Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between distance design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.


Introduction
This paper discusses the problem of numerical integration (or quadrature), which has been a fundamental task in numerical analysis, statistics, computer science including machine learning and other areas. Let P be a (known) Borel probability measure on the Euclidean space R d with support contained in an open set Ω ⊂ R d , and f be an integrand on Ω. Suppose that the integral f (x)dP (x) has no closed form solution. We consider quadrature rules that provide an approximation of the integral, in the form of a weighted sum of function values

Kernel-based quadrature rules
How can we obtain a quadrature rule whose convergence rate is faster than O(n −1/2 )? In practice, one often has prior knowledge or belief on the integrand f , such as smoothness, periodicity, sparsity, and so on. Exploiting such knowledge or assumption in constructing a quadrature rule {(w i , X i )} n i=1 may achieve faster rates of convergence, and such methods have been extensively studied in the literature for decades; see e.g. [16] and [9] for review. This paper deals with quadrature rules using reproducing kernel Hilbert spaces (RKHS) explicitly or implicitly to achieve fast convergence rates; we will refer to such methods as kernel-based quadrature rules or simply kernel quadrature. As discussed in Section 2.4, notable examples include Quasi Monte Carlo methods [25,39,16,17], Bayesian quadrature [44,9], and Kernel herding [10,5]. These methods have been studied extensively in recent years [49,8,41,42,4] and have recently found applications in, for instance, machine learning and statistics [3,30,20,9,29].
In kernel quadrature, we make use of available knowledge on properties of the integrand f by assuming that f belongs to a certain RKHS H k that possesses those properties (where k is the reproducing kernel), and then constructing weighted points {(w i , X i )} n i=1 such that the worst case error in the RKHS e n (P ; H k ) := sup is made small, where · H k is the norm of H k . The use of RKHS is beneficial when compared to other function spaces, as it leads to a closed form expression of the worst case error (1) in terms of the kernel, and thus one may explicitly use this expression for designing {(w i , X i )} n i=1 (see Section 2.3).
Note that, in a well-specified case, that is, the integrand f satisfies f ∈ H k , the quadrature error is bounded as |P n f − P f | ≤ f H k e n (P ; H k ).
This guarantees that, if a quadrature rule satisfies e n (P ; H k ) = O(n −b ) as n → ∞ for some b > 0, then the quadrature error also satisfies |P n f − P f | = O(n −b ). Take a Sobolev space H r (Ω) of order r > d/2 on Ω as the RKHS H k , for example. It is known that optimal quadrature rules achieve e n (P ; H k ) = O(n −r/d ) [37], and thus |P n f − P f | = O(n −r/d ) holds for any f ∈ H k . As we have r/d > 1/2, this rate is faster than Monte Carlo integration; this is the desideratum that has been discussed.

Misspecified settings
This paper focuses on situations where the assumption f ∈ H k is violated, that is, misspecified settings. As explained above, convergence guarantees for kernel quadrature rules often assume that f ∈ H k . However, in practice one may lack the full knowledge on the properties on the integrand, and therefore misspecification of the RKHS (via the choice of its reproducing kernel k) may occur, that is, f / ∈ H k .
Such misspecification is likely to happen when the integrand is a black box function. An illustrative example can be found in applications to computer graphics such as the problem of illumination integration (see e.g. [9]), where the task is to compute the total amount of light arriving at a camera in a virtual environment. This problem is solved by quadrature, with integrand f (x) being the intensity of light arriving at the camera from a direction x (angle). However, the value of f (x) is only given by simulation of the environment for each x, so the integrand f is a black box function. Similar situations can be found in application to statistics and machine learning. A representable example is the computation of marginal likelihood for a probabilistic model, which is an important but challenging task required for model section (see e.g. [43]). In modern scientific applications where complex phenomena are dealt with (e.g. climate science), we often encounter situations where the evaluation of a likelihood function, which forms the integrand in marginal likelihood computation, involves an expensive simulation model, making the integrand complex and even black box.
If the integrand is a black box function, there is a trade-off between the risk of misspecification and gain in the rate of convergence for kernel-based quadrature rules; for a faster convergence rate, one may want to use a quadrature rule for a narrower H k such as of higher order differentiability, while such a choice may cause misspecification of the function class. Therefore it is of great importance to elucidate their convergence properties in misspecified situations, in order to make use of such quadrature rules in a safe manner.

