Numerical weighted integration of functions having mixed smoothness

The aim of the present paper is to investigate approximation of weighted integrals over $\mathbb{R}^{d}$ for integrands lying in weighted Sobolev spaces $W^{r}_{1,w}(\mathbb{R}^{d})$ of mixed smoothness $r\in\mathbb{N}$. We want to give upper and lower bounds of the approximation error for optimal quadratures with $n$ integration nodes over the unit ball of $W^{r}_{1,w}(\mathbb{R}^{d})$.

We first introduce weighted Sobolev spaces of mixed smoothness. Let

$$w({\boldsymbol{x}}):=w_{\lambda,a,b}({\boldsymbol{x}}):=\prod_{i=1}^{d}w(x_{i}),$$

(1.1)

where

$$w(x):=w_{\lambda,a,b}(x):=\exp(-a|x|^{\lambda}+b),\ \ \lambda>1,\ \ a>0,\ \ b\in{\mathbb{R}},$$

(1.2)

is a univariate Freud-type weight. The most important parameter in the weight $w_{\lambda,a,b}$ is $\lambda$. The parameter $b$ which produces only a possitive constant in the weight $w_{\lambda,a,b}$ is introduced for a certain normalization for instance, for the standard Gaussian weight which is one of the most important weights. In what follows, we fix the parameters $\lambda,a,b$ and for simplicity, drop them from the notation. Let $1\leq p<\infty$ and $\Omega$ be a Lebesgue measurable set on ${\mathbb{R}}^{d}$. We denote by $L_{w}^{p}(\Omega)$ the weighted space of all functions $f$ on $\Omega$ such that the norm

$\displaystyle\|f\|_{L_{w}^{p}(\Omega)}:=\bigg{(}\int_{\Omega}|f({\boldsymbol{x}})w({\boldsymbol{x}})|^{p}{\rm d}{\boldsymbol{x}}\bigg{)}^{1/p}$

(1.3)

is finite. For $r\in{\mathbb{N}}$, we define the weighted Sobolev space $W^{r}_{p,w}(\Omega)$ of mixed smoothness $r$ as the normed space of all functions $f\in L_{w}^{p}(\Omega)$ such that the weak (generalized) partial derivative $D^{{\boldsymbol{k}}}f$ belongs to $L_{w}^{p}(\Omega)$ for every ${\boldsymbol{k}}\in{\mathbb{N}}^{d}_{0}$ satisfying the inequality $|{\boldsymbol{k}}|_{\infty}\leq r$. The norm of a function $f$ in this space is defined by

$\displaystyle\|f\|_{W^{r}_{p,w}(\Omega)}:=\Bigg{(}\sum_{|{\boldsymbol{k}}|_{\infty}\leq r}\|D^{{\boldsymbol{k}}}f\|_{L_{w}^{p}(\Omega)}^{p}\Bigg{)}^{1/p}.$

(1.4)

It is useful to notice that any function $f\in W^{r}_{p,w}({\mathbb{R}}^{d})$ is equivalent in the sense of the Lesbegue measure to a continuous (not necessarily bounded) function on ${\mathbb{R}}^{d}$, see Lemma 3.1 below. Hence throughout the present paper, we always assume that the functions $f\in W^{r}_{p,w}({\mathbb{R}}^{d})$ are continuous. We need this assumption for well-defined quadratures for functions $f\in W^{r}_{p,w}({\mathbb{R}}^{d})$.

Let $\gamma$ be the standard $d$-dimensional Gaussian measure $\gamma$ with the density function

$$g({\boldsymbol{x}})=(2\pi)^{-d/2}\exp(-|{\boldsymbol{x}}|_{2}^{2}/2).$$

The well-known spaces $L^{p}(\Omega;\gamma)$ and $W^{r}_{p}(\Omega;\gamma)$ which are used in many applications, are defined in the same way by replacing the norm (1.3) with the norm

$$\|f\|_{L^{p}(\Omega;\gamma)}:=\bigg{(}\int_{\Omega}|f({\boldsymbol{x}})|^{p}\gamma({\rm d}{\boldsymbol{x}})\bigg{)}^{1/p}=\bigg{(}\int_{\Omega}|f({\boldsymbol{x}})|^{p}g({\boldsymbol{x}}){\rm d}{\boldsymbol{x}}\bigg{)}^{1/p}.$$

The spaces $L^{p}(\Omega;\gamma)$ and $W^{r}_{p}(\Omega;\gamma)$ can be seen as the $L_{w}^{p}(\Omega)$ and $W^{r}_{p,w}(\Omega)$, where

$$w({\boldsymbol{x}})=w_{\lambda,a,b}({\boldsymbol{x}}),\ \ \text{with}\ \ \lambda=2/p,\ a=1/2p,\ b=-(d\log 2\pi)/2p$$

for a fixed $1\leq p<\infty$.

