degenerateness We start with a characterization of the compactness of the unit ball defined in (1.11). [P ] Lemma 2.8 The set U2 is a compact subset of L2 (Td ) if and only if the following hold: (i) For every t ∈ ]τ, +∞[, K(t) is finite. (ii) τ > 0. Proof. To prove sufficiency, suppose that (i) and (ii) hold, and fix t ∈ ]τ, +∞[. By (i), V (t) is a set of trigonometric polynomials and, consequently, a subset of L2 (Td ). In particular, using the notation (2.8), (∀f ∈ S (Td )) St (f ) ∈ L2 (Td ). Hence, by Lemma 2.5, [P ] ∀f ∈ W2 f = (f − St (f )) + St (f ) ∈ L2 (Td ). [P ] (2.15) [P ] Thus, W2 ⊂ L2 (Td ). On the other hand, (2.10) implies that U2 is a closed subset of L2 (Td ). Therefore, it is compact in L2 (Td ) if, for every ε ∈ ]0, +∞[, there exists a finite ε-net in L2 (Td ) for [P ] U2 or, equivalently, if the following following two conditions are satisfied: (iii) For every ε ∈ ]0, +∞[, there exists a finite dimensional vector subspace Gε of L2 (Td ) such that sup [P ] f ∈U2 inf g∈Gε f −g 2 (2.16) ε. 6 [P ] (iv) U2 is bounded in L2 (Td ). It follows from (2.10) that (ii)⇔(iv). On the other hand, since dim V (t) = |K(t)|, Corollary 2.6 yields (i)⇒(iii). To prove necessity, suppose that (i) does not hold. Then dim V (t˜) = |K(t˜)| = +∞ [P ] for some t˜ ∈ ]0, +∞[. By Lemma 2.7, U = f ∈ V (t˜) ∩ L2 (Td ) f 2 1/t˜ is a subset of U2 [P ] which is not compact in L2 (Td ). If (ii) does not hold, then U2 consequently, not compact in L2 (Td ). ∩ L2 (Td ) is unbounded and, The following lemma characterizes the non-degenerateness of P (D). Lemma 2.9 Then P (D) is non-degenerate if and only if (∃ C ∈ ]0, +∞[)(∀x ∈ Rd ) C max |xα |. |P (x)| (2.17) α∈ϑ(P ) Proof. As proved in [11, 15], P (D) is non-degenerate if and only if (∃ C1 ∈ ]0, +∞[)(∀x ∈ Rd ) |P (x)| |xα |. C1 (2.18) α∈ϑ(P ) Hence, since there exist constants C2 and C3 in ]0, +∞[ such that (∀x ∈ Rd ) C2 max |xα | α∈ϑ(P ) |xα | C3 max |xα |, (2.19) α∈ϑ(P ) α∈ϑ(P ) the proof is complete. Lemma 2.10 Let B be a nonempty finite subset of Rd+ and let t ∈ [0, +∞[. Then ΩB (t) is finite if and only (∀j ∈ {1, . . . , d})(∃ aj ∈ ]0, +∞[) B ∩ span(uj ) = {aj uj }. (2.20) Proof. If (2.20) holds, then ΩB (t) ⊂ dj=1 x ∈ Rd+ xj t1/aj and, consequently, ΩB (t) is bounded. Conversely, if (2.20) does not hold, there exists j ∈ {1, . . . , d} such that muj m ∈ N ⊂ ΩB (t), which shows that ΩB (t) is unbounded. [P ] Theorem 2.11 Suppose that P (D) is non-degenerate. Then U2 only if (2.20) is satisfied and 0 ∈ A. is a compact subset of L2 (Td ) if and Proof. It follows from Young’s inequality that there exists C1 ∈ ]0, +∞[ such that (∀x ∈ Rd ) |P (x)| C1 max |xα |. (2.21) α∈ϑ(P ) Hence, by Lemma 2.9, there exist C2 ∈ ]0, +∞[ such that (∀x ∈ Rd ) C2 max |xα | α∈ϑ(P ) |P (x)| C1 max |xα |. α∈ϑ(P ) 7 (2.22) [P ] Consequently, by Lemma 2.8, U2 ΩA (t) is finite and is a compact set in L2 (Td ) if and only if, for every t ∈ [0, +∞[, (2.23) inf mA (x) > 0. x∈Nd By Lemma 2.10, the first condition is equivalent to (2.20), and the second to 0 ∈ A. 3 Main result The following facts will be necessary to prove our