MATHEMATICS

S. A. SMOLYAK

INTERPOLATION AND QUADRATURE FORMULAS ON THE CLASSES \(W_s^\alpha\) AND \(E_s^\alpha\)

(Presented by Academician A. M. Kolmogorov, 20 XII 1959)

Let \(W_s^\alpha(1)\) (and, respectively, \(E_s^\alpha(1)\)) denote the class of complex-valued functions \(f(x_1,\ldots,x_s)=f(\mathbf{x})\) having period 1 in each variable and expandable in the Fourier series

\[ f(\mathbf{x})=\sum_{\mathbf{m}} C_{\mathbf{m}} e^{2\pi i(\mathbf{m},\mathbf{x})}, \tag{1} \]

where

\[ \sum_{\mathbf{m}} (\overline{m}_1\ldots \overline{m}_s)^\alpha |C_{\mathbf{m}}|^2 \leqslant 1 \quad \left( \text{respectively } |C_{\mathbf{m}}|\leqslant \frac{1}{(\overline{m}_1\ldots \overline{m}_s)^\alpha} \right). \tag{2} \]

Here \(\overline{m}=\max(1,|m|)\). For the classes \(W_s^\alpha\) and \(E_s^\alpha\), see the works \((^{1,2})\).

In the present paper upper and lower estimates are given for the quantities

\[ \Delta_1(\alpha)= \min_{\mathbf{a}}\; \inf_{\varphi_k(\mathbf{x})\in L_2}\; \sup_{f(\mathbf{x})\in W_s^\alpha(1)} \int_0^1\!\cdots\!\int \left| f(\mathbf{x})-\sum_{k=1}^{N} f\!\left(\frac{k\mathbf{a}}{N}\right)\varphi_k(\mathbf{x}) \right|^2\,d\mathbf{x} \tag{3} \]

\[ \Delta_2(\alpha)= \min_{\mathbf{a}}\; \inf_{\varphi_k(\mathbf{x})\in L_2}\; \sup_{f(\mathbf{x})\in E_s^\alpha(1)} \int_0^1\!\cdots\!\int \left| f(\mathbf{x})-\sum_{k=1}^{N} f\!\left(\frac{k\mathbf{a}}{N}\right)\varphi_k(\mathbf{x}) \right|^2\,d\mathbf{x}. \]

From \(W_s^\alpha(1)\subset E_s^\alpha(1)\) it follows at once that \(\Delta_1(\alpha)\leqslant \Delta_2(\alpha)\). Conversely, from \(f(\mathbf{x})\in E_s^\alpha(1)\) it follows that, for any \(\varepsilon>0\), \(\frac{1}{C(\varepsilon)}f(\mathbf{x})\in W_s^{\alpha-1/2-\varepsilon}(1)\), whence \(\Delta_2(\alpha)\leqslant C^2(\varepsilon)\Delta_1(\alpha-1/2-\varepsilon)\). Using this inequality, one can obtain estimates for \(\Delta_2(\alpha)\) from estimates for \(\Delta_1(\alpha)\); however, by this method only less complete results are obtained for \(\Delta_2(\alpha)\) (see Theorem 2).

Theorem 1.

\[ \frac{1}{2N^\alpha}\leqslant \Delta_1(\alpha)\leqslant C(\alpha,s)\frac{\ln^{\alpha(2s-1)}N}{N^\alpha} \quad \text{for } \alpha\geqslant 1,\ s\geqslant 2. \tag{4} \]

The upper estimate holds for \(N\) prime.

Proof. Let \(\varphi_k(\mathbf{x})\) be expanded in the Fourier series convergent in the mean,
\[ \varphi_k(\mathbf{x})=\sum_{\mathbf{n}} C_{\mathbf{n},k}e^{2\pi i(\mathbf{n},\mathbf{x})}, \]
and let the series (1) converge to \(f(\mathbf{x})\) at the points

\[ \mathbf{x}=\vec{\xi}_k=\frac{k\mathbf{a}}{N} \quad (k=1,2,\ldots,N;\ \mathbf{a}\text{ is an integer vector}). \]
Then, using the notation \(\delta_{\mathbf{m}\mathbf{n}}=0\) for \(\mathbf{m}\ne\mathbf{n}\), \(\delta_{\mathbf{m}\mathbf{m}}=1\), one can show that

\[ \int_0^1\!\cdots\!\int \left| f(\mathbf{x})-\sum_{k=1}^{N} f(\vec{\xi}_k)\varphi_k(\mathbf{x}) \right|^2 d\mathbf{x} = \sum_{\mathbf{n}} \left| \sum_{\mathbf{m}} C_{\mathbf{m}} \left\{ \sum_{k=1}^{N} C_{\mathbf{n}k} e^{2\pi i(\mathbf{m},\vec{\xi}_k)} -\delta_{\mathbf{m}\mathbf{n}} \right\} \right|^2 = \sum_{\mathbf{n}} \left| \sum_{\mathbf{m}} C_{\mathbf{m}}\lambda_{\mathbf{m}\mathbf{n}} \right|^2 . \tag{5} \]

Let us note that, for any fixed $\mathbf n$,

\[ \Delta_1(\alpha)\geq \min_{\mathbf a}\ \inf_{\varphi_k(\mathbf x)\in L_2}\ \sup_{f(\mathbf x)\in W_s^\alpha(1)} \left|\sum_{\mathbf m} C_{\mathbf m}\lambda_{\mathbf{mn}}\right|^2 = \min_{\mathbf a}\ \inf_{\varphi_k(\mathbf x)\in L_2}\sum_{\mathbf m} \frac{|\lambda_{\mathbf{mn}}|^2}{(\bar m_1\ldots \bar m_s)^{2\alpha}} . \tag{6} \]

But

\[ \begin{aligned} \sum_{\mathbf m}\frac{|\lambda_{\mathbf{mn}}|^2}{(\bar m_1\ldots \bar m_s)^{2\alpha}} &= \sum_{k,l=1}^{N} C_{\mathbf n,k}\bar C_{\mathbf n,l} \left( \sum_{\mathbf m} \frac{e^{\frac{2\pi i (k-l)(\mathbf m,\mathbf a)}{N}}} {(\bar m_1\ldots \bar m_s)^{2\alpha}} \right) \\ &\quad -\sum_{k=1}^{N} \frac{ C_{\mathbf n,k}e^{\frac{2\pi i k(\mathbf n,\mathbf a)}{N}} + \partial_{\mathbf n,k}e^{-\frac{2\pi i k(\mathbf n,\mathbf a)}{N}} } {(\bar n_1\ldots \bar n_s)^{2\alpha}} + \frac{1}{(\bar n_1\ldots \bar n_s)^{2\alpha}} . \end{aligned} \]

Denoting by $R_{\mathbf n}(\mathbf a)$ the minimum of the written expression with respect to $C_{\mathbf n1},\ldots,C_{\mathbf nN}$, after transformations we obtain

\[ R_{\mathbf n}(\mathbf a) = (\bar n_1\ldots \bar n_s)^{-2\alpha} - \frac{(\bar n_1\ldots \bar n_s)^{-4\alpha}} {\displaystyle \sum_{\substack{(\mathbf m,\mathbf a)\equiv(\mathbf n,\mathbf a)\,(\bmod N)}} (\bar m_1\ldots \bar m_s)^{-2\alpha}} = \frac{1} {\displaystyle (\bar n_1\ldots \bar n_s)^{2\alpha} + \left( \sum_{\substack{(\mathbf m,\mathbf a)\equiv(\mathbf n,\mathbf a)\\ \mathbf m\ne \mathbf n}} (\bar m_1\ldots \bar m_s)^{-2\alpha} \right)^{-1}} . \tag{7} \]

Therefore

\[ R_{\mathbf n}(\mathbf a)\geq \frac{1}{(\bar n_1\ldots \bar n_s)^{2\alpha}+(\bar m_1\ldots \bar m_s)^{2\alpha}} \tag{8} \]

for any $\mathbf m\ne \mathbf n$, $(\mathbf m,\mathbf a)\equiv(\mathbf n,\mathbf a)\pmod N$. Using Dirichlet’s principle, for $s\geq 2$ one can prove that for every $\mathbf a$ there exist $\mathbf m$ and $\mathbf n$ with these properties, and moreover $\bar m_1\ldots \bar m_s\leq \sqrt N$, $\bar n_1\ldots \bar n_s\leq \sqrt N$. Then the first of inequalities (4) will follow from (6) and (8).

To obtain the upper estimate in (4), let us note that, by virtue of (2),

\[ \sum_{\mathbf n}\left|\sum_{\mathbf m} C_{\mathbf m}\lambda_{\mathbf{mn}}\right|^2 \leq \sum_{\mathbf n}\sum_{\mathbf m} \frac{|\lambda_{\mathbf{mn}}|^2}{(\bar m_1\ldots \bar m_s)^{2\alpha}} . \tag{9} \]

Therefore, by virtue of (5), and also in view of the fact that the inner sum in (9) depends only on $C_{\mathbf n1},\ldots,C_{\mathbf nN}$,

\[ \Delta_1(\alpha) \leq \min_{\mathbf a}\ \inf_{\varphi_k(\mathbf x)\in L_2} \sum_{\mathbf n}\sum_{\mathbf m} \frac{|\lambda_{\mathbf{mn}}|^2}{(\bar m_1\ldots \bar m_s)^{2\alpha}} = \min_{\mathbf a}\sum_{\mathbf n}\inf_{C_{\mathbf n k}} \sum_{\mathbf m} \frac{|\lambda_{\mathbf{mn}}|^2}{(\bar m_1\ldots \bar m_s)^{2\alpha}} = \min_{\mathbf a}\sum_{\mathbf n} R_{\mathbf n}(\mathbf a). \tag{10} \]

Using (7), we may write

\[ \begin{aligned} \Delta_1(\alpha) &\leq \min_{\mathbf a}\sum_{\mathbf n} \left\{ (\bar n_1\ldots \bar n_s)^{-2\alpha} - \frac{(\bar n_1\ldots \bar n_s)^{-4\alpha}} {\displaystyle \sum_{\substack{(\mathbf m,\mathbf a)\equiv(\mathbf n,\mathbf a)}} (\bar m_1\ldots \bar m_s)^{-2\alpha}} \right\} \\ &= \min_{\mathbf a}\sum_{\mu=0}^{N-1} \left\{ \sum_{(\mathbf n,\mathbf a)\equiv \mu} (\bar n_1\ldots \bar n_s)^{-2\alpha} - \frac{ \displaystyle \sum_{(\mathbf n,\mathbf a)\equiv \mu} (\bar n_1\ldots \bar n_s)^{-4\alpha} } { \displaystyle \sum_{(\mathbf n,\mathbf a)\equiv \mu} (\bar n_1\ldots \bar n_s)^{-2\alpha} } \right\}. \end{aligned} \tag{11} \]

From the trivial inequality

\[ u_1+u_2+\ldots-\frac{u_1^2+u_2^2+\ldots}{u_1+u_2+\ldots} \leq 2(u_2+u_3+\ldots), \]

valid for \(u_1\geq u_2\geq\cdots\geq 0\), and from (11) we obtain

\[ \Delta_1(\alpha)\leq 2\min_{\mathbf a}\sum_{\mu=0}^{N-1} \sum_{(\mathbf n,\mathbf a)\equiv\mu}'(\bar n_1\ldots \bar n_s)^{-2\alpha}, \tag{12} \]

where the prime means that the largest term in the sum has been omitted (or one of the largest, if there are several). Put \(\varepsilon(\mathbf a;\mathbf n)=0\) if the term \((\bar n_1\ldots \bar n_s)^{-2\alpha}\) is omitted in (12), and \(\varepsilon(\mathbf a;\mathbf n)=1\) if it is not omitted. Put also \(\delta_N(z)=1\) if \(z\equiv 0\pmod N\), and \(\delta_N(z)=0\) otherwise. Finally, for convenience of notation, introduce the notation \(\bar n_1\ldots \bar n_s=|\mathbf n|\), and write \(\mathbf m<\mathbf n\) if \(\mathbf m\ne\mathbf n\) and \(|\mathbf m|\leq |\mathbf n|\). Then, from the rule for omitting terms in (12), it follows that
\(\varepsilon(\mathbf a;\mathbf n)\leq \sum_{\mathbf m<\mathbf n}\delta_N((\mathbf m-\mathbf n,\mathbf a))\), and inequality (12) may be continued as follows:

