The problem of Timan on the precise order of the best approximations of multivariate functions (Q2739140)

Let \(M_{\sigma p}(\mathbb R^n), (1\leq p\leq\infty)\) be the collection of entire functions of exponential spherical type \(\sigma >0\) which are included in \(L_p(\mathbb R^n)\) and let \(f\in L_p(\mathbb R^n).\) The best approximation of \(f\) by elements in \(M_{\sigma p}(\mathbb R^n)\) is denoted by \(A_\sigma(f)_p,\) and the modulus of continuity of \(f\) of order \(k\) in the metric of \(L_p(\mathbb R^n)\) is denoted by \(\Omega^k(f,\cdot)_p\). We recall that NEWLINE\[NEWLINE\Omega^k(f,\delta)_p=\sup_{h\in\mathbb R^n,|h|=1}\sup_{|t|\leq\delta} \|\Delta_{th}^kf\|_p,\quad \Delta_h^kf(x)=\sum_{\ell=0}^k(-1)^{\ell+k}C_k^\ell f(x+\ell k).NEWLINE\]NEWLINE The main result: \(\Omega^k(f,\frac 1\sigma)_p=O(A_\sigma(f)_p), \sigma\rightarrow \infty,\) if and only if \(\Omega^k(f,\delta)_p=O(\Omega^{k+1}(f,\delta)_p), \delta\rightarrow 0.\) This result gives an answer to a problem of Timan and in the case \(n=1\) the problem was solved previously by \textit{R. K. S. Rathore} [J. Approximation Theory 77, No. 2, 153-166 (1994; Zbl 0809.41025)].