main result. Lemma 3.1 Let B be a nonempty finite subset of Rd+ and let x ∈ Rd+ . Then mB (x) = mE(B) (x). Proof. It is clear that mE(B) (x) mB (x). Conversely, let α ∈ B E(B). On the one hand, there exists ρ ∈ ]1, +∞[ such that α = ρα ∈ ∆(B). One the other hand, by Carath´eodory’s theorem [21, Theorem 17.1], α is a convex combination of points (αj )1 j d+1 in E(B), say d+1 d+1 λj α j , α = where {λj }1 j d+1 ⊂ [0, +∞[ j=1 and λj = 1. (3.1) j=1 Hence, by Young’s inequality d+1 α α x 0. Then (∀t ∈ [2, +∞[) |ΩB (t)| tµ(B) logν(B) t. (3.8) Proof. Fix t ∈ [2, +∞[ and set ΛB (t) = x ∈ Rd+ mB (x) t . Then, as in the proof of Lemma 2.10, one can see that ΛB (t) is a bounded subset in Rd+ . If we denote by vol ΛB (t) the volume of ΛB (t), then it follows from [5, Theorem 1] that volΛB (t) tµ(B) logν(B) t. (3.9) Furthermore, proceeding as in the proof of [5, Theorem 2], one shows that |ΩB (t)| vol ΛB (t). (3.10) These asymptotic relations prove the lemma. In computational mathematics, the so-called ε-dimension nε = nε (W, X) is used to quantify the computational complexity. It is defined by nε (W, X) := inf n : ∃ Ln : sup inf f ∈W g∈Ln f −g X ε , where Ln is a linear subspace in 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. Our main result can now be stated and proved. Theorem 3.4 Suppose that P (D) is non-degenerate, that (2.20) is satisfied, and that 0 ∈ A. Then, for n sufficiently large, [P ] dn U2 , L2 (Td ) n−r(ϑ(A)) logν(ϑ(A))r(ϑ(A)) n, (3.11) ε−1/r(ϑ(A)) | log ε|ν(ϑ(A)) . (3.12) or equivalently, [P ] nε U2 , L2 (Td ) 9 Proof. Set t¯ = max{2, τ }. It follows from Corollary ?? that (∀t ∈ [t¯, +∞[) |Ωϑ(A) (t)| |K(t)|. (3.13) Since A satisfies (2.20), so does ϑ(A). Hence applying Lemma 3.3 to ϑ(A), we have tµ logν t, (∀t ∈ [t¯, +∞[) |K(t)| (3.14) where µ = µ(ϑ(A)) and ν = ν(ϑ(A)). In turn, for every m ∈ N, there exists C1 ∈ ]0, +∞[ such that dim V (m) C1 m1/r logν m. (3.15) For n ∈ N large enough, there exist m ∈ N such that C1 m1/r logν m n < C1 (m + 1)1/r logν (m + 1) C2 m1/r logν m, (3.16) where C2 ∈ ]0, +∞[ is independent from n and m. It follows from (3.15), (3.16), and Corollary 2.6 that [P ] dn (U2 , L2 (Td )) m−1 n−r logνr n. (3.17) The upper bound of (3.11) is proven. To establish the lower bound, let us recall from [26] that, for every n + 1-dimensional subspace Ln+1 of X and every ρ ∈ ]0, +∞[, we have dn (Bn+1 (ρ), X ) = ρ, where Bn+1 (ρ) = {f ∈ Ln+1 | f X ρ}. (3.18) Similarly to (3.15) and (3.16), for n ∈ N sufficiently large, there exists m ∈ N such that dim V (m) C3 m1/r logν m > n C4 m1/r logν m, (3.19) where C3 ∈ ]0, +∞[ and C4 ∈ ]0, +∞[ are independent from n and m. Consider the set U (m) = f ∈ V (m) f [P ] By Lemma 2.7, U (m) ⊂ U2 [P ] dn U2 , L2 (Td ) 2 m−1 . (3.20) and consequently, it follows from (3.18) and (3.19) that dn (U (m), L2 (Td )) m−1 n−r logνr n, (3.21) which concludes the proof. Remark 3.5 We have actually proven a bit more than Theorem 3.4. Namely, suppose that P (D) [P ] satisfies the conditions of compactness for U2 stated in Lemma 2.8 and for every n ∈ N, let m(n) be the maximal number such