Kolmogorov n widths of function classes induced by a non degenerate differential operator a convex duality approach

arXiv:1412.6400v1 [math.CA] Dec 2014 Kolmogorov n-Widths of Function Classes Induced by a Non-Degenerate Differential Operator: A Convex Duality Approach∗ Patrick L Combettes1 and Dinh D˜ ung2 Sorbonne Universités – UPMC Univ Paris 06 UMR 7598, Laboratoire Jacques-Louis Lions F-75005 Paris, France plc@ljll.math.upmc.fr Information Technology Institute Vietnam National University Hanoi, Vietnam dinhzung@gmail.com Abstract [P] Let P(D) be the differential operator induced by a polynomial P, and let U2 be the class of multivariate periodic functions f such that P(D)( f ) The problem of computing the asymp[P] [P] totic order of the Kolmogorov n-width 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 use convex analytical tools to solve it in the case when P(D) is non-degenerate Keywords asymptotic order · Kolmogorov n-widths · non-degenerate differential operator · convex duality Mathematics Subject Classifications (2010) 41A10; 41A50; 41A63 ∗ Contact author: P L Combettes, plc@ljll.math.upmc.fr, phone: +33 4427 6319, fax: +33 4427 7200 1 Introduction The aim of the present paper is to study Kolmogorov n-widths of classes of multivariate periodic functions induced 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 [14, 18] Let be a normed space, let F be a nonempty subset of such that F = −F, and let n be the class of all vector subspaces of of dimension at most n The Kolmogorov n-width of F in is dn (F, ) = inf sup inf G∈ n f ∈F g∈G f −g (1.1) This notion quantifies the error of the best approximation to the elements of F by elements in a vector subspace of of dimension at most n [18, 25, 26] In computational mathematics, the so-called ǫ-dimension nǫ (F, tational complexity It is defined by nǫ (F, ) = inf n ∈ (∃ G ∈ n) sup inf f − g ) is used to quantify the compu- ǫ f ∈F g∈G (1.2) This approximation characteristic is the inverse of dn (F, ) in the sense that the quantity nǫ (F, ) is the smallest integer nǫ such that the approximation of F by a suitably chosen approximant nǫ dimensional subspace G in gives an approximation error less than ǫ Recently, there has been strong interest in applications of Kolmogorov n-widths, and its dual Gelfand n-widths, to compressive sensing [3, 10, 11, 19] Kolmogorov n-widths and ǫ-dimensions of classes of functions with mixed smoothness have also been employed in recent high-dimensional approximation studies [5, 9] We consider functions on d which are 2π-periodic in each variable as functions defined on d = [−π, π]d Denote by L2 ( d ) the Hilbert space of square-integrable functions on d equipped with the standard scalar product, i.e., (∀ f ∈ L2 ( d ))(∀g ∈ L2 ( d )) f |g = (2π)d f (x)g(x)d x, (1.3) d and by ′ ( d ) the space of distributions on d The norm of f ∈ L2 ( d ) is f = given k ∈ d , the kth Fourier coefficient of f ∈ L2 ( d ) is fˆ(k) = f | e i〈k|·〉 Every f ∈ identified with the formal Fourier series ′ f | f and, ( d ) can be fˆ(k)e i〈k|·〉 , f = k∈ (1.4) d where the sequence ( fˆ(k))k∈ d is a tempered sequence [22, 26] By Parseval’s identity, L2 ( subset of ′ ( d ) of all distributions f for which | fˆ(k)|2 < +∞ k∈ d ) is the (1.5) d d Let α = (α1 , , αd ) ∈ d (α) d = (k1 , , kd ) ∈ d j=1 α j