In the present paper, we are interested in approximation of weighted integrals

$$\int_{{\mathbb{R}}^{d}}f({\boldsymbol{x}})w({\boldsymbol{x}})\,{\rm d}{\boldsymbol{x}}$$

(1.5)

for functions $f$ lying in the space $W^{r}_{1,w}(\mathbb{R}^{d})$. To approximate them we use quadratures of the form

$$Q_{k}f:=\sum_{i=1}^{k}\lambda_{i}f({\boldsymbol{x}}_{i}),$$

(1.6)

where ${\boldsymbol{x}}_{1},\ldots,{\boldsymbol{x}}_{k}\in{\mathbb{R}}^{d}$ are the integration nodes and $\lambda_{1},\ldots,\lambda_{k}$ the integration weights. For convenience, we assume that some of the integration nodes may coincide.

Let ${\boldsymbol{F}}$ be a set of continuous functions on ${\mathbb{R}}^{d}$. Denote by ${\mathcal{Q}}_{n}$ the family of all quadratures $Q_{k}$ of the form (1.6) with $k\leq n$. The optimality of quadratures from ${\mathcal{Q}}_{n}$ for $f\in{\boldsymbol{F}}$ is measured by

$${\rm Int}_{n}({\boldsymbol{F}}):=\inf_{Q_{n}\in{\mathcal{Q}}_{n}}\ \sup_{f\in{\boldsymbol{F}}}\bigg{|}\int_{{\mathbb{R}}^{d}}f({\boldsymbol{x}})w({\boldsymbol{x}})\,{\rm d}{\boldsymbol{x}}-Q_{n}f\bigg{|}.$$

(1.7)

We recall that the space $W^{r}_{p}(\Omega)$ is defined as the classical Sobolev space of mixed smoothness by replacing $L_{w}^{p}(\Omega)$ with $L^{p}(\Omega)$ in (1.4), where as usually, $L^{p}(\Omega)$ denotes the Lebesgue space of functions on $\Omega$ equipped with the usual $p$-integral norm.

For approximation of integrals

$$\int_{\Omega}f({\boldsymbol{x}}){\rm d}{\boldsymbol{x}}$$

over the set $\Omega$, we need natural modifications $Q_{n}^{\Omega}f$ for functions $f$ on $\Omega$, and ${\rm Int}_{n}^{\Omega}({\boldsymbol{F}})$ for a set ${\boldsymbol{F}}$ of functions on $\Omega$, of the definitions (1.6) and (1.7). For simplicity we will drop $\Omega$ from these notations if there is no misunderstanding.

We first briefly describe the main results of the present paper and then give comments on related works.

For a normed space $X$ of functions on ${\mathbb{R}}^{d}$, the boldface ${\boldsymbol{X}}$ denotes the unit ball in $X$. Throughout the present paper we make use of the notation

$$r_{\lambda}:=(1-1/\lambda)r.$$

For the set ${\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}}^{d}))$, we prove the upper and lower bounds

$$n^{-r_{\lambda}}(\log n)^{r_{\lambda}(d-1)}\ll{\rm Int}_{n}({\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}}^{d}))\ll n^{-r_{\lambda}}(\log n)^{(r_{\lambda}+1)(d-1)},$$

(1.8)

in particular, in the case of Gaussian measure

$$n^{-r/2}(\log n)^{r(d-1)/2}\ll{\rm Int}_{n}({\boldsymbol{W}}^{r}_{1}({\mathbb{R}}^{d};\gamma))\ll n^{-r/2}(\log n)^{(r/2+1)(d-1)}.$$

(1.9)

In the one-dimensional case, we prove the right convergence rate

$${\rm Int}_{n}({\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}}))\asymp n^{-r_{\lambda}}.$$

(1.10)

The difference between the upper and lower bounds in (1.8) is the logarithmic factor $(\log n)^{d-1}$.