Contributions
This paper provides convergence rates of kernel-based quadrature rules in misspecified settings, focusing on deterministic rules (i.e., without randomization). The focus of misspecification is placed on the order of Sobolev spaces: the unknown order s of the integrand f is overestimated as r, that is, s ≤ r.
Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary (see Section 3 for definition). For r > d/2, consider a positive definite kernel k r on Ω that satisfies the following assumption; Assumption 1. The kernel k r on Ω satisfies k r (x, y) := Φ(x − y), where Φ : R d → R is a positive definite function such that C 1 (1 + ξ 2 ) −r ≤Φ(ξ) ≤ C 2 (1 + ξ 2 ) −r for some constants C 1 , C 2 > 0, whereΦ is the Fourier transform of Φ. The RKHS H kr (Ω) is the restriction of H kr (R d ) to Ω (see Section 2).
The resulting RKHS H kr (Ω) is norm-equivalent to the standard Sobolev space H r (Ω). The Matérn and Wendland kernels satisfy Assumption 1 (see Section 2).
We do not specify how the weighted points are generated, but assume (2) aiming for wide applicability. Suppose that an integrand f : Ω → R has partial derivatives up to order s and they are bounded and uniformly continuous. If s ≤ r, the integrand may not belong to the assumed RKHS H kr , in which case a misspecification occurs.
Under this misspecified setting, two types of assumptions on the quadrature rule will be considered: one on the quadrature weights w 1 , . . . , w n (Section 4.1), and the other on the design points X 1 , . . . , X n (Section 4.2). In both cases, a rate of convergence of the form will be derived under some additional conditions. The results guarantee the convergence in the misspecified setting, and the rate is determined by the ratio s/r between the true smoothness s and the assumed smoothness r. As discussed in Section 2, the optimal rate of deterministic quadrature rules for the Sobolev space H r (Ω) is O(n −r/d ) [37]. If a quadrature rule satisfies this optimal rate (i.e., b = r/d), then the rate (3) becomes O(n −s/d ) for an integrand f ∈ H s (Ω) (s < r), which matches the optimal rate for H s (Ω).
The specific results are summarized as follows: • In Section 4.1, it is assumed that n i=1 |w i | = O(n c ) as n → ∞ for some constant c ≥ 0. Note that c = 0 is taken if the weights satisfy max i=1,...,n |w i | = O(n −1 ), an example of which is the equal weights w 1 = · · · = w n = 1/n. Under this assumption and other suitable conditions, Corollary 7 shows The rate O(n −bs/r ) in (3) holds if c = 0. Therefore this result provides convergence guarantees in particular for equal-weight quadrature rules, such as quasi Monte Carlo methods and kernel herding, in the misspecified setting.
• Section 4.2 uses an assumption on design points X n := {X 1 , . . . , X n } in terms of separation radius q Xn , which is defined by Corollary 9 shows that, if q X n = Θ(n −a ) as n → ∞ for some a > 0, under other regularity conditions, The best possible rate is O(n −bs/r ) when a = b/r. This result provides a convergence guarantee for quadrature rules that obtain the weights w 1 , . . . , w n to give O(n −b ) for the worst case error with X 1 , . . . , X n fixed beforehand. We demonstrate this result by applying it to Bayesian quadrature, as explained below. Our result may also provide the following guideline for practitioners: in order to make a kernel quadrature rule robust to misspecification, one should specify the design points so that the spacing is not too small.
• Section 5 discusses a convergence rate for Bayesian quadrature under the misspecified setting, demonstrating the results of Section 4.2. Given design points X n = {X 1 , . . . , X n }, Bayesian quadrature defines weights w 1 , . . . , w n as the minimizer of the worst case error (1), which can be obtained by solving a linear equation (see Section 2.4 for more detail). For points X n = {X 1 , . . . , X n } in Ω, the fill distance h X n ,Ω is defined by Assume that there exists a constant c q > 0 independent of X n such that and that h X n ,Ω = O(n −1/d ) as n → ∞. Then Corollary 11 shows that with Bayesian quadrature weights based on the kernel k r we have Note that the rate O(n −s/d ) matches the minimax optimal rate for deterministic quadrature rules in the Sobolev space of order s [37], which implies that Bayesian quadrature can be adaptive to the unknown smoothness s of the integrand f . The adaptivity means that it can achieve the rate O(n −s/d ) without the knowledge of s; it only requires the knowledge of the upper bound of the true smoothness s ≤ r.
• Section 3 establishes a rate of convergence for Bayesian quadrature in the well-specified case, which serves as a basis for the results in the misspecified case (Section 5). Corollary 5 asserts that if the the design points satisfy h X n ,Ω = O(n −1/d ) as n → ∞, then e n (P ; H kr (Ω)) = O(n −r/d ) (n → ∞).
This rate O(n −r/d ) is minimax optimal for deterministic quadrature rules in Sobolev spaces.
To the best of our knowledge, this optimality of Bayesian quadrature has not been established before, while recently there has been extensive theoretical analysis on Bayesian quadrature [8,9,40,4].
This paper is organized as follows. Section 2 provides various definitions, notation and preliminaries including reviews on kernel-based quadrature rules. Section 3 then establishes a rate of convergence for the worst case error of Bayesian quadrature in a Sobolev space. Section 4 presents the main contributions on the convergence analysis in misspecified settings, and Section 5 demonstrates these results by applying them to Bayesian quadrature. Finally Section 6 concludes the paper with possible future directions.
Preliminary results. This paper expands on preliminary results reported in a conference paper by the authors [28]. Specifically, this paper is a complete version of the results presented in Section 5 of [28]. The current paper contains significantly new topics mainly in the following points: (i) We establish the rate of convergence for Bayesian quadrature with deterministic design points, and show that it can achieve minimax optimal rates in Sobolev spaces (Section 3); (ii) We apply our general convergence guarantees in misspecified settings to the specific case of Bayesian quadrature, and reveal the conditions required for Bayesian quadrature to be robust to misspecification (Section 5); To make the contribution (ii) possible, we derive finite sample bounds on quadrature error in misspecified settings (Section 4). These results are not included in the conference paper.
We also mention that this paper does not contain the results presented in Section 4 of the conference paper [28], which deal with randomized design points. For randomized design points, theoretical analysis can be done based on an approximation theory developed in the statical learning theory literature [11]. On the other hand, the analysis in the deterministic case makes use of the approximation theory developed by [35], which is based on Calderón's decomposition formula in harmonic analysis [18]. This paper focuses on the deterministic case, and we will report a complete version of the randomized case in a forthcoming paper.
Related work. The setting of this paper is complementary to that of [41], in which the integrand is smoother than assumed. That paper proposes to apply the control functional method by [42] to Quasi Monte Carlo integration, in order to make it adaptable to the (unknown) greater smoothness of the integrand.
Another related line of research is the proposals of quadrature rules that are adaptive to less smooth integrands [13,14,15,19,22]. For instance, [19] proposed a kernel-based quadrature rule on a finite dimensional sphere. Their method is essentially a Bayesian quadrature using a specific kernel designed for spheres. They derive convergence rates for this method both in well-specified and misspecified settings, and obtain results similar to ours. The current work differs from [19] in mainly two aspects: (i) quadrature problems considered in standard Euclidean spaces, as opposed to spheres; (ii) a generic framework is presented, as opposed to the analysis of a specific quadrature rule.
Quasi Monte Carlo rules based on a certain digit interlacing algorithm [13,14,15,22] are also shown to be adaptive to the (unknown) lower smoothness of an integrand. These papers assume that an integrand is in an anisotropic function class in which every function possesses (squareintegrable) partial mixed derivatives of order α ∈ N in each variable. Examples of such spaces include Korobov spaces, Walsh spaces, and Sobolev spaces of dominating mixed smoothness (see e.g. [39,16]). In their notation, an integer d, which is a parameter called an interlacing factor, can be regarded as an assumed smoothness. Then, if an integrand belongs to an anisotropic function class with smoothness α ∈ N such that α ≤ d, the rate of the form O(n −α+ε ) (or O(n −α−1/2+ε ) in a randomized setting) is guaranteed for the quadrature error for arbitrary ε > 0. The present work differs from these works in that (i) isotropic Sobolev spaces are discussed, where the order of differentiability is identical in all directions of variables, and that (ii) theoretical guarantees are provided for generic quadrature rules, as opposed to analysis of specific quadrature methods.