that |K(m(n))| n. Then, for n sufficiently large, we have [P ] dn U2 , L2 (Td ) 1 . m(n) (3.22) 10 4 Examples First, we establish norm equivalences and, based on them, we provide some examples of [P ] dn (U2 , L2 (Td )) for non-degenerate and degenerate differential operators. Theorem 4.1 Suppose that P (D) is non-degenerate and set xα . Q: x → (4.1) α∈E(A) Then [P ] ∀f ∈ W2 f 2 [P ] W2 f Dα f 2 [Q] W2 2 2 α∈E(A) max Dα f α∈E(A) 2 2. (4.2) Moreover, the semi-norms in (4.2) are a norm if and only if 0 ∈ A. [P ] Proof. Let f ∈ W2 . It is clear that Dα f 2 2 Dα f max α∈E(A) α∈E(A) 2 2. (4.3) Parseval’s identity and Corollary 3.2 yield Dα f max 2 2 α∈E(A) |k|2α |fˆ(k)|2 max α∈E(A) k∈Zd |M (k)|2 |fˆ(k)|2 . (4.4) k∈Zd Let us decompose Zd into the subsets Zd (α), α ∈ E(A), such that Zd = Zd (α), Zd (α) ∩ Zd (α ) = ∅, α = α, (4.5) α∈E(A) and M (k) = |k α |, k ∈ Zd (α). (4.6) (Such a decomposition is easily constructed). Then we have max α∈E(A) Dα f 2 2 |k 2α ||fˆ(k)|2 = max α∈E(A) α ∈E(A) k∈Zd (α ) |k 2α | |fˆ(k)|2 α ∈E(A) k∈Zd (α ) |M (k)|2 |fˆ(k)|2 . = k∈Zd 11 (4.7) Thus, we have proven the following equation max Dα f α∈E(A) 2 2 |M (k)|2 |fˆ(k)|2 . = (4.8) k∈Zd Hence, by Corollary 3.2 and (2.10) we obtain max Dα f 2 2 f 2 [P ] . W2 (4.9) Dα f 2 2 f 2 [Q] . W2 (4.10) α∈E(A) The relation max α∈E(A) follows from the last semi-norm equivalence and the equation E(Q) = E(A). If follows from (4.2) that the semi-norms in (4.2) are a norm if and only if 0 ∈ A. 4.1 Isotropic Sobolev classes Let s ∈ N∗ . The isotropic Sobolev space H s is the Hilbert space of functions f ∈ L2 (Td ) equipped with the norm · H s which is defined by f 2 Hs = f 2 2 + f (α) 22 . (4.11) xα , (4.12) |α|=s Consider xα = P: x→1+ α∈A |α|=s where A = 0 ∪ {α : |α| = s}. If s is even, the differential operator P (D) is non-degenerate and consequently, by Theorem 4.1 the norm f H s is equivalent to one of the norms in (4.2) with E(A) = 0 ∪ α = suj 1 j d and d xsj . Q(x) = 1 + (4.13) j=1 Moreover, we have r(A) = s and ν(a) = 0, and therefore, for the unit ball U s in H s from Theorem 3.4 we again retrieve the well known result dn U s , L2 (Td ) n−s . (4.14) 12 4.2 Anisotropic Sobolev classes For β = (β1 , . . . , βd ) ∈ N∗d , the anisotropic Sobolev space H β is the Hilbert space of functions f ∈ L2 equipped with the norm f H β which is defined by d f 2 Hβ = f 2 2 j) f (βj u + 2 2. (4.15) j=1 Consider d β xα , xj j = P: x→1+ j=1 (4.16) α∈A where A = {0} ∪ {α = βj uj | j = 1, . . . , d}. If the coordinates of β are even, the differential operator P (D) is non-degenerate and consequently, by Theorem 4.1 the norm f H β is equivalent to one of the norms in (4.2) with E(A) = A and (4.17) Q(x) = P (x). We have −1 d r = r(A) = (4.18) 1/βj j=1 and ν(A) = 0, and therefore, for the unit ball U r in H β from Theorem 3.4 we again retrieve the well-known result n−r . dn U β , L2 (Td ) 4.3 (4.19) Classes of functions with a bounded mixed derivative Let α = (α1 , . . . , αd ) ∈ Nd with 0 < α1 = · · · = αν+1 < αν+2 = · · · = αd for some 0 ν d − 1. For a set e ⊂ {1, . . . , d}, let the vector α(e) ∈ Nd be defined by α(e)j = αj if j ∈ e, and α(e)j = 0 otherwise (in particular, α(∅) = 0 and α({1, . . . , d}) = α). The space W2α is the Hilbert space of functions f ∈ L2 equipped with the norm · W2α which is defined by f 2 W2α f (α(e)) 22 . = (4.20) e⊂{1,...,d} Consider xα(e) = P: x→ e⊂{1,...,d} xα , (4.21) α∈A 13 where A = α(e) e ⊂ {1, . . . , d} . If the coordinates of α are even, the differential operator P (D) is non-degenerate and consequently, by Theorem 4.1 the norm · W2α is equivalent to one of the norms in (4.2) with E(A) = A and (4.22) Q(x) = P (x). We have r(A) = α1 and ν(A) = ν, and therefore, for the unit ball U2α in W2α from Theorem 3.4 we again retrieve the result proven in [1], namely that for n sufficiently large n−α1 logνα1 n. dn U2α , L2 (Td ) (4.23) In the particular case α = r1, we have n−r log(d−1)r n. dn U2r1 , L2 (Td ) 4.4 (4.24) Classes of functions with several bounded mixed derivatives Suppose that (2.20) is satisfied and that 0 ∈ A. The space W2A is the Hilbert space of functions f ∈ L2 (Td ) equipped with the norm · W A which is defined by 2 f 2 W2A f (α) 22 . = (4.25) α∈A Notice that spaces H s , H r , and W2α are a particular cases of W2A . Now consider xα . P: x→ (4.26) α∈A If the coordinates of every α ∈ E(A) are even, the differential operator P (D) is non-degenerate and consequently, by Theorem 4.1, the norm · W A is equivalent to one of the norms in (4.2). If 2 r = r(E(A)) and ν = ν(E(A)), for the unit ball U2A in W2A from Theorem 3.4 we again retrieve the result proven in [5], namely that for n sufficiently large n−r logνr n. dn U2A , L2 (Td ) 4.5 (4.27) Classes of functions given by a differential operator [P ] We give two examples of space W2 Consider the following polynomials with non-degenerate differential operator P (D) for d = 2. P1 : x → = 8x41 − 4x31 − 3x31 x2 − 2x21 x2 − 4x1 x2 + 6x22 − 4x1 − 3x2 + 13, (4.28) P2 : x → = 6x61 + x41 x22 − 6x51 − x31 x22 + 5x42 − 14 4x32 + 3. We have A1 = {(4, 0), (3, 0), (2, 1), (2, 0), (1, 1), (0, 2), (1, 0), (0, 1), (0, 0)}, E(A1 ) = {(4, 0), (0, 2), (0, 0)}, (4.29) A2 = {(6, 0), (4, 2), (5, 0), (3, 2), (0, 4), (0, 3), (0, 0)}, E(A2 ) = {(6, 0), (4, 2), (0, 4), (0, 0)}. It is easy to verify that for i = 1, 2, Pj (D) is non-degenerate, (2.20) holds, and 0 ∈ Ai . Moreover, r(E(A1 )) = 4/3, ν(E(A1 )) = 0 and r(E(A2 )) = 8/3, ν(E(A2 )) = 1. From Theorem 3.4 we have dn U [P1 ] , L2 (T2 ) n−4/3 , (4.30) dn U [P2 ] , L2 (T2 ) n−8/3 log8/3 n. (4.31) and Let us give an example of degenerate differential operator. For P3 (x) = x41 − 2x31 x2 + x21 x22 + x21 + x22 + 1, (4.32) the differential operator P3 (D) is degenerate, although P3 (x) 1 for every x ∈ R2 , and U [P3 ] is 2 [P ] a compact set in L2 (T ). Therefore, we cannot compute dn (U 3 , L2 (T2 )) by using Theorem 3.4. However, by a direct computation we get |K(t)| t1/2 log t. Hence, by (3.22) we have dn U [P3 ] , L2 (T2 ) 4.6 n−2 log2 n. (4.33) A conjecture [P ] Suppose that U2 (i) For every t is compact in L2 (Td ). In view of Lemma 2.8, this is equivalent to the conditions: 0, K(t) is finite. (ii) τ > 0. As mentioned in (3.22), for n ∈ N sufficiently large, if m(n) is the maximal number such that |K(m(n))| n, then [P ] dn U2 , L2 (Td ) 1 . m(n) (4.34) [P ] This means that the problem of computing the asymptotic order of dn (U2 , L2 (Td )) is equivalent to the problem of computing that of |K(t)| when t → +∞. Let us formulate it as the following conjecture. 