There is a large number of works on high-dimensional unweighted integration over the unit $d$-cube ${\mathbb{I}}^{d}:=[0,1]^{d}$ for functions having a mixed smoothness (see [2, 5, 12] for results and bibliography). However, there are only a few works on high-dimensional weighted integration for functions having a mixed smoothness. The problem of optimal weighted integration (1.5)–(1.7) has been studied in [6, 7, 4] for functions in certain Hermite spaces, in particular, the space ${\mathcal{H}}_{d,r}$ which coincides with $W^{r}_{2}(\mathbb{R}^{d};\gamma)$ in terms of norm equivalence. It has been proven in [4] that

$$n^{-r}(\log n)^{(d-1)/2}\ll{\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{2}({\mathbb{R}}^{d};\gamma))\ll n^{-r}(\log n)^{d(2r+3)/4-1/2}.$$

Recently, in [1, Theorem 2.3] for the space $W^{r}_{p}(\mathbb{R}^{d},\gamma)$ with $r\in{\mathbb{N}}$ and $1<p<\infty$, we have constructed an asymptotically optimal quadrature $Q_{n}^{\gamma}$ of the form (1.6) which gives the asymptotic order

$$\sup_{f\in{\boldsymbol{W}}^{r}_{p}({\mathbb{R}}^{d};\gamma)}\bigg{|}\int_{{\mathbb{R}}^{d}}f({\boldsymbol{x}})\gamma({\rm d}{\boldsymbol{x}})-Q_{n}^{\gamma}f\bigg{|}\asymp{\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{p}({\mathbb{R}}^{d};\gamma)\big{)}\asymp n^{-r}(\log n)^{(d-1)/2}.$$

(1.11)

The results (1.9) and (1.11) show a substantial difference of the convergence rates between the cases $p=1$ and $1<p<\infty$. In constructing the asymptotically optimal quadrature $Q_{n}^{\gamma}$ in (1.11), we used a technique collaging a quadrature for the Sobolev spaces on the unit $d$-cube to the integer-shifted $d$-cubes. Unfortunately, this technique is not suitable to constructing a quadrature realizing the upper bound in (1.8) for the space $W^{r}_{1}(\mathbb{R}^{d};\gamma)$ which is the largest among the spaces $W^{r}_{p}(\mathbb{R}^{d};\gamma)$ with $1\leq p<\infty$. It requires a different technique based on the well-known Smolyak algorithm. Such a quadrature relies on sparse grids of integration nodes which are step hyperbolic crosses in the function domain ${\mathbb{R}}^{d}$, and some generalization of the results on univariate numerical integration by truncated Gaussian quadratures from [3]. To prove the lower bound in (1.8) and (1.10) we adopt a traditional technique to construct for arbitrary $n$ integration nodes a fooling function vanishing at these nodes.

It is interesting to compare the results (1.9) and (1.11) on ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{p}({\mathbb{R}}^{d};\gamma)\big{)}$ with known results on ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{p}({\mathbb{I}}^{d})\big{)}$ for the unweighted Sobolev space $W^{r}_{p}({\mathbb{I}}^{d})$ of mixed smoothness $r$. For $1<p<\infty$, there holds the asymptotic order

$${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{p}({\mathbb{I}}^{d})\big{)}\asymp n^{-r}(\log n)^{(d-1)/2},$$

and for $p=1$ and $r>1$, there hold the bounds

$$n^{-r}(\log n)^{(d-1)/2}\ll{\rm Int}_{n}({\boldsymbol{W}}^{r}_{1}({\mathbb{I}}^{d}))\ll n^{-r}(\log n)^{d-1}$$

which are so far the best known (see, e.g., [2, Chapter 8], for detail). Hence we can see that ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{p}({\mathbb{R}}^{d};\gamma)\big{)}$ and ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{p}({\mathbb{I}}^{d})\big{)}$ have the same asymptotic order in the case $1<p<\infty$, and very different lower and upper bounds in both power and logarithmic terms in the case $p=1$. The right asymptotic orders of the both ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{1}({\mathbb{I}}^{d})\big{)}$ and ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{1}({\mathbb{R}}^{d};\gamma)\big{)}$ are still open problems (cf. [2, Open Problem 1.9]).