Basic definitions and notation
We will use the following notation throughout the paper. The set of positive integers is denoted by N, and N 0 := N ∪ {0}. For α := (α 1 , . . . , α d ) T ∈ N d 0 , we write |α| := d i=1 α i . The ddimensional Euclidean space is denoted by R d , and the closed ball of radius R > 0 centered at z ∈ R d by B(z, R). For a ∈ R, ⌊a⌋ is the greatest integer that is less than a. For a set Ω ⊂ R d , diam(Ω) := sup x,y∈Ω x − y is the diameter of Ω.
Let p > 0 and µ be a Borel measure on a Borel set Ω in R d . The Banach space L p (µ) of p-integrable functions is defined in the standard way with norm f Lp(µ) = ( |f (x)| p dµ(x)) 1/p , and L ∞ (Ω) is the class of essentially bounded measurable functions on Ω with norm f L∞(Ω) := ess sup x∈Ω |f (x)|. If µ is the Lebesgue measure on Ω ⊂ R d , we write L p (Ω) := L p (µ) and further For s ∈ N and an open set Ω in R d , C s (Ω) denotes the vector space of all functions on Ω that are continuously differentiable up to order s, and C s B (Ω) ⊂ C s (Ω) the Banach space of all functions whose partial derivatives up to order s are bounded and uniformly continuous. The norm The Banach space of the continuous functions that vanish at infinity is denoted by . For function f and a measure µ on R d , the support of f and µ are denoted by supp(f ) and supp(µ), respectively. The restriction of f to a subset Ω ∈ R d is denoted by f | Ω .
Let F and F * be normed vector spaces with norms · F and · F * , respectively. Then F and F * are said to be norm-equivalent, if F = F * as a set, and there exists constants For a Hibert space H with inner product ·, · H , the norm of f ∈ H is denoted by f H .

Sobolev spaces and reproducing kernel Hilbert spaces
Here we briefly review key facts regarding Sobolev spaces necessary for stating and proving our contributions; for details we refer to [1,53,6]. We first introduce reproducing kernel Hilbert spaces. For details, see, e.g., [52,Section 4] and [55,Section 10].
Let Ω be a set. A Hilbert space H of real-valued functions on Ω is a reproducing kernel Hilbert space (RKHS) if the functional f → f (x) is continuous for any x ∈ Ω. Let ·, ·, H be the inner product of H. Then, there is a unique function k x ∈ H such that f (x) = f, k x H . The kernel defined by k(x, y) := k x (y) is positive definite, and called reproducing kernel of H. It is known (Moore-Aronszajn theorem [2]) that for every positive definite kernel k : Ω × Ω → R there exists a unique RKHS H with k as the reproducing kernel. Therefore, the notation H k is used to the RKHS associated with k.
In the following, we will introduce two definitions of Sobolev spaces, i.e., (6) and (7), as both will be used throughout our analysis. For a measurable set Ω ⊂ R d and r ∈ N, a Sobolev space W r 2 (Ω) of order r on Ω is defined by where D α f denotes the α-th weak derivative of f . This is a Hilbert space with inner-product For a positive real r > 0, another definition of Sobolev space of order r on R d is given by where the functionΦ : whereĝ(ξ) denotes the complex conjugate ofĝ(ξ).
For a measurable set Ω in R d , the (fractional order) Sobolev space H r (Ω) is defined by the restriction of H r (R d ); namely (see, e.g., [53, Eq. (1.8) and Definition 4.10]) In fact, the condition r > d/2 guarantees that the functionΦ(ξ) = (1 + ξ 2 ) −r is integrable, so thatΦ(ξ) has a (inverse) Fourier transform where Γ denotes the Gamma function and K r−d/2 is the modified Bessel function function of the third kind of order r − d/2. The function Φ is positive definite, and the kernel Φ(x − y) gives H r (R d ) as an RKHS. This kernel Φ(x − y) is essentially a Matérn kernel [31,32] with specific parameters. A Wendland kernel [54] also defines an RKHS that is norm-equivalent to H r (R d ).

Kernel-based quadrature rules
We briefly review basic facts regarding kernel-based quadrature rules necessary to describe our results. For details we refer to [9,16].
Let Ω ⊂ R d be an open set, k be a measurable kernel on Ω, and H k (Ω) be the RKHS of k with inner-product ·, · H k (Ω) . Suppose P is a Borel probability measure on R d with its support contained in Ω, and {(w i , X i )} n i=1 ⊂ (R × Ω) n is weighted points, which serve for quadrature. For an integrand f , define P f and P n f by the integral and a quadrature estimate, respectively; namely, As mentioned in Section 1, a kernel quadrature rule aims at minimizing the worst case error e n (P ; H k (Ω)) := sup Assume where the integral for m P is understood as the Bochner integral. It is easy to see that, for all f ∈ H, The worst case error (8) can then be written as and for any f ∈ H k (Ω) It follows from (10) that The integrals in (11) are known in closed form for many pairs of k and P (see e.g. Table 1 of [9]); for instance, it is known if k is a Wendland kernel and P is the uniform distribution on a ball in R d . One can then explicitly use the formula (11) in order to obtain weighted points {(w i , X i )} that minimizes the worst case error (8).
Given the design points being fixed, quadrature weights w 1 , . . . , w n are then obtained by the minimization of the worst case error (11), which can be done analytically by solving a linear system of size n. To describe this, let X 1 , . . . , X n be design points such that the kernel matrix G := (k(X i , X j )) n i,j ∈ R n×n is invertible. The weights are then given by where z := (m P (X i )) n i=1 ∈ R n , with m P defined in (9). This way of constructing the estimate P n f is called Bayesian quadrature, since P n f can be seen as a posterior estimate in a certain Bayesian inference problem with f generated as sample of a Gaussian process (see, e.g., [26] and [9]).
Quasi Monte Carlo. Quasi Monte Carlo (QMC) methods are equal-weight quadrature rules designed for the uniform distribution on a hyper-cube [0, 1] d [16]. Modern QMC methods make use of RKHSs and the associated kernels to define and calculate the worst case error in order to obtain good design points (e.g. [25,48,13,17]). Therefore, such QMC methods are instances of kernel-based quadrature rules; see [39] and [16] for a review.
Kernel herding. In the machine learning literature, an equal-weight quadrature rule called kernel herding [10] has been studied extensively [26,5,30,27]. It is an algorithm that greedily searches for design points so as to minimize the worst case error in an RKHS. In contrast to QMC methods, kernel herding may be used with an arbitrarily distribution P on a generic measurable space, given that the integral k(·, x)dP (x) admits a closed form solution with a reproducing kernel k. It has been shown that a fast rate O(n −1 ) is achievable for the worst case error, when the RKHS is finite dimensional [10]. While empirical studies indicate that the fast rate would also hold in the case of an infinite dimensional RKHS, its theoretical proof remains an open problem [5].