15 Conjecture 4.2 Suppose that, for every t ∈ [0, +∞[, K(t) is finite (the condition τ > 0 is not essential). Then there exist integers α, β, and ν such that 0 < α β 0 ν < d, and, for t large enough, |K(t)| tα/β logν t. (4.35) In view of (3.14), we know that the conjecture is true when P satisfies conditions (2.7) and (2.20). Acknowledgment. Dinh Dung’s research work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.01-2014.02, and a part of it was done when Dinh Dung was working as a research professor at and Patrick Combettes was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). Both authors thank the VIASM for providing fruitful research environment and working condition. References [1] K. I. Babenko, Approximation of periodic functions of many variables by trigonometric polynomials, Soviet Math. Dokl. 1 (1960) 513–516. [2] K. I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Soviet Math. Dokl. 1 (1960) 672–675. [3] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, A simple proof of the restricted isometry property for random matrices, Constr. Approx., 28(2008), 253–263. [4] A. 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Theory 164 (2012) 406–430. 17 [...]... the particular case α = r1, we have n−r log(d−1)r n dn U2r1 , L2 (Td ) 4.4 (4.24) Classes of functions with several bounded mixed derivatives Suppose that (2.20) is satisfied and that 0 ∈ A The space W 2A is the Hilbert space of functions f ∈ L2 (Td ) equipped with the norm · W A which is defined by 2 f 2 W 2A f (α) 22 = (4.25) α A Notice that spaces H s , H r , and W2α are a particular cases of W 2A ... class of polynomials, Proc Steklov Inst Math 91 (1967), 61–82 [16] E Novak and H Wo´zniakowski, Tractability of Multivariate Problems Vol 1: Linear Information, EMS, Z¨ urich, 2008 [17] E Novak and H Wo´zniakowski, Approximation of infinitely differentiable multivariate functions is intractable, Journal of Complexity 25(2009), 398–404 [18] E Novak and H Wo´zniakowski, Optimal order of convergence and... Mathematics (VIASM) Both authors thank the VIASM for providing fruitful research environment and working condition References [1] K I Babenko, Approximation of periodic functions of many variables by trigonometric polynomials, Soviet Math Dokl 1 (1960) 513–516 [2] K I Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Soviet Math Dokl 1 (1960)... beste Ann¨ [12] A N Kolmogorov, U aherung von Funktionen einer Funktionklasse, Ann Math 37(1936), 107–111 [13] A N Kolmogorov Selected papers, Mathematics and Mechanics, Volume I, Nauka, Moscow 1985 (in Russian) [14] A Kushpel and S A Tozoni, Entropy and widths of multiplier operators on two-point homogeneous spaces, Constr Approx 35 (2012) 137–180 [15] V