\[ \begin{aligned} \Delta_1(\alpha) &\leq 2\min_{\mathbf a}\sum_{\mathbf n}\frac{\varepsilon(\mathbf a;\mathbf n)}{|\mathbf n|^{2\alpha}} \leq 2\min_{\mathbf a}\sum_{|\mathbf n|<N/2}\frac{\varepsilon(\mathbf a;\mathbf n)}{|\mathbf n|^{2\alpha}} +2\sum_{|\mathbf n|\geq N/2}\frac1{|\mathbf n|^{2\alpha}} \\ &\leq 2\left(\min_{\mathbf a}\sum_{|\mathbf n|<N/2}\frac{\varepsilon(\mathbf a;\mathbf n)}{|\mathbf n|^2}\right)^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) \\ &\leq 2\left(\min_{\mathbf a}\sum_{|\mathbf n|<N/2}|\mathbf n|^{-2} \sum_{\mathbf m<\mathbf n}\delta_N((\mathbf m-\mathbf n,\mathbf a))^\alpha\right) +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) \\ &= 2\left(\min_{\mathbf a}\sum_{\substack{\mathbf m<\mathbf n\\ |\mathbf n|<N/2}} \frac{\delta_N((\mathbf m-\mathbf n,\mathbf a))}{|\mathbf n|^2}\right)^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right). \end{aligned} \tag{13} \]

Replacing in (13) \(\min_{\mathbf a}\) by the average over all possible integer vectors \(\mathbf a\), and using the fact that, for prime \(N\), from \(\mathbf m<\mathbf n,\ |\mathbf n|<N/2\) it follows that \(\mathbf m\not\equiv \mathbf n\pmod N\), we have

\[ \begin{aligned} \Delta_1(\alpha) &\leq 2\left[ \frac1{N^s}\sum_{a_1,\ldots,a_s=0}^{N-1} \sum_{\substack{\mathbf m<\mathbf n\\ |\mathbf n|<N/2}} \frac{\delta_N((\mathbf m-\mathbf n,\mathbf a))}{|\mathbf n|^2} \right]^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) \\ &= 2\left[ \frac1{N^s} \sum_{\substack{\mathbf m<\mathbf n\\ |\mathbf n|<N/2}} |\mathbf n|^{-2} \sum_{a_1,\ldots,a_s=0}^{N-1}\delta_N((\mathbf m-\mathbf n,\mathbf a)) \right]^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) \\ &= 2\left[ \frac1N \sum_{\substack{\mathbf m<\mathbf n\\ |\mathbf n|<N/2}} |\mathbf n|^{-2} \right]^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) \\ &\leq 2\left[ \frac1N \sum_{|\mathbf n|<N/2} \frac1{|\mathbf n|^2} \sum_{|\mathbf m|\leq |\mathbf n|}1 \right]^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) \\ &= 2\left[ \frac1N \sum_{|\mathbf n|<N/2} \frac1{|\mathbf n|^2} O\bigl(|\mathbf n|\ln^{s-1}|\mathbf n|\bigr) \right]^\alpha +O\left(\frac{\ln^{s-1}N}{N^{2\alpha-1}}\right) = O\left(\frac{\ln^{\alpha(2s-1)}N}{N^\alpha}\right), \end{aligned} \]

which proves the theorem. From the theorem just proved it immediately follows that

\[ \frac1{2N^\alpha}\leq \Delta_2(\alpha)\leq C(\alpha,s,\varepsilon)\, \frac{\ln^{(\alpha-1/2)(2s-1)}N}{N^{\alpha-1/2-\varepsilon}}, \qquad \varepsilon>0,\quad \alpha\geq \frac32+\varepsilon. \]