Convergence rates of Bayesian quadrature
This section discusses the convergence rates of Bayesian quadrature in well-specified settings. It is shown that Bayesian quadrature can achieve the minimax optimal rates for deterministic quadrature rules in Sobolev spaces. The result also serves as a preliminary to Section 5, where misspecified cases are considered.
Let Ω be an open set in R d and X n := {X 1 , . . . , X n } ⊂ Ω. The main notion to express the convergence rate is fill distance h X n ,Ω (5), which plays a central role in the literature on scattered data approximation [55], and has been used in the theoretical analysis of Bayesian quadrature in [9,40]. However, it is necessary to introduce some conditions on Ω. The first one is the interior cone condition [55,Definition 3.6], which is a regularity condition on the boundary of Ω. A cone Definition 1 (Interior cone condition). A set Ω ⊂ R d is said to satisfy an interior cone condition if there exist an angle θ ∈ (0, 2π) and a radius R > 0 such that every x ∈ Ω is associated with a unit vector ξ(x) so that the cone C(x, ξ(x), θ, R) is contained in Ω.
The interior cone condition requires that there is no 'pinch point' (i.e. a ≺-shape region) on the boundary of Ω; see also [40]. Next, the notions of special Lipschitz domain [51, p.181 such that the following conditions are satisfied: is the ball centered at x and radius ε; Examples of a set Ω having a Lipschitz boundary include: (i) Ω is an open bounded set whose boundary ∂Ω is C 1 embedded in R d ; (ii) Ω is an open bounded convex set [51, p.189].

Proposition 4.
Let Ω ⊂ R d be a bounded open set such that an interior cone condition is satisfied and the boundary ∂Ω is Lipschitz, and P be a probability distribution on R d with a bounded density function p such that supp(P ) ⊂ Ω. For r ∈ R with ⌊r⌋ > d/2, k r is a kernel on R d that satisfies Assumption 1 and H kr (Ω) is the RKHS of k r restricted on Ω. Suppose that X n := {X 1 , . . . , X n } ⊂ Ω are finite points such that G := (k r (X i , X j )) n i,j=1 ∈ R n×n is invertible, and w 1 , . . . , w n are the quadrature weights given by (12). Then there exist constants C > 0 and h 0 > 0 independent of X n , such that e n (P ; H kr (Ω)) ≤ Ch r X n ,Ω , provided that h X n ,Ω ≤ h 0 , where e n (P ; H kr (Ω)) is the worst case error for the quadrature rule Proof. The proof idea is borrowed from [9, Theorem 1]. Let f ∈ H kr (Ω) be arbitrary and fixed.
Define a function f n ∈ H kr (Ω) by It follows from the norm-equivalence that f ∈ H r (Ω) and where In fact, recalling that the weights w : Using this identity, we have where (14) follows from Theorem 11.32 and Corollary 11.33 in [55] (where we set m := 0, p := 2, q := 1, k := ⌊r⌋ and s := r − ⌊r⌋), and (15) from (13). Note that constant C 0 depends only on r, d and the constants in the interior cone condition (which follows from the fact that Theorem 11.32 in [55] is derived from Proposition 11.30 in [55]). Setting C := C 0 C 1 p ∞ completes the proof.
• Typically the fill distance h X n ,Ω decreases to 0 as the number n of design points increases. Therefore the upper bound Ch r X n Ω provides a faster rate of convergence for e n (P ; W r 2 (Ω)) by a larger value of the degree r of smoothness.
• The condition h X n ,Ω ≤ h 0 requires that the design points X n = {X 1 , . . . , X n } must cover the set Ω to a certain extent in order to guarantee the error bound to hold. This requirement arises since we have used a result from the scattered data approximation literature [55,Corollary 11.33] to derive the inequality (14) in our proof. In the literature such a condition is necessary and we refer an interested reader to Section 11 of [55] and references therein.
• The constant h 0 > 0 depends only on the constants θ and R in the interior cone condition (Definition 1). The explicit form is with ψ := 2 arcsin sin θ 4(1+sin θ) [55, p.199]. The following is an immediate corollary to Proposition 4.
Corollary 5. Assume that Ω, P and r satisfy the conditions in Proposition 4. Suppose that X n := {X 1 , . . . , X n } ⊂ Ω are finite points such that G := (k r (X i , X j )) n i,j=1 ∈ R n×n is invertible and h X n ,Ω = O(n −α ) for some 0 < α ≤ 1/d as n → ∞, and w 1 , . . . , w n are the quadrature weights given by (12) based on X n . Then we have e n (P ; H kr (Ω)) = O(n −αr ) (n → ∞), where e n (P ; H kr (Ω)) is the worst case error of the quadrature rule Remark 2.
• The result (16) implies that the same rate is attainable for the Sobolev space H r (Ω) (instead of H kr (Ω)): This follows from the normequivalence between H kr (Ω) and H r (Ω).
This rate is minimax optimal for the deterministic quadrature rules for the Sobolev space H r (Ω) on a hyper-cube [37, Proposition 1 in Section 1.3.12]. Corollary 5 thus shows that Bayesian quadrature achieves the minimax optimal rate in this setting.
• The decay rate for the fill distance h X n ,Ω = O(n −1/d ) holds when, for example, the design points X n = {X 1 , . . . , X n } are equally-spaced grid points in Ω. Note that this rate cannot be improved: if the fill distance decreased at a rate faster than O(n −1/d ), then e n (P ; H r (Ω)) would decrease more quickly than the minimax optimal rate, which is a contradiction.

