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On two methods of generalization of properties of univariate function systems to their tensor product. (English. Russian original) Zbl 0924.41012
Proc. Steklov Inst. Math. 219, 25-35 (1997); translation from Tr. Mat. Inst. Steklova 219, 32-43 (1997).
Let \(X_i\), \(1\leq i\leq d\), be finite-dimensional subspaces of \(C(T)\) and \(X= \text{span}\{ \prod_{i=1}^d f_i(x_i)\), \(f_i\in X_i\), \(1\leq i\leq d\}\) is its tensor product. If for each \(X_i\), \(1\leq i\leq d\), the following Nikol’skij inequality holds: \(\| f\| _{\infty}\leq K_i M_i^{\frac{1}{q}}\| f\| _q \), \(1\leq q\leq \infty \), \(f\in X_i\), then for each \(f\in X\) and \(\text{\textbf{1}}\leq q\leq p\leq\infty\) we have \[ \| f\| _{\mathbf p}\leq \Biggl(\prod_{i=1}^d K_i M_i^{\frac{1}{q_i}-\frac{1}{p_i}}\Biggr)\| f\| _{\mathbf q} \] Authors consider operators of the form \[ T_m:\sum_{\| s\| _1\leq m}\prod_{i=1}^d(Y^i_{s(i)}-Y^i_{s(i)-1}), \] where \(Y^i_n\) is an operator, acting on a function of the variable \(x_i\) and give an estimate of the type \(\| f-T_mf\| _{\mathbf p}\leq C2^{-rm}m^{d-1}\) for \(f\in MH^r_{p}\) with some additional conditions on \(T_m\). Applications to the Fourier width of \(MW^r_p\) and \(MH^r_p\) in \(L_p\) are also given.
For the entire collection see [Zbl 0907.00017].

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
42A10 Trigonometric approximation
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy