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Adv Comput Math (2011) 34:1–41 DOI 10.1007/s10444-009-9140-9 Optimal adaptive sampling recovery ˜ Dinh Dung Received: 29 April 2009 / Accepted: 25 August 2009 / Published online: 16 September 2009 © Springer Science + Business Media, LLC 2009 Abstract We propose an approach to study optimal methods of adaptive sampling recovery of functions by sets of a finite capacity which is measured by their cardinality or pseudo-dimension Let W ⊂ Lq , < q ≤ ∞, be a class of functions on Id := [0, 1]d For B a subset in Lq , we define a sampling recovery method with the free choice of sample points and recovering functions from B as follows For each f ∈ W we choose n sample points This choice defines n sampled values Based on these sampled values, we choose a function from B for recovering f The choice of n sample points and a recovering function from B for each f ∈ W defines a sampling recovery method SnB by functions in B An efficient sampling recovery method should be adaptive to f Given a family B of subsets in Lq , we consider optimal methods of adaptive sampling recovery of functions in W by B from B in terms of the quantity Rn (W, B )q := inf sup inf B∈B f ∈W SnB f − SnB ( f ) q Denote Rn (W, B )q by en (W)q if B is the family of all subsets B of Lq such that the cardinality of B does not exceed 2n , and by rn (W)q if B is the family of all subsets B in Lq of pseudo-dimension at most n Let < p, q, θ ≤ ∞ and α satisfy one of the following conditions: (i) α > d/ p; (ii) α = d/ p, θ ≤ min(1, q), p, q < ∞ Then for the d-variable Besov class U αp,θ (defined as the unit ball of the Besov space Bαp,θ ), there is the following asymptotic order en U αp,θ q rn U αp,θ q n−α/d Communicated by Qiyu Sun ˜ (B) D Dung Information Technology Institute, Vietnam National University, Hanoi 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam e-mail: dinhdung@vnu.edu.vn ˜ D Dung To construct asymptotically optimal adaptive sampling recovery methods for en (U αp,θ )q and rn (U αp,θ )q we use a quasi-interpolant wavelet representation of functions in Besov spaces associated with some equivalent discrete quasi-norm Keywords Adaptive sampling recovery · Quasi-interpolant wavelet representation · B-spline · Besov space Mathematics Subject Classifications (2000) 41A46 · 41A05 · 41A25 · 42C40 Introduction We are interested in problems of sampling recovery of functions defined on the unit d-cube Id := [0, 1]d Let Lq := Lq (Id ), < q ≤ ∞, denote the quasinormed space of functions on Id with the usual qth integral quasi-norm · q for < q < ∞, and the normed space C(Id ) of continuous functions on Id with the max-norm · ∞ for q = ∞ We consider sampling recoveries of functions from a class of a certain smoothness W ⊂ Lq by functions from a subset B in Lq The recovery error will be measured in the norm · q We will focus our attention to optimal methods of adaptive sampling recovery of functions in W by subsets B of a finite capacity which is measured by their cardinality or pseudo-dimension Let us first recall some well-known sampling recovery methods Suppose that f is a function in W and ξ = {xk }nk=1 are n points in Id We want to approximately recover f from the sampled values f (x1 ), f (x2 ), , f (xn ) A general sampling recovery method can be defined as Rn ( f ) = Rn (H, ξ, f ) := H( f (x1 ), , f (xn )), (1.1) where H is a given mapping from Rn to Lq Such a method is, in general, nonlinear A classical formula of linear sampling recovery of the form (1.1) is n Ln ( f ) = Ln ( , ξ, f ) := f (xk )ϕk , (1.2) k=1 where = {ϕk }nk=1 are given n functions on Id To study optimal sampling methods of recovery for f ∈ W from n their values, we can use the quantity gn (W)q := inf sup H,ξ f ∈W f − Rn (H, ξ, f ) q , where the infimum is taken over all sequences ξ = {xk }nk=1 and all mappings H from Rn into Lq Similarly in non-linear approximations [7], in non-adaptive and adaptive sampling recoveries of functions with a certain smoothness, it is convenient to take functions to be recovered from the Besov class U αp,θ which is defined Optimal adaptive sampling recovery as the unit ball of the Besov space Bαp,θ , having a fractional smoothness α > (the definition of this space is given in Section 2) Notice that in problems of sampling recovery, other classes such as well-known Sobolev and LizorkinTriebel classes, etc can be considered (see [22]) We use the notations: x+ := max(0, x) for x ∈ R; An ( f ) Bn ( f ) if An ( f ) ≤ CBn ( f ) with C an absolute constant not depending on n and/or f ∈ W, and An ( f ) Bn ( f ) if An ( f ) Bn ( f ) and Bn ( f ) An ( f ) It is known the following result (see [10, 19, 21, 22, 26] and references there) Let < p, q ≤ ∞, < θ ≤ ∞ and α > d/ p Then there is a linear sampling recovery method L∗n of the form (1.2) such that gn U αp,θ q sup f ∈U αp,θ f − L∗n ( f ) q n−α/d+(1/ p−1/q)+ (1.3) This result says that the linear sampling recovery method L∗n is asymptotically optimal in the sense that any sampling recovery method Rn of the form (1.1) does not give the rate of convergence better than L∗n Sampling recovery methods of the form (1.1) which may be linear or non-linear are non-adaptive, i.e., the points ξ = {xk }nk=1 at which the values f (x1 ), , f (xn ) are sampled, and the recovery method Rn are the same for all functions f ∈ W Let us introduce a setting of adaptive sampling recovery which will give the asymptotic order of the recovery error better than the nonadaptive sampling recovery in some cases Let B be a subset in Lq We will define a sampling recovery method with the free choice of sample points and recovering functions from B Roughly speaking, for each f ∈ W we choose a set of n sample points This choice defines a collection of n sampled values Based on the information of these sampled values, we choose a function from B for recovering f The choice of n sample points and a recovering function from B for each f ∈ W defines a sampling recovery method SnB by functions in B Let us give a precise notion of SnB Denote by I n the set of subsets ξ in Id of cardinality at most n Let V n be the set whose elements are collections of real numbers aξ = {a(x)}x∈ξ , ξ ∈ I n , a(x) ∈ R (for aξ , b η ∈ V n , we write by definition aξ = b η if and only if ξ = η and a(x) = b (x) for any x ∈ ξ ) Let In be a mapping from W into I n and P a mapping from V n into B Then the pair (In , P) generates the mapping SnB from W into B by the formula SnB ( f ) := P { f (x)}x∈In ( f ) , (1.4) which defines a general sampling recovery method with the free choice of n sample points by functions in B A non-adaptive sampling recovery method of the form (1.1) is a particular case of (1.4) for which In ( f ) ≡ ξ, f ∈ W, for some ξ = {xk }nk=1 , B = Lq and P can be treated as a mapping H from Rn into Lq We want to choose a sampling recovery method SnB so that the error of this recovery f − SnB ( f ) q is as smaller as possible Clearly, such an efficient ˜ D Dung choice should be adaptive to f The error of an optimal adaptive sampling recovery method for each f ∈ W, is measured by RnB ( f )q := inf SnB f − SnB ( f ) q , where the infimum is taken over all sampling recovery methods SnB of the form (1.4) The worst case error for the function class W is expressed by the quantity RnB (W)q := sup RnB ( f )q f ∈W Given a family B of subsets in Lq , we consider optimal sampling recoveries by B from B in terms of the quantity Rn (W, B )q := inf RnB (W)q B∈B (1.5) We assume a restriction on the sets B ∈ B , requiring that they should have, in some sense, a finite capacity In the present paper, the capacity of B is measured by its cardinality or pseudo-dimension This reasonable restriction would provide nontrivial lower bounds of asymptotic order of Rn (W, B )q for well known function classes W Denote Rn (W, B )q by en (W)q if B in (1.5) is the family of all subsets B in Lq such that |B| ≤ 2n , where |B| denotes the cardinality of B, and by rn (W)q if B in (1.5) is the family of all subsets B in Lq of pseudo-dimension at most n The quantity en (W)q is related to the entropy n-width (entropy number) εn (W)q which is the functional inverse of the classical ε-entropy introduced by Kolmogorov and Tikhomirov [18] The quantity rn (W)q is related to the nonlinear n-width ρn (W)q introduced recently by Ratsaby and Maiorov [24] (See the definition of εn (W)q and ρn (W)q in Appendix) The pseudo-dimension of a set B of real-valued functions on a set , is defined as follows For a real number t, let sgn(t) be for t > and −1 otherwise For x ∈ Rn , let sgn(x) = (sgn(x1 ), sgn(x2 ), , sgn(xn )) The pseudodimension of B is defined as the largest integer n such that there exist points a1 , a2 , , an in and b ∈ Rn such that the cardinality of the set sgn(y) : y = f (a1 ) + b , f (a2 ) + b , , f (an ) + b n , f ∈ B is 2n If n is arbitrarily large then the the pseudo-dimension of B is infinite Denote the pseudo-dimension of B by dimp (B) The notion of pseudodimension was