On optimal approximation in periodic Besov spaces
Fernando Cobos1,∗
Departamento de An´alisisMatem´atico y Matem´atica Aplicada, Facultad de Matem´aticas, Plaza de las Ciencias 3, 28040 Madrid. Spain.
Thomas K¨uhn1
Mathematisches Institut, Universit¨atLeipzig, Augustusplatz 10, 04109 Leipzig. Germany.
Winfried Sickel Mathematisches Institut, Friedrich-Schiller-Universit¨atJena, Ernst-Abbe-Platz 2, 07737 Jena. Germany.
Abstract We work with spaces of periodic functions on the d-dimensional torus. We show that estimates for L∞-approximation of Sobolev functions remain valid when 0 we replace L∞ by the isotropic periodic Besov space B∞,1 or the periodic Besov 0 space with dominating mixed smoothness S∞,1B. For t > 1/2, we also prove estimates for L2-approximation of functions in the Besov space of dominating t mixed smoothness S1,∞B, describing exactly the dependence of the involved constants on the dimension d and the smoothness t. Keywords: Approximation numbers, isotropic and mixed Besov spaces, d-dependence of the constants, Wiener algebra. 2010 MSC: 46E35, 41A25
1. Introduction
A number of problems in finance, quantum chemistry and related areas are modeled on function spaces of Sobolev type on high-dimensional domains (see, for example, [40]). For this reason there is an increasing interest nowadays in the study of linear approximation (approximation numbers) in the context of Sobolev and Besov spaces. The main point being now to determine the asymptotically optimal equivalence constants as a function of the dimension, the order of smoothness of the space and its norm. T. Ullrich and two of the present authors have investigated in [19, 20] linear approximation of functions in the isotropic periodic Sobolev spaces Hs(Td) and s d in the smaller spaces Hmix(T ) of dominating mixed smoothness, determining the exact decay rate of constants as the dimension d goes to infinity (see also the
∗Corresponding author. Email addresses: [email protected] (Fernando Cobos), [email protected] (Thomas K¨uhn), [email protected] (Winfried Sickel) 1Supported in part by MTM2017-84058-P (AEI/FEDER, UE).
Preprint submitted to Elsevier January 30, 2019 d paper by D˜ungand Ullrich [11]). The error was measured in L2(T ). Later, the present authors [8] have studied the same problems but measuring the error in the (sometimes preferred) sup-norm. A general method was established which allows to transfer results on L2-approximation into results on L∞-approximation (see [8]). In the present paper we continue that investigation working now in the con- t d text of isotropic periodic Besov spaces Bp,q(T ) and Besov spaces of dominating t d d mixed smoothness Sp,qB(T ). We show that L∞(T ) can be replaced by any 0 d 0 d of the spaces B∞,1(T ) and S∞,1B(T ) without changing the associated ap- proximation numbers. This result is unexpected and has some practical use, since the Littlewood-Paley characterization of the target spaces simplifies the computation of approximation or other s-numbers, see, e.g., the monographs by D˜ung,Temlyakov and Ullrich [12] or by Temlyakov [35], [36]. Moreover, for t > 1/2, we establish estimates for the approximation numbers of the embedding t d d Id : S1,∞B(T ) −→ L2(T ) which show the dependence of the involved constant on the dimension d and the smoothness s. We also consider the approximation of tensor products of functions with finite total variation. We start the paper by fixing the notation in Section 2. Then, in Section 3, we deal with isotropic periodic Besov spaces and, in Section 4, with Besov spaces of dominating mixed smoothness. Finally, in Section 5, we consider spaces of tensor products of functions.
2. Preliminaries
d Let d ∈ N. Given x = (x1, . . . , xd) ∈ R and 0 < p ≤ ∞ we put |x|p = 1/p Pd p j=1 |xj| if p < ∞, and |x|∞ = max1≤j≤d |xj| if p = ∞. Let T be the torus, i.e. T = [0, 2π] where the endpoints of the interval are identified. We put Td for the d-dimensional torus. We consider on Td the normalized Lebesgue −d ikx d d measure (2π) dx. So {e : k ∈ Z } is an orthonormal basis in L2(T ), where Pd kx = j=1 kjxj . d Let f ∈ L1(T ). The Fourier coefficients of f are given by Z −d −ikx d fb(k) = (2π) f(x)e dx , k ∈ Z . d T d We write Fd(w) for the Hilbert space of integrable functions on T formed by all those f having a finite norm
1/2 X 2 2 kf|Fd(w)k = w(k) |fb(k)| < ∞ . (2.1) d k∈Z
Here w(k) > 0 , k ∈ Zd , are certain weights. If 0 < s < ∞, 0 < r ≤ ∞ and Pd rs/r w(k) = 1+ j=1 |kj| , the space Fd(w) coincides with the isotropic Sobolev space Hs,r(Td). The superscript r indicates the norm we are considering in the Qd rs/r Sobolev space. When w(k) = j=1 1 + |kj| , we obtain the Sobolev space s,r d of dominating mixed smoothness Hmix(T ) (see [8] and [20]).