Main results
This section presents the main results on misspecified settings. Two results based on different assumptions are discussed: one on the quadrature weights in Section 4.1, and the other on the design points in Section 4.2. The approximation theory for Sobolev spaces developed by [35] is employed in the results.

Convergence rates under an assumption on quadrature weights
where c 1 , c 2 > 0 are constants independent of {(w i , X i )} n i=1 , f and σ.
Proof. We first derive some inequalities used for proving the assertion. It follows from normequivalence that f ∈ W s 2 (Ω), where W s 2 (Ω) is the Sobolev space defined via weak derivatives. Since Ω has a Lipschitz boundary, Stein's extension theorem [51, p.181] guarantees that there exists a bounded linear extension operator E : where C 1 is a constant independent of the choice of f . From the norm-equivalence and (19), there is a constant C 2 such that Since f ∈ L 1 (Ω), the extension also satisfies for some constant C 3 > 0. Below we writef := E(f ) for notational simplicity.
Let g σ ∈ H r (R d ) be the approximate function off defined as (49) for some constant C 4 > 0 which is independent of f .
for some constants C 5 and C ′ 5 , which are independent of σ andf . With the decomposition , each of the terms (A), (B) and (C) will be bounded below.
First, the term (A) is bounded as For the term (B), it follows from the norm equivalence and restriction that for some constant This inequality and (23) give Finally, the term (C) is bounded as Combining these three bounds, the assertion is obtained.
• The integrand f is assumed to satisfy f ∈ H s (Ω) ∩ C s B (Ω) ∩ L 1 (Ω), which is slightly stronger than just assuming f ∈ H s (Ω).
• In the upper-bound (17), the constant σ > 0 controls the trade-off between the two terms: r−s 2 e n (P ; H kr (Ω)) f H s (Ω) and c 1 ( n i=1 |w i | + 1) · σ −s f C s B (Ω) . In the proof, the integrand f is approximated by a band-limited function g σ ∈ H r (Ω), where σ is the highest spectrum that g σ possesses. Thus the trade-off in the upper-bound corresponds to the tradeoff between the accuracy of approximation of f by g σ and the penalty incurred on the regularity of g σ .
The following result, which is a corollary of Theorem 6, provides a rate of convergence for the quadrature error in a misspecified setting. It is derived by assuming certain rates for the quantity n i=1 |w i | and the worst case error e n (P ; H kr ).

Corollary 7.
Let Ω, P , r, s, k r , and H kr (Ω) be the same as Theorem 6 Setting θ = (b + c)/r, which balances the two terms in the right hand side, completes the proof.

Remark 4.
• The exponent of the rate in (25) consists of two terms: −bs/r and c(r − s)/r. The first term −bs/r corresponds to a degraded rate from the original O(n −b ) by the factor of smoothness ratio s/r, while the second term c(r − s)/r makes the rate slower. The effect of the second term increases as the constant c or the gap (r − s) of misspecification gets larger.
• The obtained rate recovers O(n −b ) for r = s (well-specified case) regardless of the value of c.
• Consider the misspecified case r > s. If c > 0, the term c(r − s)/r always makes the rate slower. It is thus better to have c = 0, as in this case we have the rate O(n −bs/r ) in the misspecified setting. The weights with max i=1,...,n |w i | = O(n −1 ), such as equal weights w i = 1/n, realize c = 0.
• As mentioned earlier, the minimax optimal rate for the worst case error in the Sobolev space . This rate is the same as the minimax optimal rate for H s (Ω), and hence implies some adaptivity to the order of differentiability.
• The assumption n i=1 |w i | = O(n c ) can be also interpreted from a probabilistic viewpoint. Assume that the observation involves noise, Y i := f (X i ) + ε i (i = 1, . . . , n), where ε i is independent noise with E[ε 2 i ] = σ 2 noise (σ noise > 0 is a constant) for i = 1, . . . , n, and that Y i are used for numerical integration. The expected squared error is decomposed as In the last expression, the first term |P n f − P f | 2 is the squared error in the noiseless case, and the second term σ 2 noise n i=1 w 2 i is the error due to noise. Since n i=1 w 2 i ≤ ( n i=1 |w i |) 2 = O(n 2c ), the error in the second term may be larger as c increases. Hence quadrature weights having smaller c are preferable in terms of robustness to the existence of noise; this in turn makes the quadrature rule more robust to the misspecification of the degree of smoothness.
Theorem 6 and Corollary 7 require a control on the absolute sum of the quadrature weights n i=1 |w i |. This is possible with, for instance, equal-weight quadrature rules that seek for good design points. However, the control of n i=1 |w i | could be difficult for quadrature rules that obtain the weights by optimization based on pre-fixed design points. This includes the case of Bayesian quadrature that optimizes the weights without any constraint. To deal with such methods, in the next section we will develop theoretical guarantees that do not rely on the assumption on the quadrature weights, but on a certain assumption on the design points.

Convergence rates under an assumption on design points
This subsection provides convergence guarantees in a misspecified settings under an assumption on the design points. The assumption is described in terms of separation radius (4), which is (the half of) the minimum distance between distinct design points. The separation radius of points X n := {X 1 , . . . , X n } ⊂ R d is denoted by q X n . Note that if X n ⊂ Ω for some Ω, then the separation radius lower bounds the fill distance, i.e., q X n ≤ h X n ,Ω . Henceforth we will consider a bounded domain Ω, and without loss of generality, we assume that it satisfies diam(Ω) ≤ 1.

Theorem 8.
Let Ω ⊂ R d be an open bounded set with diam(Ω) ≤ 1 such that the boundary is Lipschitz, P be a probability distribution on R d such that supp(P ) ⊂ Ω, r be a real number with r > d/2, and s be a natural number with s ≤ r. Let k r denote a kernel on R d satisfying Assumption 1, and H kr (Ω) the RKHS of k r restricted on Ω. For any where C > 0 is a constant depending neither on {(w i , X i )} n i=1 nor on the choice of f , and e n (P ; H kr (Ω)) is the worst case error in H kr (Ω) for Proof. By the same argument as the first part of the proof for Theorem 6, there exists an extension for some positive constants C i (i = 1, 2). Note also that f ∈ L 1 (Ω), since f ∈ C s B (Ω) and Ω is bounded. This impliesf ∈ L 1 (R d ) [51, p.181].
From the above inequalities, there is a constant C 3 > 0 independent of the choice of f such that For notational simplicity, write with Γ being the Gamma function. From Theorems A.1 and A.3 in Appendix A (which are restatements of Theorems 3.5 and 3.10 of [35]), there exists a functioñ where C s,d is a constant depending only on s and d. Combining (29) and (27) obtains , where C s,d,kr is a constant only depending on r, s, d, and k r . It follows from this inequality and (27) that where C 5 := C s,d,kr C 3 .
We are now ready to prove the assertion. In the decomposition , the term (A) is zero from (28).

