V. S. Ryaben’kii

ON TABLES AND INTERPOLATION OF FUNCTIONS FROM A CERTAIN CLASS

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

The class of functions \(E_s^\alpha(C)\) consists of all functions \(f(x_1,\ldots,x_s)\),

\[ f(x_1,\ldots,x_s)= \sum_{m_1,\ldots,m_s=-\infty}^{\infty} C(m_1,\ldots,m_s)e^{2\pi i(m_1x_1+\cdots+m_sx_s)}, \]

whose Fourier coefficients \(C(m_1,\ldots,m_s)\) satisfy the inequality

\[ |C(m_1,\ldots,m_s)| \leq \frac{C}{(\bar m_1\ldots \bar m_s)^\alpha}, \]

where \(\bar m_\nu=\max(|m_\nu|,1)\), \(\alpha>1\), and \(C\) does not depend on \(m_1,\ldots,m_s\).

Lemma 1. Let \(f\in E_s^\alpha(C)\). For every prime \(N>s\) one can specify integers \(a_1,\ldots,a_s\) such that

\[ \left| \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s)\,dx_1\ldots dx_s - \frac{1}{N}\sum_{k=1}^{N} f\left(\frac{ka_1}{N},\ldots,\frac{ka_s}{N}\right) \right| \leq A_0 C N^{-\alpha}\ln^{\alpha s}N, \]

where \(A_0=A_0(\alpha,s)\)*.

The proof is contained in the proof of Theorem 1 of the paper \((^1)\).

Lemma 2. If \(f\in E_s^\alpha(C)\), \(\bar n_1\ldots \bar n_s<N_1\), then

\[ \varphi(x_1,\ldots,x_s)\equiv f(x_1,\ldots,x_s)e^{-2\pi i(n_1x_1+\cdots+n_sx_s)} \in E_s^\alpha(A_1 C N_1^\alpha). \]

Proof. We note that

\[ \frac{\bar m}{\overline{(m-n)}\,\bar n}<A, \]

where \(A\) is a constant independent of \(m\) and \(n\). Let \(C(m_1,\ldots,m_s)\) and \(C'(m_1,\ldots,m_s)\) be the Fourier coefficients of the functions \(f\) and \(\varphi\), respectively. Then

\[ |C'(m_1,\ldots,m_s)| = |C(m_1-n_1,\ldots,m_s-n_s)| \leq \frac{C}{(\overline{m_1-n_1}\ldots \overline{m_s-n_s})^\alpha} = \]

\[ = \frac{C(\bar n_1\ldots \bar n_s)^\alpha}{(\bar m_1\ldots \bar m_s)^\alpha} \left[ \frac{\bar m_1}{\overline{(m_1-n_1)}\,\bar n_1} \cdots \frac{\bar m_s}{\overline{(m_s-n_s)}\,\bar n_s} \right]^\alpha \leq \frac{CA^{\alpha s}N_1^\alpha}{(\bar m_1\ldots \bar m_s)^\alpha}, \]

which proves the lemma.

* Here and below \(A_\nu=A_\nu(\alpha,s)\) denotes constants depending only on \(\alpha\) and \(s\).

Lemma 3. If \(1<N_1<N,\ \alpha>1\), then the estimates

\[ \sum_{\bar m_1\ldots \bar m_s\ge N_1} (\bar m_1\ldots \bar m_s)^{-\alpha} \ll A_2 N_1^{-(\alpha-1)}\ln^{s-1}N, \]

\[ \sum_{\bar m_1\ldots \bar m_s<N_1} 1 \ll A_3 N_1\ln^{s-1}N \]

hold.