P Miha˘ılov, Behavior at infinity of a certain... that the conjecture is true when P satisfies conditions (2.7) and (2.20) Acknowledgment Dinh Dung’s research work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 102.01-2014.02, and a part of it was done when Dinh Dung was working as a research professor at and Patrick Combettes was visiting the Vietnam Institute for Advanced Study in Mathematics... and approximation of functions of several variables, Mat Zametki 36 (1984) 479–491 [6] Dinh D˜ ung, Approximation of functions of several variables on a torus by trigonometric polynomials, Math USSR-Sb 59 (1988) 247–267 [7] Dinh D˜ ung, Best multivariate approximations by trigonometric polynomials with frequencies from hyperbolic crosses, J Approx Theory, 91 (1997) 205–225 [8] Dinh D˜ ung and T Ullrich,... (4.26) α A If the coordinates of every α ∈ E (A) are even, the differential operator P (D) is non-degenerate and consequently, by Theorem 4.1, the norm · W A is equivalent to one of the norms in (4.2) If 2 r = r(E (A) ) and ν = ν(E (A) ), for the unit ball U 2A in W 2A from Theorem 3.4 we again retrieve the result proven in [5], namely that for n sufficiently large n−r logνr n dn U 2A , L2 (Td ) 4.5 (4.27) Classes. .. R Baraniuk, M Davenport, R DeVore, and M Wakin, A simple proof of the restricted isometry property for random matrices, Constr Approx., 28(2008), 253–263 [4] A Chernov and Dinh D˜ ung, New estimates for the cardinality of high-dimensional hyperbolic crosses and approximations of functions having mixed smoothness, Manuscript (2013) [5] Dinh D˜ ung, The number of integral points in some sets and approximation... (in)tractability of multivariate approximation of smooth functions, Constr Appr 30(2009), 457–473 [19] A Pinkus, N-Widths in Approximation Theory, Springer-Verlag, 1985 [20] A Pinkus, Sparse representations and approximation theory, J Approx Theory 163 (2011) 388–412 [21] R T Rockafellar, Convex Analysis, Princeton, Princeton University Press, 1970 [22] H.-J Schmeisser and W Sickel, Winfried Spaces of functions... If the coordinates of α are even, the differential operator P (D) is non-degenerate and consequently, by Theorem 4.1 the norm · W2α is equivalent to one of the norms in (4.2) with E (A) = A and (4.22) Q(x) = P (x) We have r (A) = α1 and ν (A) = ν, and therefore, for the unit ball U2α in W2α from Theorem 3.4 we again retrieve the result proven in [1], namely that for n sufficiently large n−α1 logνα1 n ... supporting hyperplane of Γ(P ) is called a face of Γ(P ) The dimension of a face ranges from to d − A vertex is a 0-dimensional face The set of vertices of Γ(P ) is ϑ(P ) and the set of faces of Γ(P )... Introduction The aim of the present paper is to study Kolmogorov n-widths of classes of multivariate periodic functions given by a differential operator In order to describe the exact setting of the problem... W A which is defined by f W 2A f (α) 22 = (4.25) α A Notice that spaces H s , H r , and W2α are a particular cases of W 2A Now consider xα P: x→ (4.26) α A If the coordinates of every α ∈ E (A)