introduced by Pollard [23] and later Haussler [16] as an extension of the VC-dimension [28], suggested by Vapnik-Chervonekis for sets of indicator functions The pseudo-dimension and VC-dimension measure the capacity of a set of functions and are related to its ε-entropy They play an important role in theory of pattern recognition and regression estimation, empirical processes and computational learning theory Thus, in the probably approximately correct (PAC) learning model, if B is a set of real-valued functions on having a finite VC or pseudo-dimension, and P is a probability distribution on , then we can estimate any f ∈ B by some g to an arbitrary accuracy ε and probability − δ by just knowing its values at m randomly Optimal adaptive sampling recovery sample points from where m depends on ε and δ (see also [24, 25]) If B is a n-dimensional linear manifold of real-valued functions on , then dimp (B) = n (see [16]) We say that p, q, θ, α satisfy Condition (1.6) if < p, q, θ ≤ ∞, α < ∞, and there holds one of the following restrictions : (i) α > d/ p; (ii) α = d/ p, θ ≤ min(1, q), p, q < ∞ (1.6) The main results of the present paper are read as follows Theorem 1.1 Let p, q, θ, α satisfy Condition (1.6) Then for the d-variable Besov class U αp,θ , there is the following asymptotic order en U αp,θ q rn U αp,θ q n−α/d Let = {ϕk }k∈J be a family of elements in Lq Denote by linear set of linear combinations of n free terms from , that is ⎧ ⎫ n ⎨ ⎬ ϕ= a jϕk j : k j ∈ J n ( ) := ⎩ ⎭ n( ) the non- j=1 Consider adaptive sampling recovery methods of f ∈ W by functions from n ( ) in terms of the quantity sn (W, )q := Rn n ( ) (W)q The quantity sn (W, )q has been introduced in [14] in another equivalent form (with the notation νn (W, )q ) Let us recall it For each function f ∈ W, we choose a sequence ξ = {xs }ns=1 of n points in Id , a sequence a = {as }ns=1 of n functions on Rn and a sequence n = {ϕks }ns=1 of n functions from This choice defines a sampling recovery method given by n S( n , a, ξ, f ) := as ( f (x1 ), , f (xn ))ϕks s=1 Then the quantity sn (W, )q can be defined by sn (W, )q = sup inf f ∈W n ,a,ξ f − S( n , a, ξ, f ) q, where the infimum is taken over all sequences ξ = {xs }ns=1 of n points in Id , a = {as }ns=1 of n functions defined on Rn , and n = {ϕks }ns=1 of n functions from The optimal adaptive sampling recovery in terms of the quantity sn (W, )q is related to the quantity σn (W, )q of non-linear n-term approximation which characterizes the approximation of W by functions from n ( ) (see the definition in Appendix) The reader can find in [7, 27] surveys on various ˜ D Dung aspects of this approximation and its applications Let us recall some results in [15] on adaptive sampling recovery in regard to the quantity sn (W, )q For a given natural number r, let M be the centered B-spline of even order 2r with support [−r, r] and knots at the integer points −r, , 0, , r, and define B-spline wavelets Mk,s (x) := M(2k x − s), for a non-negative integer k and s ∈ Z Then M is the set of all Mk,s which not vanish identically on I The following result was proven in [15] Let ≤ p, q ≤ ∞, < θ ≤ ∞, and < α < min(2r, 2r − + 1/ p) Then for the the univariate Besov class U αp,θ , there is the following asymptotic order sn U αp,θ , M q n−α (1.7) To construct an asymptotically optimal adaptive sampling recovery method for sn (U αp,θ , M)q which gives the upper bound of (1.7) we used the following quasi-interpolant wavelet representation of functions in the Besov space Bαp,θ in terms of the B-spline wavelet system M associated with some equivalent discrete quasi-norm If ≤ p ≤ ∞, < θ ≤ ∞, and < α < min(2r, 2r − + 1/ p), then a function f in the Besov space Bαp,θ can be represented as a series ∞ f = ck,s ( f )Mk,s (1.8) k=0 s∈J(k) with the convergence in Bαp,θ , where J(k) is the set of s for which Mk,s not vanish identically on I, and ck,s ( f ) are functions of a finite number of values of f which does not depend on neither k, s nor f Moreover, the quasi-norm of Bαp,θ is equivalent to the discrete quasi-norm ⎞1/ p ⎫θ ⎞1/θ ⎪ ⎬ ⎜ ⎟ (α−1/ p)k ⎝ p⎠ |ck,s ( f )| ⎠ ⎝ ⎪ ⎪ ⎩ ⎭ k=0 s∈J(k) ⎛ ∞ ⎧ ⎪ ⎨ ⎛ (1.9) Such a representation can be constructed by using a quasi-interpolant See [15] for details An asymptotically optimal non-linear sampling recovery method was constructed as the sum of a linear quasi-interpolant operator at a lower B-spline dyadic scale of this representation, and non-linear adaptive operator which is the sum of greedy algorithms at