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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 $$W_s^{\alpha}$$, $$E_s^{\alpha}$$, $$H_s^{\alpha}$$.

MSC:
 41A55 Approximate quadratures 41A05 Interpolation in approximation theory