The problem of numerical integration considered in the present paper is related to the research direction of optimal approximation and integration for functions having mixed smoothness on one hand, and the other research direction of univariate weighted polynomial approximation and integration on ${\mathbb{R}}$, on the other hand. For survey and bibliography, we refer the reader to the books [2, 12] on the first direction, and [11, 9, 8] on the second one.

The paper is organized as follows. In Section 2, we prove the asymptotic order of ${\rm Int}_{n}({\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}}))$ and construct asymptotically optimal quadratures. In Section 3, we prove upper and lower bounds of ${\rm Int}_{n}({\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}}^{d}))$ for $d\geq 2$, and construct quadratures which give the upper bound. Section 4 is devoted to some extentions of the results in the previous sections to Markov-Sonin weights.

In this section, for one-dimensional numerical integration, we prove the asymptotic order of ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}})\big{)}$ and present some asymptotically optimal quadratures. We start this section with a well-known inequality in the following lemma which is implied directly from the definition (1.7) and which is quite useful for lower estimation of ${\rm Int}_{n}({\boldsymbol{F}})$.

We now consider the problem of approximation of integral (1.5) for univariate functions from $W^{r}_{1,w}({\mathbb{R}})$. Let $(p_{m}(w))_{m\in{\mathbb{N}}}$ be the sequence of orthonormal polynomials with respect to the weight $w$. In the classical quadrature theory, a possible choice of integration nodes is to take the zeros of the polynomials $p_{m}(w)$. Denote by $x_{m,k}$, $1\leq k\leq\lfloor m/2\rfloor$ the positive zeros of $p_{m}(w)$, and by $x_{m,-k}=-x_{m,k}$ the negative ones (if $m$ is odd, then $x_{m,0}=0$ is also a zero of $p_{m}(w)$). These zeros are located as

$$-a_{m}+\frac{Ca_{m}}{m^{2/3}}<x_{m,-\lfloor m/2\rfloor}<\cdots<x_{m,-1}<x_{m,1}<\cdots<x_{m,\lfloor m/2\rfloor}\leq a_{m}-\frac{Ca_{m}}{m^{2/3}},$$

(2.2)

with a positive constant $C$ independent of $m$ (see, e. g., [8, (4.1.32)]). Here $a_{m}$ is the Mhaskar-Rakhmanov-Saff number which is

$$a_{m}=a_{m}(w)=(\gamma_{\lambda}m)^{1/\lambda},\ \ \gamma_{\lambda}:=\frac{2\Gamma((1+\lambda)/2)}{\sqrt{\pi}\Gamma(\lambda/2)},$$

(2.3)

and $\Gamma$ is the gamma function. Notice that the formula (2.3) is given in [8, (4.1.4)] for the weight $w(x)=\exp\left(-|x|^{\lambda}\right)$. Inspecting the definition of Mhaskar-Rakhmanov-Saff number (see, e.g., [8, Page 116]), one easily verify that it still holds true for the general weight $w$ for any $a>0$ and $b\in{\mathbb{R}}$.

For a continuous function on ${\mathbb{R}}$, the classical Gaussian quadrature is defined as

$$Q^{\rm G}_{m}f:=\sum_{|k|\leq\lfloor m/2\rfloor}\lambda_{m,k}(w)f(x_{m,k}),$$

(2.4)

where $\lambda_{m,k}(w)$ are the corresponding Cotes numbers. This quadrature is based on Lagrange interpolation (for details, see, e.g., [11, 1.2. Interpolation and quadrature]). Unfortunately, it does not give the optimal convergence rate for functions from ${\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}})$, see Remark 2.3 below.

In [3], for the weight $w(x)=\exp\left(-|x|^{\lambda}\right)$, the authors proposed truncated Gaussian quadratures which not only improve the convergence rate but also give the asymptotic order of ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}})\big{)}$ as shown in Theorem 2.2 below. Let us introduce in the same manner truncated Gaussian quadratures for the weight $w(x)$ with any $a>0$ and $b\in{\mathbb{R}}$.