some higher dyadic scales of B-spline wavelets In the present paper, we also extend (1.7) to the case < p, q ≤ ∞ and α ≥ d/ p, and generalize it for multivariate functions on the d-cube Id In particular, important is the case < p < or < q < which are of great interest in non-linear approximations (see [7, 9]) To get d-variable B-spline wavelets we let d M(x) := M(xi ), x = (x1 , x2 , , xd ), i=1 Optimal adaptive sampling recovery and Mk,s (x) := M(2k x − s), for a non-negative integer k and s ∈ Zd Denote again by M the set of all Mk,s which not vanish identically on Id We prove the following theorem Theorem 1.2 Let p, q, θ, α satisfy Condition (1.6) and α < min(2r, 2r − + 1/ p) Then for the d-variable Besov class U αp,θ , there is the following asymptotic order sn U αp,θ , M q n−α/d We will prove a multivariate generalization of the quasi-interpolant wavelet representation (1.8–1.9) which plays an important role in the proofs of Theorems 1.1 and 1.2 (see Theorem 2.2) The methods and techniques used [15] for the proof of the representation (1.8–1.9) cannot be applied to this generalization Thus, to prove it we should overcome some difficulties by employing a different technique On the basic of this representation we construct asymptotically optimal sampling recovery methods which give the upper bound for en (U αp,θ )q , rn (U αp,θ )q and sn (U αp,θ , M)q Their lower bounds are established by the lower estimating of the smaller related quantities εn (U αp,θ )q , ρn (U αp,θ )q and σn (U αp,θ , M)q , respectively Notice that the quantities en (W)q and rn (W)q are absolute in the sense of optimal sampling recovery methods, while the quantity sn (W, )q depends on a system However, Theorems 1.1 and 1.2 show that en (U αp,θ )q , rn (U αp,θ )q and sn (U αp,θ , M)q (with the restriction α < min(2r, 2r − + 1/ p)) have the same asymptotic order For < p < q ≤ ∞, the asymptotic order of optimal adaptive sampling recovery method in terms of the quantities en (U αp,θ )q , rn (U αp,θ )q and sn (U αp,θ , M)q is better than the asymptotic order of any non-adaptive sampling recovery method of the form (1.1) It is known that the inequalities α ≥ d/ p and α > d/ p are a condition for the embedding and the compact embedding of the Besov space Bαp,θ and other function spaces of smoothness α into C(Id ), respectively The previous papers on sampling recovery considered only the case α > d/ p In Theorems 1.1 and 1.2, we receive some results also for the case α = d/ p, θ ≤ min(1, q), < p, q < ∞ of the Besov class U αp,θ In the present paper, we consider optimal adaptive sampling recoveries for the Besov class of multivariate functions Results similar to Theorems 1.1 and 1.2 are also true for the Sobolev and Lizorkin-Triebel classes of multivariate functions The paper is organized as follows In Section 2, we give a definition of quasi-interpolant form functions on Id , construct a quasi-interpolant wavelet representation in terms of the Bspline dictionary M for Besov spaces and prove some quasi-norm equivalences based on this representation, in particular, a discrete quasi-norm in terms of ˜ D Dung the coefficient functionals In Sections and 4, we prove Theorem 1.1 In Section 3, we prove the asymptotic order of rn (U αp,θ )q in Theorem 1.1 and of sn (U αp,θ , M)q in Theorem 1.2 and construct asymptotically optimal adaptive sampling recovery methods which give the upper bound for rn (U αp,θ )q and sn (U αp,θ , M)q In Section 4, we prove the asymptotic order of en (U αp,θ )q in Theorem 1.1 and construct asymptotically optimal adaptive sampling recovery methods which give the upper bound for en (U αp,θ )q In Appendix in Section 5, we give some auxiliary notions and results on non-linear approximations which are employed, in particular, in establishing the lower bounds in Theorems 1.1 and 1.2 Quasi-interpolant wavelet representations in Besov spaces Let = {λ( j)} j∈Pd (μ) be a finite even sequence, i.e., λ(− j) = λ( j), where Pd (μ) := { j ∈ Zd : | ji | ≤ μ, i = 1, 2, , d} We define the linear operator Q for functions f on Rd by Q( f, x) := ( f, s)M(x − s), (2.1) λ( j) f (s − j) (2.2) s∈Zd where ( f, s) := j∈Pd (μ) The operator Q is bounded in C(Rd ) and Q( f ) C(Rd ) ≤ f C(Rd ) for each f ∈ C(Rd ), where |λ( j)| = j∈Pd (μ) Moreover, Q is local in the following sense There is a positive number δ > such that for any f ∈ C(Rd ) and x ∈ Rd , Q( f, x) depends only on the value f (y) at a finite number of points y with |yi − xi | ≤ δ, i = 1, 2, d We d will require