In this article, we continue to study the approximation power of Smolyak sparse grid sampling recovery for multivariate function classes with small mixed smoothness Sp,θrB([0,1]d) and Sp,θrF([0,1]d) based on the multivariate Faber representation [1, 2] of f. The advantage of such a representation is the fact that the coefficient functionals only use discrete functional values of f, see (2.1) and (2.3) below, which allow for various applications. For instance, Kempka et al. [3] used the Faber system to analyze the path regularity of the Brownian motion, where a small smoothness setting is also required.

It provides a powerful tool for studying various situations of the sampling recovery problem where errors are measured in L_q. Surprisingly, the proposed method turned out to be sharp in several regimes. A systematic discretization of multivariate functions with mixed smoothness in terms of Faber coefficients is given [1, 2, 4, 5], see also [6] for further history. In this article, we study an endpoint situation for the sampling recovery problem on the cube [0, 1]^d, where r = 1/p in the source space and the fine parameter θ ≤ 1 is sufficiently small. This still allows for an embedding into the continuous functions (therefore making function evaluations possible). However, the embedding into L_q is only compact if q < ∞.

Recent observations regarding the problem of optimal sampling recovery of function classes in L₂ bring classes with mixed smoothness to the focus again since several newly developed techniques only work for Hilbert-Schmidt operators [7–9] or, more generally, in situations where certain asymptotic characteristics (approximation numbers) are square summable [10, 11]. We need new techniques for situations where this is not the case. In Temlyakov and Ullrich [12, 13], the authors consider the range of small smoothness where one is far away from square summability of the corresponding widths. However, we have the compact embedding into L_∞ for those examples. This embedding seems to be of crucial importance when (non-linear) sampling widths (_ϱm)m, defined by

with X = L₂ are related to certain asymptotic characteristics such as Kolmogorov (_dm)m or best m-term trigonometric widths (_σm)m in L_∞. It has been shown by Temlyakov [14], Bartel et al. [9] and Jahn et al. [15] that the following inequalities hold for relevant function classes F

Here, we have some constants C, D > 0 and oversampling parameter 1 <b ≤ 2 in the first relation. We speak about linear sampling widths denoted by (ϱmlin)m if the reconstruction mapping R is linear. The second relation might not need a compact embedding into L_∞ at first glance. However, the proof heavily relies on certain compactness properties of the embedding, see [15]. Clearly, a compact embedding into L_∞ allows us to use the decaying Kolmogorov widths for controlling the linear sampling widths. As discussed by Temlyakov and Ullrich [12], sparse grid techniques [16] perform asymptotically worse by a log factor for classes with mixed smoothness compactly embedded into L_∞.

In this article, we continue our research in this direction. Note that there are several relevant (multivariate) function classes F which are continuously but not compactly embedded into L_∞. Thus, at least the first relation in (1.1) is useless. In fact, only few results have been published on reconstructing functions from samples, which only satisfy a Besov regularity with smoothness r = 1/p or Sobolev type regularity with r = 1 and p = 1. Our aim is to investigate this problem systematically in the Fourier analytic context. As a first step, we prove the following relation for the Faber coefficients dj,k2(f) of f, namely

in case 1/2 ≤ p ≤ ∞. Our contribution is a proof that works for Fourier analytic defined spaces and allows for incorporating also the extreme cases S1/2,1/22B(ℝd) and S∞,10B(ℝd) [in contrast to Dũng [17], Theorem 3.2]. It represents an extreme case of the considered limit situation, in which the Faber approximation can benefit from the highest regularity of smoothness equals 2. The above relation directly implies that the truncated Faber representation still works well when we consider errors in L_q with q < ∞. We make progress toward the solution of an open problem mentioned [[18], Section 3.2]. The univariate class Bp,11/p([0,1]) and its approximation by equidistant samples in [0, 1] have been considered by Oswald [19] in the beginning of the 80s. In 2011, Dũng [[17], Theorem 3.2] obtained results for the multivariate situation in the framework of Besov spaces with a bounded mixed difference. In contrast to the spaces considered by Dũng [17], we highlight that the Besov and Triebel-Lizorkin spaces considered here are defined using Fourier analytic building blocks. Note that in the considered limiting situation and p < 1, it is not yet known whether these spaces coincide with the ones considered by Dũng [see [20], Remark 2.3.4/2].