The term (C) is upper-bounded as
These bounds complete the proof.
• From q X n ≤ h X n , the separation radius q X n typically converges to zero as n → ∞. For the upper bound in (26), the factor q −(r−s) X n in the first term diverges to infinity as n → ∞, while the second term goes to zero. Thus q X n should decay to zero in an appropriate speed depending on the rate of e n (P ; H kr (Ω)), in order to make the quadrature error small in the misspecified setting.
• Note that as the gap between r and s becomes large, the effect of the separation radius becomes serious; this follows from the expression q Based on Theorem 8, we establish below a rate of convergence in a misspecified setting by assuming a certain rate of decay for the separation radius as the number of design points increases.

Corollary 9.
Let Ω, P, r, s, k r , H kr (Ω) be the same as in Theorem 8. Suppose {(w i , X i )} n i=1 ∈ (R × Ω) n is design points such that e n (P ; H kr (Ω)) = O(n −b ) and q X n = Θ(n −a ) for some b > 0 and a > 0, respectively, as n → ∞. Then for any f ∈ C s B (Ω) ∩ H s (Ω), we have In particular, the rate in the right hand side is optimized when a = b/r, which gives Proof. Plugging e n (P ; H kr (Ω)) = O(n −b ) and q X n = Θ(n −a ) into (26) yields which proves (31). The second assertion is obvious.
Remark 6. As stated in the assertion, the best rate for the bound is achieved when a = b/r. The resulting rate in (32) coincides with that of Corollary 7 (see (25)) with c = 0. Therefore observations similar to those for Theorem 6 can be made with the rate in (32).

Bayesian quadrature in misspecified settings
To demonstrate the results of Section 4, a rate of convergence for Bayesian quadrature in misspecified settings is derived. To this end, an upper-bound on the integration error of Bayesian quadrature is first provided, when the smoothness of an integrand is overestimated. It is obtained by combining Theorem 8 in Section 4 and Proposition 4 in Section 3.

Theorem 10.
Let Ω ⊂ R d be a bounded open set with diam(Ω) ≤ 1 such that an interior cone condition is satisfied and the boundary is Lipschitz, P be a probability distribution on R d with a bounded density function p such that supp(P ) ⊂ Ω, r be a real number with ⌊r⌋ > d/2, and s be a natural number with s ≤ r. Suppose that k r is a kernel on R d satisfying Assumption 1, X n := {X 1 , . . . , X n } ⊂ Ω is design points such that G := (k r (X i , X j )) n i,j=1 ∈ R n×n is invertible, and w 1 , . . . , w n are the Bayesian quadrature weights in (12) based on k r . Assume that there exist constants c q > 0 and δ > 0 independent of X n , such that 1 − s/r < δ ≤ 1 and Then there exist positive constants C and h 0 independent of X n , such that for any provided that h X n ,Ω ≤ h 0 .
Proof. Under the assumptions, Theorem 8 gives that where C 1 > 0 is a constant, and e n (P ; H kr (Ω)) is the worst case error of {(w i , X i )} n i=1 in H kr (Ω). On the other hand, Proposition 4 implies that there exist constants C 2 > 0 and h 0 > 0 independent of the choice of X n , such that e n (P ; H kr (Ω)) ≤ C 2 h r X n ,Ω , provided that h X n ,Ω ≤ h 0 . Note also that (33) implies that From q X n ≤ h X n ,Ω and the above inequalities, it follows that where C 1 , C 2 and C 3 are positive constants independent of the choice of design points X n , and we used q X n ≤ h X n ,Ω in (⋆), 0 < h X n ≤ 1 and 0 < r − (r − s)/δ ≤ s in ( †).
• The condition (33) implies that where c ′ := c −1/δ q is independent of X n . This condition is stronger for a larger value of δ, requiring that distinct design points should not be very close to each other. Note that the lower-bound 1 − s/r < δ is necessary for the upper-bound of the error (34) to have a positive exponent, while the upper-bound δ ≤ 1 follows from q X n ≤ h X n ,Ω , which holds by definition. The constraint 1 − s/r < δ and (38) thus imply that a stronger condition is required for X n as the degree of misspecification gets more serious (i.e., as the ratio s/r gets smaller).
• If the condition (33) is satisfied for δ = 1, then the design points X n are called quasi-uniform [47,Section 7.3]. In this case, the bound in (34) is This is the same order of approximation as that of Proposition 4 when r = s. Proposition 4 provides an error bound for Bayesian quadrature in a well-specified case, where one knows the degree of smoothness s of the integrand. Therefore, (39) suggests that, if the design points are quasi-uniform, then Bayesian quadrature can be adaptive to the (unknown) degree of the smoothness s of the integrand f , even in a situation where one only knows its upper-bound r ≥ s.
We obtain the following as a corollary of Theorem 10. The proof is obvious, and omitted.
Corollary 11. Let Ω, P, r, s, k r , X n , G and w i (i = 1, . . . , n) be the same as Theorem 10. Assume that there exist constants c q > 0 and δ > 0 independent of X n , such that 1 − s/r < δ ≤ 1 and h X n ,Ω ≤ c q q δ X n , and further h X n ,Ω = O(n −α ) as n → ∞ for some In particular, the best possible rate in the right hand side is achieved when δ = 1 and α = 1/d, Remark 8.
• The rate O(n −s/d ) in (40) matches the minimax optimal rate of deterministic quadrature rules for the worst case error in the Sobolev space H s (Ω) with Ω being a cube [37, Proposition 1 in Section 1.3.12]. Therefore, it is shown that the optimal rate may be achieved by Bayesian quadrature, even in the misspecified setting (under a slightly stronger assumption that f ∈ H s (Ω) ∩ C s B (Ω)). In other words, Bayesian quadrature may achieve the optimal rate adaptively, without knowing the degree s of smoothness of a test function: one just needs to know its upper bound r ≥ s.
• The main assumptions required for the optimal rate (40) are that (i) h X n ,Ω = O(n −1/d ) and that (ii) h X n ,Ω ≤ c q q δ X n for δ = 1. Recall that (i) is the same assumption that is required for the optimal rate O(n −r/d ) in the well-specified setting f ∈ H r (Ω) (Corollary 5). On the other hand, (ii) is the one required for the finite sample bound in Theorem 10. Both these assumptions are satisfied, for instance, if X 1 , . . . , X n are grid points in Ω.