Q to reproduce the space P2r−1 of polynomials of order at most 2r − in each variable xi , that is, d Q( p) = p, p ∈ P2r−1 d , is called a quasiAn operator Q of the form (2.1–2.2) reproducing P2r−1 d interpolant in C(R ) There are many ways to construct quasi-interpolants A method of construction via Neumann series was suggested by Chui and Diamond [4] (see also [3, p 100–109]) De Bore and Fix [5] introduced another quasi-interpolant based on the values of derivatives The reader can see also the books [3, 6] for surveys on quasi-interpolants Optimal adaptive sampling recovery Let = [a, b ]d be a d-cube in Rd Denote by L p ( ) the quasi-normed space of functions on with the usual pth integral quasi-norm · p, for < p < ∞, and the normed space C( ) of continuous functions on with the max-norm · ∞, for p = ∞ If τ be a number such that < τ ≤ min( p, 1), then for any sequence of functions { fk } there is the inequality τ fk ≤ p, fk τ p, (2.3) We will introduce Besov spaces of smooth functions and give necessary knowledge of them The reader can read this and more details about Besov spaces in the books [1, 8, 20] Let ωl ( f, t) p := sup |h|0 t ωl ( f, t) p , The Besov quasi-norm is defined by B( f ) = f Bαp,θ := f p + | f | Bαp,θ We will assume that continuous functions to be recovered are from the Besov space Bαp,θ with the restriction on the smoothness α ≥ 1/ p which is a condition for the embedding of this space into C(Id ) If { fk }∞ k=0 is a sequence whose component functions fk are in L p , for β < p, θ ≤ ∞ and β ≥ we use the b θ (L p ) “quasi-norms” ∞ { fk } β b θ (L p ) := 1/θ 2βk fk p θ k=0 with the usual change to a supremum when θ = ∞ When { fk }∞ k=0 is a positive sequence, we replace fk p by | fk | and denote the corresponding quasi-norm ˜ D Dung 10 by { fk } b β We will need the following discrete Hardy inequality Let {ak }∞ k=0 θ and {ck }∞ k=0 be two positive sequences and let for some M > 0, τ > 1/τ ∞ |ck | ≤ M |ak | τ (2.4) k=m Then for β > 0, θ > 0, {ck } β bθ ≤ CM {ak } (2.5) β bθ with C = C(β, θ) (see, e.g, [8]) For the Besov space Bαp,θ , there is the following quasi-norm equivalence B( f ) ωl f, 2−k B1 ( f ) := p b αθ + f p If Q of is a quasi-interpolant of the form (2.1–2.2), for h > and a function f on Rd , we define the operator Qh by Q( f ; h) = σh ◦ Q ◦ σ1/ h ( f ), where σh ( f, x) = f (x/ h) By definition it is easy to see that ( f, k; h)M(h−1 x − k), Q( f, x; h) = k where λ( j) f (h(k − j)) ( f, k; h) := j∈Pd (μ) The operator Q(·; h) has the same properties as Q: it is a local bounded d Moreover, it linear operator in Rd and reproduces the polynomials from P2r−1 gives a good approximation of smooth functions [6, p 63–65] We will also call it a quasi-interpolant for C(Rd ) The quasi-interpolant Q(·; h) is not defined for a function f on Id , and therefore, not appropriate for an approximate sampling recovery of f from its sampled values at points in Id An approach to construct a quasi-interpolant for a function on Id is to extend it by interpolation Lagrange polynomials This approach has been proposed in [15] for the univariate case Let us recall it For a non-negative integer m, we put x j = j2−m , j ∈ Z If f is a function on I, let 2r−1 U m ( f, x) := f (x0 ) + 2sm s 2−m f (x0 ) s! s=1 2r−1 Vm ( f, x) := f (x2m −2r+1 ) + s=1 2sm s−1 (x − x j ), j=0 s 2−m f (x2m −2r+1 ) s! s−1 (x − x2m −2r+1+ j) j=0 (2.6) be the (2r − 1)th Lagrange polynomials interpolating f at the 2r left end points x0 , x1 , , x2r−1 , and 2r right end points x2m −2r+1 , x2m −2r+3 , , x2m , of the Optimal adaptive sampling recovery 27 with regard to the decomposition (3.10) for an adaptive approximation of each component function qk ( f ) in the kth scale subspaces (k) for k¯ < k ≤ k∗ The greedy algorithms Gk are constructed as follows We reorder the k indexes s ∈ J(k) as {s j}mj=1 so that |ck,s1 ( f )| ≥ |ck,s2 ( f )| ≥ · · · |ck,sn ( f )| ≥ · · · |ck,mk ( f )|, (3.14) and then take the first largest n term for an approximation of qk ( f ) by forming the linear combination nk Gk (qk ( f )) := ck,s j ( f )Mk,s j (3.15) j=1 For all f ∈ U there is the inequality qk ( f ) − Gk (qk ( f )) k = k¯ + 1, , k∗ , 2−αk 2δk n−δ k , q (3.16) where δ := d(1/ p − 1/q) This estimate was proven in [14] for the case d = by using Lemma 5.1 The case d > can be proven in a completely similar way Further, by (2.18–2.19) and (3.11) we have for all f ∈ U qk ( f ) 2−(α−δ)k , q k = k∗ + 1, +2, (3.17) Consider the mapping G from U into B defined by (3.12) Let B