Quadrature and interpolation formulas for tensor products of certain classes of functions. (English. Russian original) Zbl 0202.39901

Sov. Math., Dokl. 4, 240-243 (1963); translation from Dokl. Akad. Nauk SSSR 148, 1042-1045 (1963).
From the text: Let \(K\) be a normed linear space and let \(\tau_i\), \(\tau_i^{(j)}\) be certain of its elements. Let \(K^s\) be the tensor product of \(K\) by itself \(s\) times, i. e., the space of formal finite sums of the form
\[ \sum_{1\leq p\leq N} \lambda_p \tau_p^{(1)}\otimes\dots\otimes \tau_p^{(s)} \]
(where the \(\lambda_p\) are numbers), for which addition and multiplication by numbers is defined in a trivial manner and which has been factored with respect to all relations of the form
\[ \left(\sum_{p=1}^{N_1}\lambda_p^{(1)} \tau_p^{(1)}\right)\otimes\dots\otimes\left(\sum_{p=1}^{N_s}\lambda_p^{(s)} \tau_p^{(s)}\right) = \sum_{p_1=1}^{N_1}\dots \sum_{p_s=1}^{N_s} \lambda_{p_1}^{(1)}\dots \lambda_{p_s}^{(s)}\tau_{p_1}^{(1)}\dots\tau_{p_s}^{(s)}. \]
We shall consider those \(K^s\) for which it is possible to introduce a norm such that
\[ \| \tau_1 \otimes\dots\otimes \tau_s\| = \| \tau_1\| \cdots \| \tau_s\|. \]
Then we have the following
Theorem. Let \(\vartheta_\nu\) \((\nu=0,1,\dots,q)\) and \(I\) be elements of \(K\) such that for \(\alpha>0\), \(\| I\|\leq B\), \(\| \vartheta_\nu\|\leq B\), \(\| I-\vartheta_\nu\|\leq A\cdot 2^{-\nu\alpha}\), \(\theta_0=\vartheta_0\), \(\theta_\nu=\vartheta_\nu-\vartheta_{\nu-1}\) \((\nu\geq 1)\). Then
\[ \| I \otimes\dots\otimes I - \sum_{\nu_1+\dots +\nu_s\leq q} \theta_{\nu_1}\otimes\dots\otimes \theta_{\nu_s}\| \leq C(A,B,s,\alpha) \frac{q^{s-1}}{2^{\alpha q}}. \]
Moreover, the indicated sum is in fact a linear combination of terms \(\vartheta_{\nu_1}\otimes\dots\otimes \vartheta_{\nu_s}\) with \(q-s\leq \nu_1+ \dots+ \nu_s\leq q\). From this simple theorem may be deduced a series of interesting corollaries concerning quadrature and interpolation formulas for certain classes of functions, for instance \(W_s^{\alpha}\), \(E_s^{\alpha}\), \(H_s^{\alpha}\).


41A55 Approximate quadratures
41A05 Interpolation in approximation theory