Notation. In general, ℕ denotes the natural numbers, ℕ₀ = ℕ∪{0}, ℕ₋₁ = ℕ₀ ∪ {−1}, ℤ denotes the integers, ℝ denotes the real numbers, and ℂ denotes the complex numbers. The letter d is always reserved for the underlying dimension in ℝ^d, ℤ^d, etc. For a ∈ ℝ, we denote a₊: = max{a, 0}. For 0 <p ≤ ∞ and x ∈ ℝd, we denote |x|p=(∑i=1d|xi|p)1/p with the usual modification in the case p = ∞. We further denote x₊ : = ((_x1)+, …, (_xd)+) and |x|₊ : = |x₊|₁. By (x₁, …, x_d) > 0, we mean that each coordinate is positive. If X and Y are two (quasi-)normed spaces, the (quasi-)norm of an element x in X will be denoted by ||x|X||. The symbol X↪Y indicates that the identity operator is continuous. For two sequences a_n and b_n, we will write a_n ≲ b_n if there exists a constant c > 0 such that a_n ≤ cb_n for all n. We will write a_n ≍ b_n if a_n ≲ b_n and b_n ≲ a_n.

As a main tool, we will use decompositions of functions in terms of a Faber series expansion.

In the sequel we deal with two tensor domains. On the one hand the d-variate unit cube I^d = [0, 1]^d and on the other hand the d-variate Euclidean space ℝ^d. We use the notation

with the usual modification in case p = ∞. The space C(I^d) is often used as a replacement for L∞(Id). It denotes the collection of all continuous and bounded d-variate functions equipped with the uniform norm. The computation of the Fourier transform (and its inverse) of an L₁-integrable d-variate function is performed by the integrals (ξ ∈ ℝd)

where ξ · x := ξ₁x₁ + ⋯ + ξ_dx_d. To begin with, we recall the concept of a dyadic decomposition of the unity. The space C0∞(ℝ) consists of all infinitely many times differentiable compactly supported functions.

Definition 3.1. Let Φ(ℝ) be the collection of all systems φ={φn(x)}n=0∞⊂C0∞(ℝ) satisfying

Now we fix a system φ = {_{φn}n ∈ ℕ0} ∈ Φ(ℝ). for ℓ=(ℓ1,…,ℓd)∈ℕ0d, let the building blocks f_ℓ be given by

Because of the Paley-Wiener theorem, the functions f_ℓ are entire analytic functions and therefore continuous. The goal of this section is to derive bounds for fixed “levels” j of the Faber expansion (2.3) of such a bandlimited function f_ℓ. To be more precise, we aim at bounds for ||∑k∈ℤddj,k2(fℓ)vj,k(·)||p. Clearly, due to the compact support of v_j,k, we may replace v_j,k by the characteristic function χ_j,k of the parallelepiped [2-j1k1,2-j1(k1+1)]×⋯×[2-jdkd,2-jd(kd+1)]. Note that for any continuous function f ∈ C(ℝd)

where 0 < p ≤ ∞. To perform this, we need some tools from harmonic analysis. We state a mixed version of the Peetre maximal inequality, proved in [[20], 1.6.4].

Lemma 3.2. [Peetre maximal inequality] Let 0 < p ≤ ∞ and a > 1/p (a > 0 in case p = ∞). Furthermore, let f∈L1(ℝd) such that suppFf⊂[-b1,b1]×⋯×[-bd,bd], where b=(b1,…,bd)∈ℝ+d. Then there is a constant c > 0, only depending on a and p but not on f and b, such that

The following univariate pointwise estimate connecting differences between bandlimited functions and Peetre maximal operator is taken from [[22], Lemma 3.3.1].

Lemma 3.3. Let a, b > 0 and f ∈ L₁(ℝ) with suppFf⊂[-b,b]. Then there exists a constant C > 0 such that

The bound may be slightly improved when replacing pointwise estimates with estimates involving L_p norms. We have the following.

Lemma 3.4. Let j, ℓ ∈ ℕ₀, a > 0, 0 < p ≤ ∞, and f ∈ C(ℝ). Then we have

independent of x ∈ ℝ, j, ℓ, and f.

Proof. We start with a pointwise estimate

Taking L_p-norms on both sides gives

Finally, we trivially observe

In the next lemma, we combine both univariate bounds and derive a multivariate estimate via iteration with respect to coordinate directions. Let f ∈ L₁(ℝd) and f_j+ℓ denotes the bandlimited function from (3.1).

Lemma 3.5. Let 0 < p ≤ ∞, j∈ℕ-1d, and ℓ ∈ ℤ^d. Then we have

Proof. Step 1. To provide a technically transparent proof of this lemma, we start with the univariate case (d = 1). In the second part of this proof, we deal with the multivariate case, which requires more involved notation. For x ∈ ℝ, we define

Let x ∈ [2^−jk, 2^−j(k + 1)]. For this x, we have

Since χ_j,k(x) do not overlap, we receive

By Lemma 3.4, we find

for some a > 0, which is at our disposal.