Discussion
In this paper, we have discussed the convergence properties of kernel quadratures with deterministic design points in misspecified settings. In particular, we have focused on settings where quadrature weighted points are generated based on misspecified assumptions on the degree of smoothness, that is, the situation where the integrand is less smooth than assumed.
We have revealed conditions for quadrature rules under which adaptation to the unknown lesser degree of smoothness occurs. In particular we have shown that a kernel quadrature rule is adaptive if the sum of absolute weights remains constant, or if the spacing between design points is not too small (as measured by the separation radius). Moreover, by focusing on Bayesian quadratures as working examples, we have shown that they can achieve minimax optimal rates of the unknown degree of smoothness, if the design points are quasi-uniform. We expect that this result provides a practical guide for developing kernel quadratures that are robust to the misspecification of the degree of smoothness; such robustness is important in modern applications of quadrature methods, such as numerical integration in sophisticated Bayesian models, since they typically involve complicated or black box integrands and thus misspecification is likely to happen.
There are several important topics to be investigated as part of future work.
Other RKHSs. This paper has dealt with Sobolev spaces as RKHSs of kernel quadrature. However, there are many other important RKHSs of interest where similar investigation can be carried out. For instance, Gaussian RKHSs (i.e. the RKHSs of Gaussian kernels) have been widely used in the literature on Bayesian quadrature. Such an RKHS consists of functions with infinite degree of smoothness. This makes theoretical analysis challenging: our analysis relies on the approximation theory by [35], which only applies to the standard Sobolev spaces. Similarly, the theory of [35] is also not applicable to Sobolev spaces with dominating mixed smoothness, which have been popular in the QMC literature. In order to analyze quadrature rules in these RKHSs, we therefore need to extend the approximation theory of [35] to such spaces. This will be an important but challenging theoretical problem. Sequential (adaptive) quadrature. Another important direction is the analysis for kernel quadratures that sequentially select design points. Such methods are also called adaptive, since the selection of the next point X n+1 depends on the function values f (X 1 ), . . . , f (X n ) of the already selected points X 1 , . . . , X n . Note that the adaptability here is different from that of the current paper where we used it in the context of adaptability of quadrature to unknown degree of smoothness. For instance, the WSABI algorithm by [24] is an example of adaptive Bayesian quadrature which is considered as state-of-the-art for the application of Bayesian model evidence calculation. Such adaptive methods have been known to be able to outperform non-adaptive methods in the following case: the hypothesis space is imbalanced or non-convex (see e.g. Section 1 of [38]). In the worst case error, the hypothesis space is the unit ball in the RKHS H, which is balanced and convex and so adaptation does not help. In fact, it is known that the optimal rate can be achieved without adaptation. However, if the hypothesis space is imbalanced (i.e. f being in the hypothesis space does not imply that −f is in the hypothesis space), then adaptive methods may perform better. For instance, the WSABI algorithm focuses on non-negative integrands, which means that the hypothesis is imbalanced and thus adaptive selection helps. Our analysis in this paper has focused on the worst case error defined by the unit ball in an RKHS, which is balanced and convex. A future direction is thus to consider the setting of imbalanced or non-convex hypothesis spaces, such as the one consisting of non-negative functions, which will enable us to analyze the convergence behavior of sequential or adaptive Bayesian quadrature in misspecified settings.
Random design points. We have focused on deterministic quadrature rules in this paper. In the literature, however, the use of random design points has also been popular. For instance, the design points of Bayesian quadrature might be i.i.d. with a certain proposal distribution or generated as an MCMC sequence. Likewise, QMC methods usually apply randomization to deterministic design points. Our forthcoming paper will deal with such situations and provide more general results than the current paper.

MK and KF are supported in part by MEXT Grant-in-Aid for Scientific Research on Innovative Areas (25120012) and MEXT KAKENHI (17K12654). BKS is partially supported by NSF-DMS-1713011.
A Key results from [35] Here we review some key results from [35], which are needed in the proofs for our results.
For σ > 0, below we denote by B σ a subset of L 2 (R d ) such that each f ∈ B σ has a spectral density whose support is contained in the (closed) ball B(0, σ) with radius σ, i.e., This is a Paley-Weiner class of band-limited functions. Thus the functions in B σ are analytic (and thus they are continuous), and vanish at infinity. Therefore The following theorem is a restatement of Theorem 3.5 of [35].
Theorem A.1. Let X n := {X 1 , . . . , X n } ⊂ R d be n distinct points with separation radius q X n := .
Then for any f ∈ C 0 (R d ) ∩ L 2 (R d ), there exists f σ ∈ B σ that satisfies In the above theorem, f σ is an interpolant of f on X n . Thus the theorem guarantees that such a f σ can be taken as a band-limited function with a sufficiently large band-length σ. More precisely, the lower bound σ 0 for σ is proportional to the reciprocal of the separation radius q X n . This means that the band-length σ should increase as the minimum distance between distinct design points decreases.
The following proposition is a restatement of Proposition 3.7 of [35], which establishes an upper-bound on the L 1 -error for the approximate function defined in (49)-see Appendix B.2.
and g σ is the approximate function defined in (49). Then for any σ > 0, The following theorem, which is Theorem 3.10 in [35], provides an upper-bound on the approximation error of the interpolant f σ .
, f σ is the interpolant from Theorem A.1 with σ > 0 and X n := {X 1 , . . . , X n } satisfies the conditions in Theorem A.1. Then there is a constant C |α|,s,d that depends only on |α|, s and d such that The following proposition, which is Proposition 3.11 in [35], provides an upper-bound on a Sobolev norm of the interpolant f σ .
We have the following comments on Propositions A.2, A.4 and Theorem A.3.
• In the original statement of Proposition 3.7 in [35], the assumption f ∈ L 1 (R d ) is missing.
However, since this assumption is required for the function g σ to be well- • In the original statement of Proposition 3.11 in [35], the condition σ ≥ 1 is required. This condition is implicitly satisfied by σ in Proposition A.4 as the condition on σ in Theorem A.1 implies σ ≥ 1, which can be seen from the fact that q X n ≤ 1/2 (follows from the assumption diam(X n ) ≤ 1) and the definition of the lower-bound σ 0 of σ.