2 The Wiener algebra A(Td) consists of all integrable functions on Td with absolutely convergent Fourier series. The norm on A(Td) is
d X kf|A(T )k = |fb(k)| . d k∈Z The present authors characterized in [8, Theorem 3.1] boundedness and com- d d d pactness of embeddings of Fd(w) in A(T ), C(T ) or L∞(T ): Theorem 2.1. The following conditions are equivalent.
d (i) Fd(w) ,→ A(T ) compactly d (ii) Fd(w) ,→ A(T ) boundedly d (iii) Fd(w) ,→ C(T ) compactly d (iv) Fd(w) ,→ C(T ) boundedly d (v) Fd(w) ,→ L∞(T ) compactly d (vi) Fd(w) ,→ L∞(T ) boundedly (vii) P w(k)−2 < ∞ d k∈Z
Owing to this characterization, subsequently we assume that Fd(w) is defined by weights satisfying P w(k)−2 < ∞. d k∈Z Let X,Y be Banach spaces and T : X → Y be a bounded linear operator from X into Y . For n ∈ N, the n-th approximation number an(T ) of T is defined by an(T ) = an(T : X → Y ) = inf{kT − Rk : rank R < n}. Here rank R is the dimension of the range of R. See [27] and [28] for properties of these numbers. The following result was proved in [8, Theorem 3.4]. It provides an exact formula for the approximation numbers of the embeddings of Theorem 2.1.
P −2 Theorem 2.2. Let Fd(w) be given by weights satisfying w(k) < ∞, and d k∈Z let (σ ) denote the non-increasing rearrangement of (1/w(k)) d . Moreover, j j∈N k∈Z let d d d Gd = A(T ) or C(T ) or L∞(T ) . Then one has for all n ∈ N,
∞ 1/2 X 2 an(Id : Fd(w) → Gd) = σj . (2.2) j=n
3 3. Embeddings of isotropic Besov spaces There is a rich literature on Besov spaces. We refer, for example, to the monographs by Peetre [26] and Triebel [38]. Special emphasis to the periodic situation is given in the monograph by Schmeisser and Triebel [31]. As there, we shall also use the Fourier-analytic approach to introduce these spaces. However, let us mention that they can be described also in terms of moduli of smoothness (differences) or in terms of best approximation by trigonometric polynomials in the sense of equivalent norms. Clearly, for switching to these other characteriza- tions we have to pay a price, expressed by a constant in the related inequalities, which depends in general on d. This will not be done here, but it would be of certain interest. Later on we will be forced to deal with spaces with negative smoothness s, that is to say, spaces of tempered distributions. This requires some preparations. Let D(Td) denote the collection of all periodic infinitely differentiable complex- valued functions. In particular, f(x) = f(y) if x − y = 2πk, k ∈ Zd. The locally convex topology in D(Td) is generated by the semi-norms α k f kα = sup |D f(x)| , d x∈T d α where α ∈ N0 with N0 = N ∪ {0}, and D f is the distributional derivative of f of order α. By D0(Td) we denote the topological dual of D(Td), i.e., the set of all linear functionals g on D(Td) such that X |g(f)| ≤ cN k f kα |α|≤N
d 0 d for all f ∈ D(T ) and for some N ∈ N0 and cN > 0. We equip D (T ) with the weak topology, i.e.,
d g = lim gj ⇐⇒ g(f) = lim gj(f) for all f ∈ D( ) . j→∞ j→∞ T
The Fourier coefficients of g ∈ D0(Td) are defined by −d −ikx d gb(k) = (2π) g(e ) , k ∈ Z . 0 d For every g ∈ D (T ) there are a constant cg > 0 and a natural number Kg with
Kg d |gb(k)| ≤ cg (1 + |k|) , k ∈ Z . Furthermore, every g ∈ D0(Td) can be represented by its Fourier series,
X ikx 0 d g = gb(k) e (convergence in D (T )). d k∈Z For all these facts we refer to [31, 3.2.1 and 3.2.2], but see also [14, Chapter 12]. Starting point of the Fourier-analytic approach to Besov spaces is a smooth d dyadic decomposition of unity. By Br we denote the Euclidean ball in R of ∞ d radius r, centered at the origin. Let ψ ∈ C0 (R ) be a real-valued non-negative function such that
d supp ψ ⊂ B3/2 , ψ(ξ/2)−ψ(ξ) ≥ 0 for all ξ ∈ R and ψ(ξ) = 1 for ξ ∈ B1 . (3.1)
4 We define ϕ(ξ) = ψ(ξ/2) − ψ(ξ),
−j+1 ϕ0(ξ) = ψ(ξ) and ϕj(ξ) := ϕ(2 ξ) , j ∈ N . (3.2) Then ∞ X d ϕj(ξ) = 1 for all ξ ∈ R (3.3) j=0 and supp ϕj ⊂ B3 ·2j−1 \ B2j−1 , j ∈ N . d In addition we mention that in every point x ∈ R at most two of these functions 0 d ϕj, j ∈ N0, do not vanish. For g ∈ D (T ) we define
X ikx gj(x) := ϕj(k) gb(k) e . (3.4) d k∈Z
Since ϕj has compact support, the gj are trigonometric polynomials.