In case ℓ ≤ 0, we may continue arguing pointwise. First of all, we have

Using Lemma 3.3 and the fact that ℓ ≤ 0, we obtain

Combining (3.3) and (3.4) gives

Choosing a > 1/p and applying the Peetre maximal inequality in Lemma 3.2 gives the result for d = 1.

Step 2. We deal with the multivariate case and start with a pointwise estimate of

where we apply the above procedure in every direction. In order not to drown in notation, we introduce the following direction-wise maximal operator

where ei=(0,…,0︸i-1,1,0,…,0︸d-i). Clearly, for x ∈ ℝd, we have

Here we use the fact that we have in every direction

including the case j = −1, where the difference is replaced by the function value at the respective point. This case is included in ∏i:ℓi>0𝔐ji since for ℓ_i < 0 there is nothing to prove in this case. We use the triangle inequality in order to estimate the difference by point evaluations. Taking sup over the step length of this absolutely valued point evaluations leads to the direction-wise maximal operator. In case ℓ_i ≤ 0, we keep the direction-wise difference. Nevertheless, in order to get rid of the characteristic function, we have to additionally apply a direction-wise sup also in this case. Since it holds that

with |f| ≤ |g|, we first estimate

pointwise from above using Lemma 3.3 iteratively. This gives

for any a > 0. The maximal operators (∏i:ℓi≤0𝔐jii) go on the same coordinates. Since ℓ_i ≤ 0 we clearly also have

It remains to apply (∏i:ℓi>0𝔐jii) and take the L_p-norm, p < ∞. Here we use Lemma 3.4 iteratively:

Finally, we use the trivial estimate [already known from (3.2)] to replace the operators 𝔐ji+ℓii by the Peetre maximal function. This gives

if we choose a > 1/p. For the sake of completeness, let us additionally consider the case p = ∞. We have

if we choose a > 0.

For the definition of the corresponding function spaces on ℝd, we refer to [1, 20, 23]. The corresponding function spaces on [0, 1]^d are defined via restrictions of functions on ℝd [see [1], Section 3.4]. In this section, we mainly focus on the definition of Besov and Triebel-Lizorkin spaces with dominating mixed (in the sequel only called mixed) smoothness on ℝd since they are crucial for our subsequent analysis. We closely follow [[20], Chapter 2] and use the building blocks f_j(·) defined in (3.1).

Definition 4.1. [Mixed Besov and Triebel-Lizorkin spaces] (i) Let 0 < p, θ ≤ ∞, and r > (1/p − 1)₊. If θ ≤ min{p, 1}, we admit r = (1/p − 1)₊. Then Sp,θrB(ℝd) is defined as the collection of all f ∈ L_{max{p, 1}}(ℝd) such that

is finite (with the usual modification if θ = ∞).

(ii) Let 0 < p < ∞, 0 < θ ≤ ∞, and r > (1/p − 1)₊. Then Sp,θrF(ℝd) is defined as the collection of all f ∈ L_{max{p, 1}}(ℝd) such that

is finite (with the usual modification if θ = ∞).

It is noted that this definition is independent of the chosen system φ in the context of equivalent (quasi-)norms. Moreover, in the case min{p, θ} ≥ 1, the defined spaces are Banach spaces, whereas they are quasi-Banach spaces in the case min{p, θ} < 1. For details, we refer to [[20], Section 2.2.4]. In the next lemma, there appears the condition r > (1/p − 1)₊, which is caused by the parameter range in our Definition 4.1. All the subsequent embeddings, of course, also hold true for the general situation, where r ∈ ℝ. We have the following elementary embeddings [see [20], Section 2.2.3].

Lemma 4.2. Let 0 < p < ∞ (including p = ∞ in the B-case), 0 < θ ≤ ∞, and r > (1/p − 1)₊. If θ ≤ min{p, 1}, we admit r = (1/p − 1)₊ in the B-case. Furthermore, let A ∈ {B, F}.

and especially the limiting case

Proof. The embeddings (i), (iia), (iiia), (iiib), and (iv) are standard and can be found in [[20], Chapter 2] especially we refer to [[20], Remark 2, p. 132], which includes the limiting case p = ∞ in (iiib). As for the Jawerth-Franke type embeddings, we refer to [[24], Theorem 1.2 and 1.4] and the summary [[6], Lemma 3.4.2 and 3.4.3]. See also [[6], Rem. 3.4.4] for further references, especially for the mixed smoothness case. Finally, the embedding in (iiic) is a consequence of (iic) and (iiib).