As usual, we set |α| = ′ derivative of f ∈ d ( ′ and let f ∈ ( d ) We set (∀ j ∈ {1, , d}) α j = ⇒ k j = and, given z = (z1 , , zd ) ∈ ) is the distribution f (α) ′ ∈ ( d d , we set z α = (1.6) αj d j=1 z j The αth ) given through the identification (ik)α fˆ(k)e i〈k|·〉 f (α) = k∈ (1.7) d (α) The differential operator Dα on ′ ( d ) is defined by Dα : f → (−i)|α| f (α) Now let A ⊂ nonempty finite set, let (cα )α∈A be nonzero real numbers, and define a polynomial by cα x α P: x → d be a (1.8) α∈A The differential operator P(D) on ′ ( d ) induced by P is cα Dα P(D) = (1.9) α∈A Set [P] W2 ′ = f ∈ ( d ) P(D)( f ) ∈ L2 ( [P] denote the seminorm of f ∈ W2 f [P] W2 d ) , (1.10) by = P(D)( f ) , (1.11) and let [P] U2 [P] = f ∈ W2 f [P] W2 (1.12) [P] [P] The problem of computing asymptotic orders of dn (U2 , L2 ( d )) in the general case when W2 is compactly embedded into L2 ( d ) has been open for a long time; see, e.g., [24, Chapter III] for details Our main contribution is to solve it for a non-degenerate differential operator P(D) (see Definition 2.4) Using convex-analytical tool, we establish the asymptotic order [P] d n U2 , L ( d ) ≍ n−̺ (log n)ν̺ , (1.13) where ̺ and ν depend only on P The first exact values of n-widths of univariate Sobolev classes were obtained by Kolmogorov [14] [P] (see also [15, pp 186–189]) The problem of computing the asymptotic order of dn (U2 , L2 ( d )) 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 ( d ) of these classes were established in [1] Further work on asymptotic orders and hyperbolic cross approximation can be found in [7, 8, 24] and recent developments in [16, 21, 23, 27] In [6], the strong asymptotic order of dn (U2A, L2 ( d )) 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) The remainder of the paper is organized as follows In Section 2, we provide as auxiliary results [P] Jackson-type and Bernstein-type inequalities for trigonometric approximations of functions from W2 [P] We also characterize the compactness of U2 in L2 ( d ) and the non-degenerateness of P(D) In [P] Section 3, we present the main result of the paper, namely the asymptotic order of dn U2 , L2 ( d ) in the case when P(D) is non-degenerate In Section 4, we derive norm equivalences relative to · W [P] [P] and, based on them, we provide examples of n-widths dn (U2 , L2 ( operators d )) for non-degenerate differential Preliminaries 2.1 Notation, standing assumption, and definitions We set = {0, 1, , }, ∗ = {1, 2, , }, + = [0, +∞[, and and let Φ and Ψ be functions from Θ to Then we write (∀θ ∈ Θ) ++ = ]0, +∞[ Let Θ be an abstract set, Φ(θ ) ≍ Ψ(θ ) (2.1) if there exist γ1 ∈ ++ and γ2 ∈ ++ such that (∀θ ∈ Θ) γ1 Φ(θ ) j ∈ {1, , d}, u j denotes the j standard unit vector of d and j = λu j λ ∈ Ψ(θ ) γ2 Φ(θ ) For every (2.2) ++ the jth standard strict ray d Definition 2.1 Let B be a nonempty finite subset of spanned by B, ∆(B) = α ∈ B The convex hull conv(B) of B is the polyhedron λα λ ∈ [1, +∞[ ∩ conv(B) = {α} , (2.3) and ϑ(B) is the set of vertices of conv(∆(B)) In addition, (∀t ∈ +) ΩB (t) = k ∈ d max kα α∈B t (2.4) Throughout the paper, the convention 00 is adopted and the following standing assumption is made Assumption 2.2 A is a nonempty finite subset of cα x α P: x → α∈A d and τ = inf |P(k)| k∈ and (cα )α∈A are nonzero real numbers We set (2.5) d Moreover, for every t ∈ d K(t) = k ∈ +, we set |P(k)| V (t) = and t f ∈ ′ ( d ) fˆ(k)e i〈k|·〉 f = (2.6) k∈K(t) Remark 2.3 If ∈ A, then ∈ ϑ(A) and ∆(conv(A)) = ∆(A), so that ϑ(conv(A)) = ϑ(A) Now suppose that t ∈ ]τ, +∞[ Then K(t) = ∅ and dim V (t) = card K(t), where card K(t) denotes the cardinality of K(t) In addition, if card 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 conv(A) The intersection of conv(A) with a supporting hyperplane of conv(A) is a face of conv(A); Σ(A) is the set of intersections of A with a face of conv(A) The differential operator P(D) is non-degenerate if P and, for every σ ∈ Σ(A), Pσ : d → : x → α∈σ cα x α not vanish outside the coordinate planes of d , i.e., d d ∀x ∈ xj = ⇒ ∀σ ∈ Σ(A) P(x)Pσ (x) = (2.7) j=1 Remark 2.5 Suppose that P is non-degenerate and let α ∈ ϑ(A) Then it follows from (2.7) that all the components of α are even 2.2 Trigonometric approximations We first prove a Jackson-type inequality Lemma 2.6 Let t ∈ ′ ∀f ∈ ( d ++ ) and define a linear operator S t : ′ ( d )→ ′ ( d ) by fˆ(k)e i〈k|·〉 St ( f ) = (2.8) k∈K(t) [P] Let f ∈ W2 and suppose that t > τ Then the distribution f − S t ( f ) represents a function in L2 ( f − St ( f ) t −1 f [P] W2 Proof Set g = f − S t ( f ) Then g ∈ f [P] W2 k∈ ) and (2.9) ′ ( d ) On the other hand, Parseval’s identity yields |P(k)|2 | fˆ(k)|2 = d (2.10) d Hence, | fˆ(k)|2 |ˆ g (k)|2 = k∈ d k∈ d \K(t) sup k∈ t −2 f |P(k)|2 | fˆ(k)|2 |P(k)|−2 d \K(t) k∈ d \K(t) [P] , W2 (2.11) which means that f − S t ( f ) represents a function in L2 ( d ) for which (2.9) holds Corollary 2.7 Let t ∈ ]τ, +∞[ Then sup f f −g inf g∈V (t) f −g∈L ( d ) [P] ∈U2 t −1 (2.12) Next, we prove a Bernstein-type inequality Lemma 2.8 Let t ∈ ]τ, +∞[ and let f ∈ V (t) ∩ L2 ( f t f [P] W2 d ) Then (2.13) Proof By (2.10), we have [P] W2 f |P(k)|2 | fˆ(k)|2 = k∈K(t) k∈K(t) | fˆ(k)|2 sup |P(k)|2 t2 f 2, (2.14) k∈K(t) which establishes (2.13) 2.3 Compactness and non-degenerateness We start with a characterization of the compactness of the unit ball defined in (1.12) [P] Lemma 2.9 The set U2 is a compact subset of L2 ( d ) if and only if the following hold: (i) For every t ∈ ]τ, +∞[, K(t) is finite (ii) τ > 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 ( d ) In particular, using the notation (2.8), (∀ f ∈ ′ ( d )) S t ( f ) ∈ L2 ( d ) Hence, by Lemma 2.6, [P] ∀ f ∈ W2 [P] Thus, W2 ⊂ L2 ( f = ( f − S t ( f )) + S t ( f ) ∈ L2 ( d [P] U2 d ) (2.15) [P] ) On the other hand, (2.10) implies that U2 d is a closed subset of L2 ( fore, is compact in L2 ( ) if, for every ǫ ∈ ++ , it has a finite ǫ-net in L2 ( the following following two conditions are satisfied: (iii) For every ǫ ∈ sup [P] f ∈U2 ++ , inf g∈Gǫ d ǫ ) There- ) or, equivalently, if there exists a finite-dimensional vector subspace Gǫ of L2 ( f −g d d ) such that (2.16) [P] (iv) U2 is bounded in L2 ( d ) It follows from (2.10) that (ii)⇔(iv) On the other hand, since dim V (t) = card K(t), Corollary 2.7 yields (i)⇒(iii) To prove necessity, suppose that (i) does not hold Then dim V (˜t ) = card K(˜t ) = +∞ [P] for some ˜t ∈ ++ By Lemma 2.8, U = f ∈ V (˜t ) ∩ L2 ( d ) f 1/˜t is a subset of U2 which is not compact in L2 ( compact in L2 ( d ) d [P] ) If (ii) does not hold, then U2 ∩ L2 ( d ) is unbounded and, consequently, not The following lemma