Inequalities of Jackson type and multipliers in \(L_ p\) (Q1907447)

The author's aim is to determine the set of all collections of the parameters \((p, q, r, m, d)\) and vectors \(\alpha\in I^d\) such that the following inequality holds: \[ E_{n-1} (f)_q\leq Yn^{-r+ (1/p- 1/q)}+ \max_j |\Delta^m_{2\pi \alpha_j/ n} f^{(r)} |_p, \tag{1} \] where \(Y> 0\), \(f\in L^r_p\) and the further notations are traditional. Among others he proves that if \(r\geq m\gamma\), \(1< p\leq \infty\), and \(1\leq q< \infty\), or if \(r> m\gamma+ \gamma\) and \(1\leq p, q\leq \infty\), where \(\gamma>0\), \(r,m,d\in \mathbb{N}\) and \(d\geq 1/\gamma\), then inequality (1) holds provided that \((\alpha_0, \dots, \alpha_d)\) has the property \[ \liminf_{k\in \mathbb{N}} k^\gamma \max_{1\leq \ell\leq d} |k\alpha_\ell/ \alpha_0 |>0, \tag{2} \] where \(|\alpha |:= \min(\{ \alpha \}, 1-\{ \alpha \})\), and \(\{ \alpha \}\) denotes the fractional part of \(\alpha\). He also verifies that if \(0< r< m\gamma\) and \(1\leq p, q\leq \infty\), or if \(r= m\gamma\), \(d=1\), then there exists a vector \((\alpha_0, \dots, \alpha_d)\) satisfying (2) such that (1) fails. Two theorems concerning the properties of multipliers in the spaces \(L_p\) are also proved.