Definition 3.1. Let 1 ≤ p, q ≤ ∞ and s ∈ R. Then the periodic Besov space s d 0 d Bp,q(T ) is the collection of all g ∈ D (T ) such that
∞ q1/q s d ψ X jsq X ikx d k g |Bp,q(T )k := 2 ϕj(k) gb(k) e Lp(T ) < ∞ . (3.5) d j=0 k∈Z Remark 3.2. (i) Of course, the norm in (3.5) depends on the generating func- tion ψ of the chosen smooth decomposition of unity (ϕj)j. All these norms are equivalent. If necessary, we indicate the dependence on ψ by a superscript as s d in (3.5). Otherwise we simply write k g |Bp,q(T )k. s d (ii) The spaces Bp,q(T ) are Banach spaces such that
d s d 0 d D(T ) ,→ Bp,q(T ) ,→ D (T ) , see [31, Theorem 3.5.1]. (iii) Let us also mention that we have, in the sense of equivalent norms,
s,r d s d H (T ) = B2,2(T ) , where the equivalence constants depend on ψ, s, d and r.
By convention if the equivalence class of a measurable function f contains a continuous representative, then we call f itself continuous and work with the continuous representative. Lemma 3.3. We have the following chain of continuous embeddings
d/2 d d 0 d d B2,1 (T ) ,→ A(T ) ,→ B∞,1(T ) ,→ C(T ) , (3.6) where the last two embedding operators have norm one.
5 d/2 d d Proof. The embedding B2,1 (T ) ,→ A(T ) is well-known. We refer for example to the book of Triebel [37, page 141 in Section 3.4.3]. Concerning the second embedding, by (3.3) we have for g ∈ A(Td) the inequality
∞ ∞ X X ikx d X X d ϕj(k) gb(k) e L∞(T ) ≤ ϕj(k) |gb(k)| ≤ k g |A(T )k . d d j=0 k∈Z j=0 k∈Z
Here we used the non-standard condition ϕj ≥ 0, see (3.1). 0 d d Finally, we consider the embedding B∞,1(T ) ,→ C(T ). For any g ∈ 0 d B∞,1(T ), the continuous representative in the equivalence class of g is just the Fourier series of g. It follows
∞ X X ikx |g(x)| = ϕj(k) gb(k) e d k∈Z j=0 ∞ X X ikx d 0 d ≤ ϕj(k) gb(k) e L∞(T ) ≤ k g |B∞,1(T )k . d j=0 k∈Z In both cases this proves that the norm of the corresponding embedding operator is ≤ 1. By considering the constant function g(x) = 1, it is immediate that these norms are ≥ 1 as well.
Remark 3.4. (i) The embeddings stated in (3.6) cannot be improved in the d/2 d d/2 d framework of periodic Besov spaces. Indeed, if we replace B2,1 (T ) by B2,q (T ) d d/2 d for some q > 1, then an embedding into A(T ) cannot exist, since B2,q (T ) 0 d contains unbounded functions, see [37, 3.4.3] and [33]. Also B∞,1(T ) cannot 0 d d 0 d be replaced by B∞,q(T ) with q < 1 in the embedding A(T ) ,→ B∞,1(T ). There are explicit counterexamples, we omit the details. (ii) We did not indicate the chosen norm in (3.6) in the Besov spaces. The 0 d ψ statement is true with respect to all norms of the type k · |B∞,1(T )k , see (3.5).
As an immediate consequence of Lemma 3.3 and Theorem 2.2 we obtain
P −2 Theorem 3.5. Let Fd(w) be given by a weight w satisfying w(k) < ∞ d k∈Z and let (σ ) be the non-increasing rearrangement of (1/w(k)) d . Then one j j∈N k∈Z has for all n ∈ N
0 d d an(Id : Fd(w) → B∞,1(T )) = an(Id : Fd(w) → L∞(T )) d = an(Id : Fd(w) → C(T )) ∞ d X 21/2 = an(Id : Fd(w) → A(T )) = σj . j=n
0 d ψ 0 d This result is true for any norm of the type k · |B∞,1(T )k on B∞,1(T ). See Definition 3.1.
6 4. Embeddings of Besov spaces of dominating mixed smoothness As mentioned in the Introduction, Sobolev and Besov spaces of dominating mixed smoothness are helpful when dealing with approximation problems in high dimensions. These spaces are much smaller than their isotropic counter- parts. The behaviour of the approximation numbers of embeddings of isotropic d spaces into Lp(T ) is closer to the one-dimensional case than to the d-dimensional situation. ∞ Let ψ ∈ C0 (R) be a real-valued non-negative function satisfying properties (3.1) in the case d = 1. The associated smooth dyadic decomposition of unity ∞ on R is denoted by (ϕj)j=0, see (3.2). We shall work with tensor products of d d this decomposition: For ` = (`1, . . . , `d) ∈ N0 and x = (x1, . . . , xd) ∈ R , we put d Y ϕ`(x) = ϕ`j (xj) . j=1 Observe that X d ϕ`(x) = 1 for all x ∈ R , d `∈N0 d d and that now for every x ∈ R at most 2 functions ϕ` do not vanish. Definition 4.1. Let 1 ≤ p, q ≤ ∞ and t ∈ R. Then the periodic Besov space t d 0 d Sp,qB(T ) of dominating mixed smoothness is the collection of all g ∈ D (T ) such that the norm X X q1/q t d |`|1tq ikx d k g |Sp,qB(T )k := 2 ϕ`(k) gb(k)e Lp(T ) (4.1) d d `∈N0 k∈Z is finite. Remark 4.2. (i) Of course, the norm in (4.1) depends on the generating func- tion ψ of the chosen smooth decomposition of unity (ϕj)j. Again all these norms are equivalent. If necessary, we indicate the dependence on ψ as in (3.5) t d ψ by k g |Sp,qB(T )k . t d (ii) The spaces Sp,qB(T ) are Banach spaces such that
d t d 0 d D(T ) ,→ Sp,qB(T ) ,→ D (T ) .
(iii) Let us also mention that, in the sense of equivalent norms,
s,r d s d Hmix(T ) = S2,2B(T ) , where the equivalence constants depend on ψ, s, r and d. (iv) One of the most attractive features of these spaces is the following. If
d Y d g(x) = gj(xj) , x = (x1, . . . , xd) ∈ T , j=1 then d t d ψ Y t ψ k g |Sp,qB(T )k = k gj |Bp,q(T)k . j=1
7 (v) Besov spaces of dominating mixed smoothness are investigated since the seventies of the last century, we refer to the monographs of Amanov [3] and of Schmeisser, Triebel [31] as well as to the article of Lizorkin and Nikol’skij [22].
Lemma 4.3. We have the chain of continuous embeddings
d 0 d d A(T ) ,→ S∞,1B(T ) ,→ C(T ) , (4.2) where the embedding operators are of norm one.
Proof. For g ∈ A(Td), using (3.3), we have
X X ikx d ϕ`(k) gb(k) e L∞(T ) d d `∈N0 k∈Z X h X i d ≤ ϕ`(k) |gb(k)| ≤ k g |A(T )k . d d k∈Z `∈N0
0 d d Next, we turn to the embedding S∞,1B(T ) ,→ C(T ). It follows h i X X ikx |g(x)| = ϕ`(k) gb(k) e d d k∈Z `∈N0
X X ikx d 0 d ≤ ϕ`(k) gb(k) e L∞(T ) = k g |S∞,1B(T )k . d d `∈N0 k∈Z In both cases this proves that the norm of the associated embedding operator is ≤ 1. Considering g(x) = 1, it is immediate that these norms are ≥ 1 as well. Similarly as for isotropic Besov spaces (see Theorem 3.5), we can derive the following consequence from Theorem 2.2, where we can use any norm of the 0 d ψ type k g |Sp,qB(T )k . P −2 Theorem 4.4. Let Fd(w) be given by a weight w satisfying w(k) < ∞, d k∈Z and let (σ ) be the non-increasing rearrangement of (1/w(k)) d . Then one j j∈N k∈Z has for all n ∈ N
0 d d an(Id : Fd(w) → S∞,1B(T )) = an(Id : Fd(w) → L∞(T )) d = an(Id : Fd(w) → C(T )) ∞ d X 21/2 = an(Id : Fd(w) → A(T )) = σj . j=n
Next we deal with another situation where now one has some advantage from the flexibility in the target space. Duality will be an important tool for our arguments. Therefore we define
d n 0 d d o B(T ) = g ∈ D (T ): k g |B(T )k = sup |g(k)| < ∞ . d b k∈Z Obviously we have, with equality of norms, the duality