A.1 The Sobolev norm of the interpolant f σ
Here we provide an upper-bound on the Sobolev (RKHS) norm of the interpolant f σ in Theorem A.1. The result essentially follows from an argument in p.298 of [35], but we prove it for completeness.
Lemma A.5. Let r ∈ R, r > d/2 and s ∈ N, r ≥ s. Let k r be a kernel on R d such that , f σ is the interpolant from Theorem A.1 with σ > 0 and X n := {X 1 , . . . , X n } satisfies the conditions in Theorem A.1. Then we have where C s,d,kr is a constant only depending on r, s, d, and k r (note that the dependency on the kernel k r is via the constant C 1 ).
Proof. Note that, since σ satisfies the conditions in Theorem A.1, it follows that σ ≥ 1. We then have Therefore, by using Proposition A. 4, it follows that

B.1 Fundamental lemma
In the proof of Theorem 6, we used Proposition 3.7 of [35], which assumes the existence of a function ψ : R d → R satisfying the properties in Lemma B.1. Since the existence of this function is not proved in [35], we will first prove it for completeness. Lemma B.1 is a variant of Lemma 1.1 of [18], from which we borrowed the proof idea.
Lemma B.1. Let s ∈ N. Then there exists a function ψ : R d → R satisfying the following properties: (a) ψ is radial; Proof. Define a function u ∈ L 1 (R d ) as the inverse Fourier transform of a functionû ∈ L 1 (R d ) defined byû Thenû is radial, Schwartz, and satisfies supp(û) ⊂ B(0, 1). Also note that u is real-valued, sincê u is symmetric.
Let m ∈ N satisfy m > s/2. Define a function h : where ∆ denotes the Laplacian defined by ∆f : . Note that we have (see e.g. p.117 of where C m is a constant depending only on m. From this expression, it follows thatĥ is radial and Schwartz (and so is h), and that supp(ĥ) ⊂ B(0, 1). Thus the function h satisfies the required properties (a) (b) and (c). Later we will define the function ψ in the assertion based on h.
We next show that h satisfies the property (d). Let β ∈ N d 0 be any multi-index satisfying |β| ≤ s, and let p β (x) := x β be a monomial. It follows that p β h is Schwartz, and thus p β h ∈ L 1 (R d ). Then we have which follows from p β h ∈ L 1 (R d ) and from the definition of Fourier transform. Note that we have (see e.g. Theorem 5.16 of [55]) The mixed partial derivative in the right side can be expanded as where, in the last equality, we used the Leibniz rule for mixed partial derivatives, γ ≤ β is defined by that γ i ≤ β i for all i = 1, . . . , d, and β γ : Using the multinomial theorem, the mixed partial derivative ∂ γ ξ 2m in the above equation can be further expanded as Note that we have Also note that, since |α| = m and |γ| ≤ |β| ≤ s < 2m, we have |γ| < 2|α|. This implies that there exists at least one index ℓ ∈ {1, . . . , d} such that 2α ℓ > γ ℓ . For this ℓ we then have From this and (46), it follows that ∂ γ ξ 2m ξ=0 = 0, and thus (45) gives that ∂ βĥ (0) = 0. Therefore, from (43) and (44), it holds that which is the property (d).

B.2 Approximation via Calderón's formula.
If ψ ∈ L 1 is radial and satisfies (41), Calderón's formula [18,Theorem 1.2] guarantees that any f ∈ L 2 can be written as where Note that the integral in (47) is improper, and should be interpreted in the following L 2 sense: if 0 < ε < δ < ∞ and f ε,δ (x) := δ ε (ψ t * ψ t * f )(x) dt t , then f − f ε,δ L 2 → 0 as ε → +0 and δ → ∞ independently. It is easy to verify from (48) that Let ψ be the function in Lemma B.1. Following Section 3.2 of [35], we consider the following approximation of f based on Calderón's formula (47): The integral in (49) is also improper and should be interpreted as follows. Let δ > 1/σ and define g σ,δ := Then g σ in (49) is defined to be a function in L 2 such that lim δ→∞ g σ − g σ,δ L 2 = 0. Such g σ exists (as a limit of g σ,δ ), as shown in Lemma B.4 below. Since there is no proof of this result in [35], we provide a proof for the sake of completeness. To this end, we first need the following lemma.
Proof. For p ∈ {1, 2}, note that where in the last line we used the assumption f ∈ L p and the fact ψ ∈ L 1 , which is a consequence of ψ being a Schwartz function (see Lemma B.1).
In the above derivation, Fubini's theorem is applicable since ψ t * ψ t * f ∈ L 1 (which follows from ψ ∈ L 1 , f ∈ L 1 and Minkowski's inequality; see the proof of Lemma B.2).
We are now ready to show that the improper integral in (49) is well-defined as a limit of g σ,δ in L 2 .
The following lemma provides an expression for the Fourier transform of the function g σ ∈ L 2 .
Lemma B.5. Assume f ∈ L 1 ∩ L 2 , and let g σ ∈ L 2 be the function in Lemma B.4. Then the Fourier transform of g σ is given bŷ Proof. As shown in Lemma B.4, the function g σ is given as a limit in L 2 of functions g σ,δ as δ → ∞. Therefore, the Fourier transform of g σ is given as a limit in L 2 of the Fourier transforms of g σ,δ : lim δ→∞ ĝ σ −ĝ σ,δ L 2 = 0.

B.3 The Sobolev norm of the approximate function
In the main body of the paper, we use the following lemma, which is not provided in [35].
Lemma B.6. Let r, s ∈ R, r, s > 0 such that r ≥ s and let σ > 0 be a constant. If f ∈ H s (R d ) ∩ L 1 (R d ), the function g σ defined in (49) satisfies where C > 0 is a constant independent of f and σ.