Throughout this paper, we fix a number $\theta$ with $0<\theta<1$, and denote by $j(m)$ the smallest integer satisfying $x_{m,j(m)}\geq\theta a_{m}$. It is useful to remark that

$$d_{m,k}\,\asymp\,\frac{a_{m}}{m}\asymp m^{1/\lambda-1},\ \ |k|\leq j(m);\quad x_{m,j(m)}\,\asymp\,m^{1/\lambda},$$

(2.5)

where $d_{m,k}:=x_{m,k}-x_{m,k-1}$ is the distance between consecutive zeros of the polynomial $p_{m}(w)$. These relations were proven in [3, (13)] for the weight $w(x)=\exp\left(-|x|^{\lambda}\right)$. From their proofs there, one can easily see that they are still hold true for the general case of the weight $w$. By (2.2) and (2.5), for $m$ sufficiently large we have that

$$Cm\leq j(m)\leq m/2$$

(2.6)

with a positive constant $C$ depending on $\lambda,a,b$ and $\theta$ only.

For a continuous function on ${\mathbb{R}}$, consider the truncated Gaussian quadrature

$$Q^{\rm{TG}}_{2j(m)}f:=\sum_{|k|\leq j(m)}\lambda_{m,k}(w)f(x_{m,k}).$$

(2.7)

Notice that the number $2j(m)$ of samples in the quadrature $Q^{\rm{TG}}_{2j(m)}f$ is strictly smalller than $m$ – the number of samples in the quadrature $Q^{\rm G}_{m}f$. However, due to (2.6) it has the asymptotic order as $2j(m)\asymp m$ when $m$ going to infinity.

The lower bound in (2.8) is already contained in Theorem 3.8 below. Since its proof is much simpler for the case $d=1$, let us proccess it separately. In order to prove the lower bound in (2.8) we will apply Lemma 2.1. Let $\left\{\xi_{1},...,\xi_{n}\right\}\subset{\mathbb{R}}$ be arbitrary $n$ points. For a given $n\in{\mathbb{N}}$, we put $\delta=n^{1/\lambda-1}$ and $t_{j}=\delta j$, $j\in{\mathbb{N}}_{0}$. Then there is $i\in{\mathbb{N}}$ with $n+1\leq i\leq 2n+2$ such that the interval $(t_{i-1},t_{i})$ does not contain any point from the set $\left\{\xi_{1},...,\xi_{n}\right\}$. Take a nonnegative function $\varphi\in C^{\infty}_{0}([0,1])$, $\varphi\not=0$, and put

$$b_{0}:=\int_{0}^{1}\varphi(y){\rm d}y>0,\quad b_{s}:=\int_{0}^{1}|\varphi^{(s)}(y)|{\rm d}y,\ s=1,...,r.$$

Define the functions $g$ and $h$ on ${\mathbb{R}}$ by

$$g(x):=\begin{cases}\varphi(\delta^{-1}(x-t_{i-1})),&\ \ x\in(t_{i-1},t_{i}),\\ 0,&\ \ \text{otherwise},\end{cases}$$

and

$$h(x):=(gw^{-1})(x).$$

Let us estimate the norm $\left\|{h}\right\|_{W^{r}_{1,w}({\mathbb{R}})}$. For a given $k\in{\mathbb{N}}_{0}$ with $0\leq k\leq r$, we have

$$h^{(k)}=(gw^{-1})^{(k)}=\sum_{s=0}^{k}\binom{k}{s}g^{(k-s)}(w^{-1})^{(s)}.$$

(2.10)

By a direct computation we find that for $x\in{\mathbb{R}}$,

$$(w^{-1})^{(s)}(x)=(w^{-1})(x)(\operatorname{sign}(x))^{s}\sum_{j=1}^{s}c_{s,j}(\lambda,a)|x|^{\lambda_{s,j}},$$

(2.11)

where $\operatorname{sign}(x):=1$ if $x\geq 0$, and $\operatorname{sign}(x):=-1$ if $x<0$,

$$\lambda_{s,s}=s(\lambda-1)>\lambda_{s,s-1}>\cdots>\lambda_{s,1}=\lambda-s,$$

(2.12)

and $c_{s,j}(\lambda,a)$ are polynomials in the variables $\lambda$ and $a$ of degree at most $s$ with respect to each variable. Hence, we obtain

$$h^{(k)}(x)w(x)=\sum_{s=0}^{k}\binom{k}{s}g^{(k-s)}(x)(\operatorname{sign}(x))^{s}\sum_{j=1}^{s}c_{s,j}(\lambda,a)|x|^{\lambda_{s,j}}$$

(2.13)

which implies that

$$\int_{{\mathbb{R}}}|h^{(k)}w|(x){\rm d}x\leq C\max_{0\leq s\leq k}\ \max_{1\leq j\leq s}\int_{t_{i-1}}^{t_{i}}|x|^{\lambda_{s,j}}|g^{(k-s)}(x)|{\rm d}x.$$

