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 \(\Omega_r(f, 2^{- n})_p\).


42B99 Harmonic analysis in several variables