However, one can obtain a sharper result by observing that from
\(f(\mathbf x)\in E_s^{\alpha+1/2}(1)\) it follows that

\[ \sum_{\mathbf m}'\left| \frac{(\bar m_1\ldots \bar m_s)^\alpha C_{\mathbf m}} {\sqrt{\ln(\bar m_1+3)\ldots \ln(\bar m_s+3)}\, \ln\ln(\bar m_1+3)\ldots \ln\ln(\bar m_s+3)} \right|^2 \leq C_0^2(s), \]

and, considering the class of functions \(W_s^\alpha(C_0(s))\), defined by the last inequality. Defining for it \(\Delta_1'(\alpha)\) by an equality analogous to (3), and carrying out the estimates in the same way as in the preceding theorem, one can obtain

\[ \Delta_1'(\alpha)\leqslant O\left(\frac{\ln^{(2s-1)(\alpha+1)+1/2}N\cdot \ln^{2s}\ln N}{N^\alpha}\right), \qquad \alpha\geqslant 1,\quad s\geqslant 2. \]

Hence, since \(\Delta_2(\alpha)\leqslant \Delta_1'(\alpha-\tfrac12)\), we obtain

Theorem 2*.

\[ \frac{1}{2N^\alpha}\leqslant \Delta_2(\alpha)\leqslant C(\alpha,s)\frac{\ln^{(2s-1)\alpha+s}N\cdot \ln^{2s}\ln N}{N^{\alpha-1/2}}, \qquad \text{for } \alpha\geqslant \tfrac32,\quad s\geqslant 2. \]

The theorems proved make it possible, using the parallelepipedal grids introduced by N. M. Korobov, to interpolate values of functions from the classes \(W_s^\alpha\) and \(E_s^\alpha\) with greater accuracy than is allowed by ordinary interpolation formulas with a uniform grid of interpolation nodes, which in our case can give a mean-square error of order \(N^{-2\alpha/s}\).

From the theorems proved there follows, in particular, the possibility of constructing quadrature formulas for integration with an arbitrary square-summable weight

\[ \int_0^1\cdots\int f(x)\rho(x)\,dx \approx \sum_{k=1}^{N} p_k(\rho) f\left(\frac{ka}{N}\right) \tag{14} \]

with a sufficiently good remainder term. Indeed, if, for example, \(f(x)\in W_s^\alpha(1)\) and \(\rho(x)\in L_2\), and

\[ \int_0^1\cdots\int \left|f(x)-\sum_{k=1}^{N} f\left(\frac{ka}{N}\right)\varphi_k(x)\right|^2 dx \leqslant O\left(\frac{\ln^{(2s-1)\alpha}N}{N^\alpha}\right), \]

then

\[ \left| \int_0^1\cdots\int f(x)\rho(x)\,dx - \sum_{k=1}^{N} f\left(\frac{ka}{N}\right) \int_0^1\cdots\int \varphi_k(x)\rho(x)\,dx \right| \leqslant O\left(\frac{\ln^{(s-1/2)\alpha}N}{N^{\alpha/2}}\right), \]

i.e. this quadrature formula can no longer be substantially improved, since the following general theorem holds:

Theorem 3. Let \(\rho(x)\in W_s^\beta(1)\), with \(\beta\leqslant \alpha\), \(\alpha+\beta\geqslant 1\), \(s\geqslant 2\). Then there exist numbers \(p_k(\rho)\) and an integer vector \(a\) such that, for every \(f(x)\in W_s^\alpha(1)\),

\[ \left| \int_0^1\cdots\int f(x)\rho(x)\,dx - \sum_{k=1}^{N} p_k(\rho) f\left(\frac{ka}{N}\right) \right| \leqslant C(\alpha,\beta,s)\frac{\ln^{(\alpha+\beta)(s-1/2)}N}{N^{(\alpha+\beta)/2}}. \tag{15} \]

On the other hand, there exists a function \(\rho(x)\in W_s^\beta(1)\) such that, whatever \(a\) and \(p_k(\rho)\) may be, there will always be an \(f(x)\in W_s^\alpha(1)\) such that the left-hand side of (15) is greater than \(\tfrac12 N^{-(\alpha+\beta)/2}\).

The proof is carried out analogously to the proof of Theorem 1.

Moscow State University
named after M. V. Lomonosov

Received
14 XII 1959

References

1. N. M. Korobov, DAN, 124, No. 6, 1207 (1959).
2. S. A. Smolyak, DAN, 131, No. 1 (1960).
3. V. S. Ryabenkii, DAN, 131, No. 5 (1960).

* After the completion of this work, it became known to the author that V. S. Ryabenkii had independently obtained a somewhat more accurate upper estimate for \(\Delta_2(\alpha)\) (see (9)).