From (2.12), the inequality $n^{1/\lambda}\leq x\leq(2n+2)n^{1/\lambda-1}$ and

$$\int_{t_{i-1}}^{t_{i}}|g^{(k-s)}(x)|{\rm d}x=b_{k-s}\delta^{-k+s+1}=b_{k-s}n^{(k-s-1)(1-1/\lambda)},$$

we derive

	$\displaystyle\int_{{\mathbb{R}}}|h^{(k)}w|(x){\rm d}x$	$\displaystyle\leq C\max_{0\leq s\leq k}\int_{t_{i-1}}^{t_{i}}|x|^{\lambda_{s,s}}|g^{(k-s)}(x)|{\rm d}x$
		$\displaystyle\leq C\max_{0\leq s\leq k}\left(n^{1/\lambda}\right)^{s(\lambda-1)}\int_{t_{i-1}}^{t_{i}}|g^{(k-s)}(x)|{\rm d}x$
		$\displaystyle\leq C\max_{0\leq s\leq k}n^{s(\lambda-1)/\lambda}n^{(k-s-1)(1-1/\lambda)}$
		$\displaystyle=Cn^{(1-1/\lambda)(k-1)}\leq Cn^{(1-1/\lambda)(r-1)}.$

If we define

$$\bar{h}:=C^{-1}n^{-(1-1/\lambda)(r-1)}h,$$

then $\bar{h}$ is nonnegative, $\bar{h}\in{\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}})$, $\sup\bar{h}\subset(t_{i-1},t_{i})$ and

	$\displaystyle\int_{{\mathbb{R}}}(\bar{h}w)(x){\rm d}x$	$\displaystyle=C^{-1}n^{-(1-1/\lambda)(r-1)}\int_{t_{i-1}}^{t_{i}}g(x){\rm d}x$
		$\displaystyle=C^{-1}n^{-(1-1/\lambda)(r-1)}b_{0}\delta\gg n^{-(1-1/\lambda)r}$

Since the interval $(t_{i-1},t_{i})$ does not contain any point from the set $\left\{\xi_{1},...,\xi_{n}\right\}$, we have $\bar{h}(\xi_{k})=0$, $k=1,...,n$. Hence, by Lemma 2.1,

$${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}})\big{)}\geq\int_{{\mathbb{R}}}\bar{h}(x)w(x){\rm d}x\gg n^{-r_{\lambda}}.$$

The lower bound in (2.8) is proven.      

In this section, for high-dimensional numerical integration ($d\geq 2$), we prove upper and lower bounds of ${\rm Int}_{n}\big{(}{\boldsymbol{W}}^{r}_{1,w}({\mathbb{R}}^{d})\big{)}$ and construct quadratures based on step-hyperbolic-cross grids of integration nodes which give the upper bounds. To do this we need some auxiliary lemmata.

Fix $\tau>\lambda$ and define for ${\boldsymbol{x}}=(x_{1},x_{2})$,

$$v({\boldsymbol{x}}):=\exp(-a|x_{1}|^{\tau}+b)\exp(-a|x_{2}|^{\tau}+b).$$

We preliminarly prove that $W^{r}_{1,w}({\mathbb{T}}^{2})$ is continuously embbeded into the space $C_{v}({\mathbb{T}}^{2})$ where ${\mathbb{T}}^{2}:=[-T,T]^{2}$, $T$ is any positive number and $C_{v}({\mathbb{T}}^{2})$ is the Banach space of continuous functions $f$ on ${\mathbb{T}}^{2}$ equipped with the norm

$$\left\|{f}\right\|_{C_{v}({\mathbb{T}}^{2})}:=\max_{{\boldsymbol{x}}\in{\mathbb{T}}^{2}}|(vf)({\boldsymbol{x}})|.$$

Since the subspace $C^{\infty}_{0}({\mathbb{T}}^{2})$ of infinitely differentiable functions with compact support is dense in both the Banach spaces $C({\mathbb{T}}^{2})$ and $W^{1}_{1,w}({\mathbb{T}}^{2})$, to prove this continuous embbeding, it is sufficient to show the inequality

$$\left\|{f}\right\|_{C_{v}({\mathbb{T}}^{2})}\ll\left\|{f}\right\|_{W^{1}_{1,w}({\mathbb{T}}^{2})},\ \ f\in C^{\infty}_{0}({\mathbb{T}}^{2}).$$

(3.1)

For ${\boldsymbol{k}}\in{\mathbb{N}}_{0}^{2}$, denote by $D^{\boldsymbol{k}}g$ the ${\boldsymbol{k}}$th partial derivative of $g$. Taking a function $f\in C^{\infty}_{0}({\mathbb{T}}^{2})$, we have that for ${\boldsymbol{x}}\in{\mathbb{T}}^{2}$,

$$(vf)({\boldsymbol{x}})=\int_{-T}^{x_{1}}\int_{-T}^{x_{2}}D^{(1,1)}(vf)({\boldsymbol{t}}){\rm d}{\boldsymbol{t}},$$

and

	$\displaystyle D^{(1,1)}(vf)({\boldsymbol{x}})$	$\displaystyle=v({\boldsymbol{x}})\big{[}D^{(1,1)}f({\boldsymbol{x}})-a\tau\operatorname{sign}(x_{1})|x_{1}|^{\tau-1}D^{(1,0)}f({\boldsymbol{x}})$
		$\displaystyle\ \ \ -a\tau\operatorname{sign}(x_{2})|x_{2}|^{\tau-1}D^{(0,1)}f({\boldsymbol{x}})+a^{2}\tau^{2}\operatorname{sign}(x_{1})|x_{1}|^{\tau-1}\operatorname{sign}(x_{2})|x_{2}|^{\tau-1}f({\boldsymbol{x}})\big{]}.$

Hence by using the inequality $\tau>\lambda$ we derive (3.1):

	$\displaystyle\left\|{f}\right\|_{C_{v}({\mathbb{T}}^{2})}$	$\displaystyle\ll\int_{{\mathbb{T}}^{2}}\big{|}\big{(}vD^{(1,1)}f\big{)}({\boldsymbol{x}})\big{|}{\rm d}{\boldsymbol{x}}+\int_{{\mathbb{T}}^{2}}\big{|}\big{(}vD^{(1,0)}f\big{)}({\boldsymbol{x}})\big{|}|x_{1}|^{\tau-1}{\rm d}{\boldsymbol{x}}$
		$\displaystyle+\int_{{\mathbb{T}}^{2}}\big{|}\big{(}vD^{(0,1)}f\big{)}({\boldsymbol{x}})\big{|}|x_{2}|^{\tau-1}{\rm d}{\boldsymbol{x}}+\int_{{\mathbb{T}}^{2}}\big{|}\big{(}vf\big{)}({\boldsymbol{x}})\big{|}|x_{1}x_{2}|^{\tau-1}{\rm d}{\boldsymbol{x}}$
		$\displaystyle\ll\int_{{\mathbb{T}}^{2}}\big{|}\big{(}wD^{(1,1)}f\big{)}({\boldsymbol{x}})\big{|}{\rm d}{\boldsymbol{x}}+\int_{{\mathbb{T}}^{2}}\big{|}\big{(}wD^{(1,0)}f\big{)}({\boldsymbol{x}})\big{|}{\rm d}{\boldsymbol{x}}$
		$\displaystyle+\int_{{\mathbb{T}}^{2}}\big{|}\big{(}wD^{(0,1)}f\big{)}({\boldsymbol{x}})\big{|}{\rm d}{\boldsymbol{x}}+\int_{{\mathbb{T}}^{2}}\big{|}\big{(}wf\big{)}({\boldsymbol{x}})\big{|}{\rm d}{\boldsymbol{x}}=\left\|{f}\right\|_{W^{1}_{1,w}({\mathbb{T}}^{2})}$

From the continuous embbeding of $W^{r}_{1,w}({\mathbb{T}}^{2})$ into $C_{v}({\mathbb{T}}^{2})$ it follows that any function $f\in W^{r}_{1,w}({\mathbb{T}}^{2})$ is equivalent in sense of the Lebesgue measure to a continuous (not necessarily bounded) function on ${\mathbb{T}}^{2}$. Hence we obtain the claim of the lemma for $p=1$ since $T$ has been taken arbitrarily and the restriction of a function $f\in W^{r}_{1,w}({\mathbb{R}}^{2})$ to ${\mathbb{T}}^{2}$ belongs to $W^{r}_{1,w}({\mathbb{T}}^{2})$.