be the set of B-splines of the form (3.1) Then G( f ) ∈ B By Lemma 3.1 B ⊂ Qm , where m is given in (3.2) According to (3.5) there is an absolute constant C such that dimp (Qm ) ≤ Cm Therefore, dimp (B) ≤ Cm ¯ k∗ and a sequence {nk }k∗ with nk ≤ mk , k = k¯ + 1, , k∗ Let us select k, ¯ k=k+1 We define an integer k¯ from the condition ¯ ¯ C1 2dk ≤ n < C2 2dk , (3.18) where C1 , C2 are absolute constants which will be chosen below The number of sampled values defining G( f ) does not exceed k∗ k¯ m := (2 + 1) + (2μ + 2r) d nk d (3.19) ¯ k=k+1 Notice that under the hypotheses of Theorem 3.1 we have < δ < α Further, we fix a number ε satisfying the inequalities < ε < (α − δ)/δ ∗ An appropriate selection of k and ∗ {nk }kk=k+1 ¯ (3.20) is k∗ := [ε−1 log(λn)] + k¯ + (3.21) and ¯ nk = [λn2−ε(k−k) ], k = k¯ + 1, k¯ + 2, , k∗ , (3.22) with a positive constant λ Here [t] denotes the integer part of the number t It is easy to find constants C1 , C2 in (3.18) and λ in (3.22) so that ˜ D Dung 28 nk ≤ mk , k = k¯ + 1, , k∗ , Cm ≤ n and m ≤ n Therefore, the sampling recovery method SnB ( f ) = G( f ) is of the form (1.4) and dimp (B) ≤ n Let us estimate f − SnB ( f ) q Fix a number < τ ≤ min( p, 1) From (2.3), (3.16), (3.17) we derive that for all functions f ∈ U k∗ τ q f − SnB ( f ) ≤ qk ( f ) − Gk (qk ( f )) τ q + ¯ k=k+1 qk ( f ) τ q k>k∗ k∗ δ 2−τ αk 2τ δk n−τ + k ¯ k=k+1 ∞ 2−τ αk 2τ δk (3.23) k=k∗ +1 By using (3.18), (3.20–3.22) and the inequality α > δ, we can continue the last inequality as follows n −τ δ −τ (α−δ)k¯ k∗ ¯ ∗ 2−τ (α−δ+δε)(k−k) + 2−τ (α−δ)k ¯ k=k+1 ¯ ∞ ∗ 2−τ (α−δ)(k−k ) k=k∗ +1 ∗ n−τ δ 2−τ (α−δ)k + 2−τ (α−δ)k n−τ α/d (3.24) Thus, the inequality (3.4) for the case p < q is proven Corollary 2.3 already gives the upper bound of sn (U αp,θ , M)q in Theorem 1.2 for the case where < q ≤ p ≤ ∞ We now construct an adaptive sampling method giving the upper bound for the remaining case where < p < q ≤ ∞ Let k∗ Sn n (M) ( f, x) = nk ak,s ( f )Mk,s (x) + ¯ s∈J(k) ck,s j ( f )Mk,s j (x) (3.25) ¯ j=1 k=k+1 ¯ k∗ and a sequence of non-negative integers where non-negative integers k, k∗ {nk }k=k+1 are defined by the conditions (3.18), (3.20–3.22) In a way completely ¯ similar to the proof of Theorem 3.1, we find constants C1 , C2 in (3.18) and λ in (3.22) so that the numbers of sampled values and of B-splines Mk,s in Sn n (M) ( f ) not exceed n, and therefore, prove the following theorem Theorem 3.2 Let p, q, θ, α satisfy Condition (1.6), α < 2r and p < q Then we have sn U αp,θ , M q ≤ sup f ∈U αp,θ f − Sn n (M) ( f ) q n−α/d Proof of Theorem 1.2 The upper bound of (1.7) for the case where p < q is in Theorem 3.2 and in Corollary 2.3 for the case where q ≤ p The lower bound Optimal adaptive sampling recovery 29 follows from the inequality sn (U αp,θ , M)q ≥ σn (U αp,θ , M)q and Theorem 5.1 in Appendix Adaptive sampling recovery by sets of finite cardinality m For < p ≤ ∞, denote by l m p the space of all sequences x = {xk }k=1 of numbers, equipped with the quasi-norm 1/ p m {xk } lm p = x := lm p |xk | p k=1 with the change to the max norm when p = ∞ Denote by Bm p the unit ball in m m m lm Let E = {e } be the canonical basis in l , i e., x = k k=1 p q k=1 xk ek For a ε > 0, and t ∈ R we define the function [P]ε (t) := ε[t/ε] sgn(t) Further, let the operator [P]ε in lqm be defined by [P]ε (x) := {[P]ε (xk )}m k=1 for x = {xk }m k=1 Lemma 4.1 Let < p, q ≤ ∞ Then for any n ≥ m, there is a constant n C = C( p) such that |[P]ε (Bm p )| ≤ and sup x − [P]ε (x) q x∈Bm p ≤ Cm1/q−1/ p 2−n/m , where ε = Cm1/q−1/ p 2−n/m m m For x = {xk }m k=1 ∈ lq , we let the set {k j } j=1 be ordered so that |x j1 | ≥ |x j2 | ≥ · · · |x js | ≥ · · · ≥ |x jm | Then, for a ε > and n ≤ m we define the operator n G [S]n,ε (x) := [P]ε (xk j )ek j j=1 Lemma 4.2 Let < p ≤ q ≤ ∞ Then for any positive integer n ≤ m, there is a G n m (Bm constant C = C( p) such that |[S]n,ε p )| ≤ n and G sup x − [S]n,ε (x) q x∈Bm p where ε = Cn1/q−1/ p Lemmas 4.1 and 4.2 were proven in [14] ≤ Cn1/q−1/ p , ˜ D Dung 30 Theorem 4.1 Let p, q, θ, α satisfy Condition (1.6) Then for the d-variable Besov class U αp,θ , there is the following asymptotic order en (U αp,θ )q n−α/d (4.1) If in addition, α < 2r, we can explicitly construct a subset B in n (M) having |B| ≤ 2n , and a sampling recovery method SnB of the form (1.4), such that sup f ∈U αp,θ f − SnB ( f ) n−α/d q (4.2) Proof The lower bound of (4.1) follows from the inequality en (U αp,θ )q ≥ εn (U αp,θ )q and Theorem 5.5 Let us prove (4.2) and therefore, the upper bound of (4.1) As in the proof of Theorem 3.1, it is sufficient to prove the theorem for U := U αp,∞ In a way completely similar to the proof of (3.8–3.11), we can prove the following representation of any f ∈ U : ∞ f = qk ( f ) (4.3) k=0 with the component functions qk ( f ) = ck,s ( f )Mk,s (x) (4.4) s∈J(k) from the subspace the condition qk ( f ) (k) and ck,s ( f ) given in (2.40) Moreover, qk ( f ) satisfy p 2−dk/ p {ck,s ( f )} p,k 2−αk , k = 0, 1, 2, (4.5) On the basic of this representation we will explicitly construct a subset B of cardinality at most 2n , and a sampling recovery method SnB of the form (1.4) for which there holds (4.2) ¯ Given a positive integer n, we take a positive integer k¯ = k(n) satisfying the condition ¯ ¯ C1 2dk ≤ n ≤ C2 2dk , (4.6) where C1 , C2 are absolute constants whose value will be chosen below Put δ = d(1/ p − 1/q) We first consider the case p < q We have < δ < α Fix a number ε satisfying the inequalities < ε < min{1, (α − δ)/δ} (4.7) ∗ Let the sequence {nk }kk=0 be given by ¯ nk := mk 2d(1−ε)(k−k) + ¯ mk 2d(1+ε)(k−k) ¯ for ≤ k ≤ k, ¯ for k < k ≤ k∗ , (4.8) Optimal adaptive sampling recovery 31 where mk := |J(k)| = (2k + 2r − 1)d It is easy to check that nk > for any ¯ + ε)/ε − k0 , where k0 = k0 (ε, d) is a positive constant Since (1 + k ≤ k(1 ε)/ε > α/(α − δ), we can fix a number γ so that (1 + ε)/ε > γ > α/(α − δ) Put ¯ Then for k¯ large enough, we have nk > for any k ≤ k∗ k∗ = [γ k] ¯ Then nk ≥ mk By Lemma 4.1, there is a constant C = C( p) Let ≤ k ≤ k nk k such that |[P]εk (Bm p )| ≤ and x − [P]εk (x) m lq k −nk /mk ≤ Cm−δ x k lpk , m (4.9) −nk /mk where εk = Cm−δ We define a subset Bk of Lq and a mapping Sk : U → k Bk as follows Consider the coefficients {ck,s ( f )}s∈J(k) in the representation (4.4) of qk ( f ) as an element of lqmk Define c∗k,s ( f )Mk,s , Sk ( f ) := s∈J(k) where {c∗k,s ( f )}s∈J(k) = [P]εk ({ck,s ( f )}s∈J(k) ) Let Bk := Sk (U) nk k |[P]εk (Bm p )| ≤ and, by (4.5), (4.9) for all functions f ∈ U qk ( f ) − Sk ( f ) Then |Bk | ≤ 2−αk 2−nk /mk q Hence, by (4.8) we obtain for all functions f ∈ U qk ( f ) − Sk ( f ) 2−αk 2−2 q ¯ (1−ε)(k−k) (4.10) Let k¯ < k ≤ k∗ Then nk < mk By Lemma 4.2, there is a constant C = C( p) nk mk k and such that |[S]nGk ,εk (Bm p )| ≤ nk x − [S]nGk ,εk (x) m lq k ≤ Cn−δ x k lpk , m (4.11) where εk = Cn−δ k ¯ we define a subset Bk of Lq and a mapping Similarly to the case ≤ k ≤ k, Sk : U → Bk as follows Consider the coefficients {ck,s ( f )}s∈J(k) in the representation (4.4) of qk ( f ) as an element of lqmk Define c∗k,s ( f )Mk,s , Sk ( f ) := s∈J(k) where {c∗k,s ( f )}s∈J(k) = [S]nGk ,εk ({ck,s ( f )}s∈J(k) ) Let Bk := Sk (U) Then we have nk mk k and by (4.5), (4.11) for all functions f ∈ U |Bk | ≤ |[S]nGk ,εk (Bm p )| ≤ nk qk ( f ) − Sk ( f ) q ¯ ¯ 2−αk 2−β(k−k) , (4.12) where β = α − dδ(1 + ε) > Finally, let k > k∗ From (2.18–2.19) and (4.5) it follows that for all functions f ∈U qk ( f ) q 2−(α−δ)k , k = k∗ , k∗ + 1, (4.13) ˜ D Dung 32 For approximating functions f ∈ U define the mapping S by k∗ S := Sk k=0 We have by (4.3) k∗ f − S( f ) = ∞ {qk ( f ) − Sk ( f )} + qk ( f ) k=k∗ +1 k=0 Fix a number < τ ≤ min(q, 1) By (4.10), (4.12), (4.13) and the inequality γ > α/(α − δ), we get the following estimates for all functions f ∈ U: k∗ f − S( f )) τ q ≤ qk ( f ) − Sk ( f ) ∞ τ q + qk ( f ) τ q k=k∗ +1 k=0 2−τ αk 2−τ ¯ (1−ε)(k−k) ¯ 0≤k≤k¯ ¯ 2−τ αk 2−τβ(k−k) + + ∗ ¯ kk∗ ¯ (1−ε)(k−k) 0≤k≤k¯ ¯ ¯ + 2−τ αk 2−τβ(k−k) ∗ ¯ kk∗ ¯ 2−τ αk n−τ α/d (4.14) Thus, we have proven the following inequality sup f − S( f ) q f ∈U n−α/d k∗ k=0 Notice that S is a mapping from U into B := sampled values defining S( f ) does not exceed (4.15) Bk , and the number of k∗ ¯ m := (2k + 1)d + (2μ + 2r)d nk (4.16) ¯ k=k+1 Moreover, by (4.8) we obtain k∗ ¯ 0≤k≤k¯ k=0 ¯ ¯ 2−dε(k−k) 2dk + log |Bk | log |B| ≤ ¯ 2−dε(k−k) 2dk + log ∗ ¯ k 0, we have [12] σn+m (W, , Y) ≤ σn (W, , X)σm (SX, , Y) From the last inequality and Lemma 5.1 we obtain Lemma 5.2 Let < q ≤ ∞ Then we have for any positive integer n < m m ≥ (m − n)1/q σn Bm ∞ , E , lq Lemma 5.3 Let < p, θ ≤ ∞ and < α < min(2r, 2r − + 1/ p) Then for each f ∈ (m), there holds the Bernstein inequality f Bαp,θ ≤ C2αm f p, where C does not depending on f and m This lemma was proven in [9] Theorem 5.1 Let < p, q, θ ≤ ∞ and < α < min(2r, 2r − + 1/ p, 2r) Then for the d-variable Besov class U αp,θ , we have σn U αp,θ , M q n−α/d α ⊂ βU αp,θ for some multiplier β, Proof Because of the inclusion U := U ∞,θ it suffices to prove the theorem for the case p = ∞ Let V := { f ∈ (k) : α f ∞ ≤ 1} From Lemma 5.3 it follows that λ2−αk V is a subset in U ∞,θ for some constant λ This implies the inequality σn (U, M)q 2−αk σn (V, M)q (5.2) Denote by (k)q the normed space of all functions f ∈ (k), equipped with the norm Lq Notice that Pk is a linear projection from the space Lq onto the subspace (k)q and Pk (M) = Mk , where Mk ⊂ M consists of all Mk,s ∈ M for a non-negative integer k Hence, it is easy to verify that σn (V, M)q σn (V, Mk , (k)q ) (5.3) Observe that m := |J(k)| = dim (k)q = (2 + 2r − 1) negative inter n, define m = m(n) from the condition k n We let the mapping from 2dk d m > 2n (k)q onto lqm be defined by ( f ) := {as }s∈J(k) For a nondk (5.4) ˜ D Dung 36 for f ∈ (k)q and f = s∈J(k) as Mk,s Clearly, (V) = Bm (Mk ) = E and ∞, ( (k)q ) = lqm , where E is the canonical basis in lqm We have by (2.18)– m and ( f ) l∞ f q 2−dk/q ( f ) lqm Hence, we obtain by (2.19) f ∞ Lemma 5.2 σn (V, Mk , (k)q ) m 2−dk/q σn Bm ∞ , E , lq 2−dk/q (m − n)1/q Combining the last estimates and (5.2–5.4) completes the proof of the theorem Theorem 5.2 Let < p, q, θ ≤ ∞ and d(1/ p − 1/q)+ < α < min(2r, 2r − + 1/ p) Then for the d-variable Besov class U αp,θ , there is the following asymptotic order σn U αp,θ , M q n−α/d (5.5) Proof The lower bound is in Theorem 5.1 The upper bound of (5.5) can be proven in a way completely similar to the proof of Theorem 1.2 by using the representation (2.20–2.21) Let < q < ∞ and μ be a probability distribution on Denote by Lq ( , μ) the quasi-normed linear space of real-valued functions on , equipped with the quasi-norm 1/q f Lq ( ,μ) := | f (x)|q dμ For a subset B of the quasi-normed linear space X, let Mε (B, X) be the cardinality of the maximal ε-separated subset of B (a set A is called εseparated, if f − f X > ε for any f, f ∈ A such that f = f ) For the proof of the following lemma see [17] and [14] Lemma 5.4 Let < q ≤ 1, and μ be a probability distribution on Then if B ⊂ Lq ( , μ) is a set of pseudo-dimension n such that | f (x)| ≤ λ for every f ∈ B and x ∈ , we have Mε (B, Lq ( , μ)) ≤ e(n + 1)(4eλ/ε)n Lemma 5.5 Let < q ≤ and m ≥ 32 log(e41+1/q )n Then for any subset D in lqm of pseudo-dimension ≤ n, m 1/q E Bm ∞ , D, lq ≥ (m/8) Proof We will use a technique in a proof of Lemma in [24] on a similar inequality for VC-dimension and q = Consider a sequence x = {xk }m k=1 as Optimal adaptive sampling recovery 37 a function on = {1, 2, , m} Then, it is easy to see that m−1/q · lqm = · Lq ( ,μ) for some probability distribution μ Let D be any subset in lqm of pseudo-dimension ≤ n By Lemma 5.3 we obtain n Mε D, lqm ≤ e(n + 1) 4em1/q /ε (5.6) m/16 By Lemma in [14], there is a subset ⊂ Bm such ∞ of cardinality at most that for any x, y ∈ , x = y, we have x−y lqm ≥ (m/2)1/q (5.7) Given arbitrary δ > 0, we put , D, lqm + δ γ =E There exists a mapping S : → D such that for any x ∈ x − S(x) lqm ≤ γ From (5.7), it follows that for any x, y ∈ S(x) − S(y) q lqm ≥ x−y q lqm − x − S(x) q lqm − y − S(y) q lqm ≥ 2(ε∗ )q − 2γ q , where ε∗ = (m/4)1/q Suppose that γ ≤ 2−1/q ε∗ Then for any x, y ∈ = S( ), we have x − y ≥ ε∗ This means that | | = | | > 2m/16 and therefore, Mε∗ ( , lqm ) > 2m/16 By (5.6), we obtain Mε∗ , lqm ≤ e(n + 1)(4e/ε∗ )n Consequently, 2m/16 < e(n + 1) 4em1/q /ε∗ n n = e(n + 1) e41+1/q Hence, we can see that for arbitrary n ≥ 1, m < 16n log e41+1/q + 16 log(e(n + 1)) On the other hand, obviously, for any m ≥ 32 log(e41+1/q )n there holds the opposite inequality m > 16n log e41+1/q + 16 log(e(n + 1)) This contradiction shows that γ > 2−1/q ε∗ for arbitrary δ > and, therefore, m E Bm ∞ , D, lq ≥ E , D, lqm ≥ 2−1/q ε∗ = (m/8)1/q Lemma 5.6 Let < p, q ≤ ∞ Then we have for any positive integer n < m m C p,q A p,q (m, n) ≤ ρn Bm ≤ A p,q (m, n), p , lq where A p,q (m, n) = n1/q−1/ p , (m − n)1/q−1/ p , for p < q, for p ≥ q, ˜ D Dung 38 and C p,q = 1/16(32 log(e41+1/q ))1/ p−1/q , 1/16, for p < q, for p ≥ q Proof This lemma has been proven in [24] for the case ≤ p, q ≤ ∞ For the case < p, q ≤ ∞, the upper bound immediately follows from Lemma 5.1 and Lemma in [24], and the lower bound can be proven in a way similar to the proof of the lower bound for the case ≤ p, q ≤ ∞, in [24] by replacing Lemma in [24] by Lemma 5.5 when < q < Theorem 5.3 Let < p, q, θ ≤ ∞ and α > Then for the d-variable Besov class U αp,θ , we have ρn U αp,θ n−α/d q Proof As in the proof of Theorem 5.1 it 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