Proof. The lemma is proved elementarily by induction on \(s\). Denote

\[ N_1=[\sqrt N\,\ln^{s/2}N], \]

\[ \widetilde C(m_1,\ldots,m_s) =\frac1N\sum_{k=1}^{N} f\left(\frac{ka_1}{N},\ldots,\frac{ka_s}{N}\right) e^{-2\pi i\,\frac{a_1m_1+\cdots+a_sm_s}{N}\,k} \tag{1} \]

\[ P(x_1,\ldots,x_s) = \sum_{\bar m_1\ldots \bar m_s<N_1} \widetilde C(m_1,\ldots,m_s) e^{2\pi i(m_1x_1+\cdots+m_sx_s)}. \]

The trigonometric polynomial (1) may be regarded as an interpolation polynomial for the function \(f(x_1,\ldots,x_s)\), constructed from the table of values of this function at the points
\[ \left(\frac{ka_1}{N},\ldots,\frac{ka_s}{N}\right),\quad k=1,2,\ldots,N. \]
The grid formed by these points in the space \(x_1,\ldots,x_s\) will be called parallelepipedal.

Theorem. If \(f\in E_s^\alpha(C)\), then the estimate

\[ \int_0^1\cdots\int_0^1 \left|f(x_1,\ldots,x_s)-P(x_1,\ldots,x_s)\right|^2 \,dx_1\cdots dx_s = O\!\left(N^{-\alpha+1/2}\ln^{(\alpha+1/2)s-1}N\right) \]

holds.

Proof. Applying abbreviated notation, we obtain

\[ \int |f(\mathbf x)-P(\mathbf x)|^2\,d\mathbf x = \sum_{\bar m_2\ldots \bar m_s<N_1} |C(\mathbf m)-\widetilde C(\mathbf m)|^2 + \sum_{\bar m_1\ldots \bar m_s\ge N_1} |C(\mathbf m)|^2, \tag{2} \]

where

\[ \widetilde C(\mathbf m)=\widetilde C(m_1,\ldots,m_s), \]

\[ C(\mathbf m)= \int_0^1\cdots\int_0^1 f(x_1,\ldots,x_s) e^{-2\pi i(x_1m_1+\cdots+x_sm_s)} \,dx_1\cdots dx_s. \]

By virtue of the definition of \(\widetilde C(\mathbf m)\), using Lemmas 1 and 2, we obtain

\[ |C(\mathbf m)-\widetilde C(\mathbf m)| \ll A_4 C N_1^\alpha N^{-\alpha}\ln^{\alpha s}N. \tag{3} \]

Observing that
\[ |C(\mathbf m)|^2\ll C^2(\bar m_1\ldots \bar m_s)^{-2\alpha}, \]
from (2) and (3), according to Lemma 3, we obtain

\[ \int |f(\mathbf x)-P(\mathbf x)|^2\,d\mathbf x \ll A_5C^2\left( N_1^{2\alpha+1}N^{-2\alpha}\ln^{2\alpha s+s-1}N + N_1^{-(2\alpha-1)}\ln^{s-1}N \right) = \]

\[ = O\!\left(N^{-(\alpha-1)/2}\ln^{(\alpha+1/2)s-1}N\right), \]

where the constant in the \(O\) depends on \(\alpha\), \(s\), and \(C\).

Let us compare the tables of values of functions \(f\in E_s^\alpha(C)\) at the points of a parallelepipedal grid with the tables of values of functions \(f\in E_s^\alpha(C)\) at the nodes of a cubic grid whose sides are parallel to the coordinate axes and have length \(h=1/n\). The number of all grid points lying in the unit cube,

i.e., \(N \cong n^s\). Any interpolation formula constructed from tables of values of functions \(f \in E_s^\alpha(C)\) at the nodes of a cubic grid has a remainder term of order no greater than \(O(N^{-2\alpha/s})\), which decreases with increasing \(s\). To prove this assertion it suffices to note that the function \(f_1(x_1,\ldots,x_s) \equiv 0\) and the function
\[ f_2(x_1,\ldots,x_s)=2C\frac{\sin n\pi x_1}{n^\alpha}\in E_s^\alpha(C) \]
have identical tables (identically zero), although
\[ \int |f_1(\mathbf{x})-f_2(\mathbf{x})|^2\,d\mathbf{x} =\frac{2C^2}{n^{2\alpha}} =O\left(N^{-\frac{2\alpha}{s}}\right). \]

After the completion of this work it became known to me that, independently of me, S. A. Smolyak was engaged in closely related questions (see \((^2)\)).

Received
10 XII 1959

CITED LITERATURE

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