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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’skiĭ-type inequalities) 42A10 Trigonometric approximation 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
##### Keywords:
Jackson type estimate; Nikol’skij inequality