## Multivariate trigonometric polynomial approximations with frequencies from the hyperbolic cross.(English. Russian original)Zbl 0839.42006

Math. Notes 56, No. 3, 900-918 (1994); translation from Mat. Zametki 56, No. 3, 36-63 (1994).
The authors prove several interesting theorems pertaining to the subject given by the title. The statements and proofs of theorems need a lot of notations and notions, and hence it is not possible for the reviewer to state them succintly. Using their notations and citing their words: “The main purpose of the present paper is to provide a characterization of the spaces $${\mathcal A}^\alpha_q(L_p(\mathbb{T}^d))$$ by introducing new moduli of smoothness for functions in $$L_p(\mathbb{T}^d)$$,” their main result reads as follows: Let $$r= 1,2,\dots$$, for $$1< p< \infty$$, $$0< q\leq \infty$$, $$0< \alpha< r$$. Then $$f\in {\mathcal A}^\alpha_q(L_p(\mathbb{T}^d)$$, $$({\mathcal T}_n))$$ if and only if the quantity $\begin{cases} (\sum^\infty_{k= 0} [2^{k\alpha} \Omega^*_r(f, 2^{- k})_p]^q)^{1/q},\quad & 0< q< \infty,\\ \sup_{0\leq k< \infty} 2^{k\alpha} \Omega^*_r(f, 2^{- k})_p,\quad & q= \infty,\end{cases}\tag{$$*$$}$ is finite. Moreover, expression $$(*)$$ with $$\Omega_r(f, 2^{- k})_p$$ in place of $$\Omega^*_r(f, 2^{- k})_p$$ after adding $$|f|_p$$ is equivalent to $$|f|_{{\mathcal A}^\alpha_q(L_p(\mathbb{T}^d), ({\mathcal T}_n))}$$.
They also prove direct and inverse theorems which compare the error $$E_{2^n}'(f)_p$$ of the approximation by polynomials with harmonics of hyperbolic cross with $$\Omega_r(f, 2^{- n})_p$$.

### MSC:

 42B99 Harmonic analysis in several variables