characterizes the non-degenerateness of P(D) Lemma 2.10 P(D) is non-degenerate if and only if (∃ γ ∈ ++ )(∀x ∈ d ) |P(x)| γ max |x α | (2.17) α∈ϑ(A) Proof As proved in [12, 17], P(D) is non-degenerate if and only if (∃ γ1 ∈ ++ )(∀x d ∈ ) |P(x)| |x α | γ1 (2.18) α∈ϑ(A) Hence, since there exist γ2 ∈ (∀x ∈ d ) and γ3 ∈ ++ |x α | γ2 max |x α | α∈ϑ(A) ++ such that γ3 max |x α|, (2.19) α∈ϑ(A) α∈ϑ(A) the proof is complete Lemma 2.11 Let B be a nonempty finite subset of ΩB (t) = k ∈ d max kα d and let t ∈ + Then (2.20) t α∈B is finite if and only if (∀ j ∈ {1, , d}) j B∩ = ∅ (2.21) Proof If (2.21) holds, then (∀ j ∈ {1, , d})(∃ a j ∈ ΩB (t) ⊂ d j=1 k∈ d kj t 1/a j ++ ) ajuj ∈ B ∩ m∈ ⊂ ΩB (t), which shows that ΩB (t) is [P] Theorem 2.12 Suppose that P(D) is non-degenerate Then U2 only if (2.21) is satisfied and ∈ A Proof Let us prove that there exists γ1 ∈ d |P(k)| Hence, (2.4) implies that and, therefore, ΩB (t) is bounded Conversely, if (2.21) does not hold, then there exists j ∈ {1, , d} such that mu j unbounded ∀k ∈ j ++ is a compact subset of L2 ( d ) if and such that γ1 max |kα | (2.22) α∈ϑ(A) Since there exists γ1 ∈ ∀k ∈ d ++ |P(k)| such that γ1 max |kα |, (2.23) α∈A and since (2.22) trivially holds if there exists j ∈ {1, , d} such that k j = 0, it is enough to show that ∗d ∀α ∈ A ∀k ∈ kα max kβ , (2.24) β∈ϑ(A) and a fortiori that d + ∀α ∈ A ∀x ∈ 〈α | x〉 max β | x (2.25) β∈ϑ(A) Indeed, since α ∈ conv(ϑ(A)), by Carathéodory’s theorem [20, Theorem 17.1], α is a convex combination of points (β j )1 j d+1 in ϑ(B), say d+1 d+1 λjβ j, α= where (λ j )1 j d+1 ∈ d+1 + λ j = (2.26) λ j max β | x = max β | x (2.27) and j=1 j=1 Therefore d+1 d + ∀x ∈ 〈α | x〉 = d+1 λj βj | x j=1 β∈ϑ(A) j=1 Hence, Lemma 2.10 asserts that there exists γ2 ∈ ∀k ∈ d γ2 max |kα | |P(k)| α∈ϑ(A) [P] Consequently, by Lemma 2.9, U2 finite and ++ β∈ϑ(A) such that γ1 max |kα | (2.28) α∈ϑ(A) is a compact set in L2 ( d ) if and only if, for every t ∈ +, inf max kα > k∈ d ΩA(t) is (2.29) α∈A In view of Lemma 2.11, the first condition is equivalent to (2.21) and the second to ∈ A Main result 3.1 Convex-analytical results Several important convex-analytical facts underly our analysis (see [4, 20] for background on convex analysis) We start with the following corollary Corollary 3.1 Suppose that P(D) is non-degenerate Then ∀k ∈ d |P(k)| ≍ maxα∈ϑ(A) |kα | Proof Combine (2.28) and Lemma 2.10 Next, we investigate the geometry of our problem from the view-point of convex duality Let C be a subset of d Recall that the polar set of C is C⊙ = x ∈ d (∀α ∈ C) 〈α | x〉 , (3.1) and the indicator function of C is d ιC : if x ∈ C; 0, → ]−∞, +∞] : x → (3.2) +∞, otherwise Moreover, if C is convex and ∈ C, the Minkowski gauge of C is the lower semicontinuous convex function mC : d → ]−∞, +∞] : x → inf ξ ∈ d Finally, the domain of a function ϕ : ++ → ]−∞, +∞] is dom ϕ = x ∈ Lemma 3.2 Let B be a nonempty finite subset of 0∈B and (∀ j ∈ {1, , d}) Set = (1, , 1) ∈ d x ∈ ξC B∩ j d + (3.3) d ϕ(x) < +∞ such that = ∅ (3.4) , let µ(B) be the optimal value of the problem d x j, maximize x∈B ⊙ (3.5) j=1 and set ̺(B) = max ρ ∈ Then ̺(B) ∈ ++ ++ ρ1 ∈ conv(B) (3.6) and µ(B) = 1/̺(B) Proof It follows from (3.4) that d + ∩ B⊙ = d + x∈ ∩ d 〈x | α〉 (3.7) α∈B is a nonempty compact set and hence (3.5) does have a solution Now fix j ∈ {1, , d} Then (∃ a j ∈ ++ ) a j u j ∈ B Hence x j = (1/a j )u j ∈ B ⊙ and therefore µ(B) = max x∈B⊙ 〈x | 1〉 xj |1 = 1/a j > Altogether µ(B) ∈ ++ Likewise, (3.4) implies that ̺(B) ∈ ++ Let us set ϕ = mconv(B) and d Furthermore, the conjugate of ψ = ι{1} Then it follows from (3.4) that dom ϕ = dom mconv(B) = + ϕ is ϕ ∗ = ι(conv(B))⊙ = ιB⊙ [4, Propositions 14.12 and 7.14(vi)] and the conjugate of ψ is ψ∗ = 〈· | 1〉 d ++ , Hence, since ∈ int dom ϕ = Proposition 15.13] yields dom ψ ∩ int dom ϕ = ∅ and the Fenchel duality formula [4, d µ(B) = max x∈B ⊙ xj j=1 = − 〈−x | 1〉 x∈B ⊙ = − ιB⊙ (x) + 〈−x | 1〉 d x∈ = − ϕ ∗ (x) + ψ∗ (−x) d x∈ ϕ(α) + ψ(α) = inf α∈ d mconv(B) (α) + ι{1} (α) = inf α∈ d = mconv(B) (1) = inf ξ ∈ = sup ρ ∈ ++ ++ ∈ ξconv(B) ρ1 ∈ conv(B) (3.8) We conclude that µ(B) = 1/̺(B) To illustrate the duality principles underlying Lemma 3.2, we consider two examples Example 3.3 We consider the case when d = and B = {(6, 0), (0, 6), (4, 4), (0, 0)} (see Figure 1) Then (3.4) is satisfied, µ(B) = 1/4, and ̺(B) = The set of solutions to (3.5) is the set S represented by the solid red segment: S = (x , x ) ∈ [1/12, 1/6]2 x + x = 1/4 Example 3.4 In this example we consider the case when B = {(0, 6), (2, 4), (4, 0), (0, 0)} Then (3.4) is satisfied, µ(B) = 3/8, and ̺(B) = 8/3 The set of solutions to (3.5) reduces to the singleton S = {(1/4, 1/8)} Lemma 3.5 Let B be a nonempty finite subset of (∀ j ∈ {1, , d}) B∩ j d + and suppose that = ∅ (3.9) Let µ(B) be the optimal value of the problem d maximize x∈B ⊙ x j, (3.10) j=1 and let ν(B) be the dimension of its set of solutions Then µ(B) ∈ (∀t ∈ [2, +∞[) card ΩB (t) ≍ t µ(B) log t ν(B) 10 ++ and (3.11) ̺(B)1 − conv(B) | µ(B) − − 12 B⊙ µ(B) | | 12 Figure 1: Graphical illustration of Example 3.3: In gray, the Newton polyhedron (top) and its polar (bottom) The dashed lines are the hyperplanes delimiting the polar set B ⊙ and the dotted line represents the optimal level curve of the objective function x → 〈x | 1〉 in (3.5) The solid red segment depicts the solution set of (3.5) 11 − conv(B) ̺(B)1 | −4 − B⊙ | | | µ(B) | Figure 2: Graphical illustration of Example 3.4: In gray, the Newton polyhedron (top) and its polar (bottom) The dashed lines are the hyperplanes delimiting the polar set B ⊙ and the dotted line represents the optimal level curve of the objective function x → 〈x | 1〉 in (3.5) The red dot locates the unique solution to (3.5) 12 Proof The fact that µ(B) ∈ ++ was proved as in Lemma 3.2 Now fix t ∈ [2, +∞[ and set ΛB (t) = d x∈ + maxα∈B x α t Then, as in the proof of Lemma 2.11, one can see that ΛB (t) is a bounded d subset of + If we denote by vol ΛB (t) the volume of ΛB (t), then it follows from [6, Theorem 1] that vol ΛB (t) ≍ t µ(B) (log t)ν(B) (3.12) Furthermore, proceeding as in the proof of [6, Theorem 2], one shows that card ΩB (t) ≍ vol ΛB (t) (3.13) These asymptotic relations prove the claim 3.2 Main result: asymptotic order of Kolmogorov n-width Our main result can now be stated and proved Theorem 3.6 Suppose that P(D) is non-degenerate and that ∈ A and (∀ j ∈ {1, , d}) A∩ j = ∅ (3.14) Let µ be the optimal value of the problem d x j, maximize x∈ϑ(A)⊙ (3.15) j=1 let ν be the dimension of its set of solutions, and set ̺ = max ρ ∈ Then µ = 1/̺ ∈ ++ [P] d ) ≍ n−̺ log n Equivalently, using (1.2), for ǫ ∈ [P] n ǫ U2 , L ( d (3.16) and, for n sufficiently large, ++ d n U2 , L ( ρ1 ∈ conv(ϑ(A)) ++ ν̺ (3.17) sufficiently small, ) ≍ ǫ −1/̺ | log ǫ|ν (3.18) Proof Since A satisfies (3.14), so does ϑ(A) Hence the fact that µ = 1/̺ ∈ ++ follows from Lemma 3.2 We also note that the equivalence between (3.17) and (3.18) follows from (1.1) and (1.2) To show (3.17), set ¯t = max{2, τ} Then we derive from Corollary 3.1 that (∀t ∈ [¯t , +∞[) card Ωϑ(A) (t) ≍ card K(t) (3.19) Applying Lemma 3.5 to ϑ(A) yields (∀t ∈ [¯t , +∞[) dim V (t) = card K(t) ≍ t 1/̺ log t 13 ν (3.20) Hence, for every n ∈ large enough, there exists t ∈ ν γ3 t 1/̺ log t γ1 dim V (t) ++ depending on n such that ν n < γ3 (t + 1)1/̺ log(t + 1) γ2 dim V (t + 1) γ4 t 1/̺ log t ν , (3.21) where γ1 , γ2 , γ3 , and γ4 are strictly positive real parameters that are independent from n and t Therefore, n ≍ t 1/̺ log t ν (3.22) or, equivalently, t −1 ≍ n−̺ log n ν̺ (3.23) It therefore follows from (1.1) and Corollary 2.7 that [P] d n U2 , L ( d ν̺ t −1 ≍ n−̺ log n ) , (3.24) which establishes the upper bound in (3.17) To establish the lower bound, let us recall from [25] that, for every n + 1-dimensional vector subspace Gn+1 of L2 ( d ) and every η ∈ ++ , we have dn Bn+1 (η), L2 ( d ) = η, Arguing as in (3.20)–(3.23), for n ∈ dim V (t) where γ5 ∈ ++ γ5 t 1/̺ log t and γ6 ∈ U(t) = f ∈ V (t) ++ f ν Bn+1 (η) = f ∈ Gn+1 where f L2 ( sufficiently large, there exists t ∈ >n γ6 t 1/̺ log t ν d) ++ η (3.25) such that , (3.26) are independent from n and t Now set t −1 (3.27) [P] By Lemma 2.8, U(t) ⊂ U2 Consequently, it follows from (3.25)–(3.27) and (3.23) that [P] d n U2 , L ( d ) dn U(t), L2 ( d ) t −1 ≍ n−̺ log n ν̺ , (3.28) which concludes the proof of (3.17) Next, let us prove (3.18) Given a sufficiently small ǫ ∈ ++ , take t ∈ ++ such that < t − < ǫ −1 t and dim V (t) > From the above results, it can be seen that [P] n ǫ U2 , L ( dim V (t) − d ) dim V (t) (3.29) which, together with (3.20), proves (3.18) Remark 3.7 We have actually proven a bit more than Theorem 3.6 Namely, suppose that P(D) satis[P] fies the conditions of compactness for U2 stated in Lemma 2.9 and, for every n ∈ , let t(n) be the largest number such that card K(t(n)) n Then, for n sufficiently large, we have [P] d n U2 , L ( d ) ≍ t(n) (3.30) 14 Examples We first establish norm equivalences and use them to provide examples of asymptotic orders of [P] dn U2 , L2 ( d ) for non-degenerate and degenerate differential operators Theorem 4.1 Suppose that P(D) is non-degenerate and set x α Q: x → (4.1) α∈ϑ(A) Then [P] ∀ f ∈ W2 f [P] W2 ≍ f [Q] W2 Dα f ≍ 2 α∈ϑ(A) ≍ max Dα f α∈ϑ(A) 2 (4.2) Moreover, the seminorms in (4.2) are norms if and only if ∈ A [P] Proof Let f ∈ W2 It is clear that Dα f 2 ≍ max Dα f α∈ϑ(A) α∈ϑ(A) 2 (4.3) Parseval’s identity and Corollary 3.1 yield max Dα f α∈ϑ(A) Now let ( d 2 |k|2α | fˆ(k)|2 = max α∈ϑ(A) d k∈ k∈ (α))α∈ϑ(A) be a partition of max |kβ | = |kα |, β∈ϑ(A) k∈ d max |kα | d d α∈ϑ(A) | fˆ(k)|2 (4.4) such that (α) (4.5) Then max Dα f α∈ϑ(A) 2 |k2α | | fˆ(k)|2 = max α∈ϑ(A) α′ ∈ϑ(A) k∈ d (α′ ) ′ |k2α | | fˆ(k)|2 α′ ∈ϑ(A) k∈ (4.6) d (α′ ) max |kα |2 | fˆ(k)|2 = k∈ α∈ϑ(A) d k∈ α∈ϑ(A) d Thus, max Dα f α∈ϑ(A) 2 max |kα |2 | fˆ(k)|2 = (4.7) 15 Hence, appealing to Corollary 3.1 and (2.10), we obtain max Dα f α∈ϑ(A) 2 ≍ f [P] W2 (4.8) 2 ≍ f [Q] W2 (4.9) The relation max Dα f α∈ϑ(A) follows from the last seminorm equivalence and the identity ϑ(ϑ(A)) = ϑ(A) Therefore, we derive from (4.2) that the seminorms in (4.2) are norms if and only if ∈ A 4.1 Isotropic Sobolev classes Let s ∈ ∗ The isotropic Sobolev space H s is the Hilbert space of functions f ∈ L2 ( the norm · Hs : f → f 2 2 f (α) + d ) equipped with (4.10) |α|=s Consider xα = P: x → 1+ x α, (4.11) α∈A |α|=s where A = {0}∪ α ∈ d |α| = s If s is even, it follows directly from Lemma 2.10 that the differential operator P(D) is non-degenerate, and consequently, by Theorem 4.1, · H s is equivalent to one of the norms appearing in (4.2) with ϑ(A) = {0} ∪ su j j d and d x sj Q: x → + (4.12) j=1 Moreover, we have ̺(A) = s/d and ν(a) = Therefore, we retrieve from Theorem 3.6 the well-known result d n U s , L2 ( d ) ≍ n−s/d , (4.13) where U s denotes the closed unit ball in H s This result is a direct generalization of the first result on n-widths established by Kolmogorov in [14] 4.2 Anisotropic Sobolev classes ∗d Given β = (β1 , , βd ) ∈ equipped with the norm , the anisotropic Sobolev space H β is the Hilbert space of functions f ∈ L2 d · Hβ : f → f 2 j f (β j u ) + 2 (4.14) j=1 16 Consider the polynomial d βj P: x → 1+ x α, xj = (4.15) α∈A j=1 where A = {0} ∪ β j u j j d If the coordinates of β are even, the differential operator P(D) is non-degenerate Consequently, by Theorem 4.1, · H β is equivalent to one of the norms in (4.2) with ϑ(A) = A and Q = P (4.16) We have   ̺ = ̺(A) =  −1 d  1/β j  (4.17) j=1 and ν(A) = 0, and therefore, from Theorem 3.6 we retrieve the known result [13] d n U β , L2 ( d ) ≍ n−̺ , (4.18) where U ̺ denotes the unit ball in in H β 4.3 Classes of functions with a bounded mixed derivative Let α = (α1 , , αd ) ∈ d with < α1 = · · · = αν+1 < αν+2 = · · · = αd for some ν ∈ {0, , d − 1} Given a set e ⊂ {1, , d}, let the vector α(e) ∈ d be defined by α(e) j = α j if j ∈ e, and α(e) j = otherwise (in particular, α(∅) = and α({1, , d}) = α) The space W2α is the Hilbert space of functions f ∈ L2 equipped with the norm · W2α : f (α(e)) f → 2 (4.19) e⊂{1, ,d} Consider x α(e) = P: x → e⊂{1, ,d} x α, (4.20) α∈A where A = α(e) e ⊂ {1, , d} If the coordinates of α are even, the differential operator P(D) is non-degenerate and hence, by Theorem 4.1, · W2α is equivalent to one of the norms in (4.2) with ϑ(A) = A and Q = P We have ̺(A) = α1 and ν(A) = ν, and therefore, from Theorem 3.6 we recover the result proven in [1], namely that for n sufficiently large dn U2α , L2 ( d ) ≍ n−α1 log n να1 , (4.21) where U2α denotes the unit ball in W2α In the particular case when α = ̺1, we have ̺1 d n U2 , L ( d ) ≍ n−̺ log n (d−1)̺ (4.22) 17 4.4 Classes of functions with several bounded mixed derivatives Suppose that (3.14) is satisfied Let W2A be the Hilbert space of functions f ∈ L2 ( the norm · W2A : f (α) f → 2 d ) equipped with (4.23) α∈A Notice that spaces H s , H r , and W2α are a particular cases of W2A Now consider x α P: x → (4.24) α∈A If the coordinates of every α ∈ ϑ(A) are even, the differential operator P(D) is non-degenerate and it follows from Theorem 4.1 that · W A is equivalent to one of the norms in (4.2) If ̺ = ̺(ϑ(A)) and ν = ν(ϑ(A)), we again retrieve from Theorem 3.6 the result proven in [6], namely that for n sufficiently large dn U2A, L2 ( d ) ≍ n−̺ log n ν̺ , (4.25) where U2A denotes the unit ball in W2A 4.5 Classes of functions induced by a differential operator [P] We give two examples of spaces W2 Consider the polynomials with non-degenerate differential operator P(D) for d = P1 : x → 8x 14 − 4x 13 − 3x 13 x − 2x 12 x − 4x x + 6x 22 − 4x − 3x + 13 P2 : x → 6x 16 + x 14 x 22 − 6x 15 − x 13 x 22 + 5x 24 − 4x 23 + We have  A1   ϑ(A ) A2   ϑ(A2 ) (4.26) = {(4, 0), (3, 0), (2, 1), (2, 0), (1, 1), (0, 2), (1, 0), (0, 1), (0, 0)} = {(4, 0), (0, 2), (0, 0)} = {(6, 0), (4, 2), (5, 0), (3, 2), (0, 4), (0, 3), (0, 0)} (4.27) = {(6, 0), (4, 2), (0, 4), (0, 0)} It is easy to verify that P1 (D) and P2 (D) are non-degenerate and that (3.14) holds Moreover, ̺(ϑ(A1 )) = 4/3, ν(ϑ(A1 )) = 0, ̺(ϑ(A2 )) = 8/3, and ν(ϑ(A2 )) = We derive from Theorem 3.6 that dn U [P1 ] , L2 ( ) ≍ n−4/3 , dn U [P2 ] , L2 ( ) ≍ n−8/3 log n (4.28) and 8/3 (4.29) 18 Let us give an example of a degenerate differential operator For P3 : x → x 14 − 2x 13 x + x 12 x 22 + x 12 + x 22 + 1, (4.30) the differential operator P3 (D) is degenerate, although P3 on , and U [P3 ] is a compact set in [P3 ] L2 ( ) Therefore, we cannot compute dn (U , L2 ( )) by using Theorem 3.6 However, by a direct computation we get card K(t) ≍ t 1/2 log t Hence, (3.30) yields dn U [P3 ] , L2 ( 2 ) ≍ n−2 log n (4.31) 4.6 A conjecture [P] Suppose that U2 is compact in L2 ( (i) For every t ∈ +, d ) In view of Lemma 2.9, this is equivalent to the conditions: K(t) is finite (ii) τ > As mentioned in (3.30), for every n ∈ that card K(t(n)) n, then [P] d n U2 , L ( d ) ≍ t(n) sufficiently large, if t(n) ∈ ++ is the maximal number such (4.32) [P] This means that the problem of computing the asymptotic order of dn (U2 , L2 ( d )) is equivalent to the problem of computing that of card K(t) when t → +∞ Let us formulate it as the following conjecture Conjecture 4.2 Suppose that, for every t ∈ + , K(t) is finite (the condition τ > is not essential) Then there exist integers α, β, and ν such that < α β, ν < d, and, for t large enough, card K(t) ≍ t α/β log t ν (4.33) In view of (3.20), we know that the conjecture is true when P satisfies conditions (2.7) and (3.9) 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 and Patrick Combettes was visiting at the Vietnam Institute for Advanced Study in Mathematics (VIASM) Both authors thank the VIASM for providing fruitful research environment and working condition They also thank the LIA CNRS Formath Vietnam for providing travel support References [1] K I Babenko, Approximation of periodic functions of many variables by trigonometric polynomials, Soviet Math Dokl (1960) 513–516 19 [2] K I Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Soviet Math Dokl (1960) 672–675 [3] R Baraniuk, M Davenport, R DeVore, and M Wakin, A simple proof of the restricted isometry 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P is conv (A) The intersection of conv (A) with a supporting hyperplane of conv (A) is a face of conv (A) ; Σ (A) is the set of intersections of A with a face of conv (A) The differential operator P(D)... and approximation of functions having mixed smoothness, http://arxiv.org/abs/1309.5170 [6] Dinh D˜ ung, The number of integral points in some sets and approximation of functions of several variables,