On trigonometric series with \((K,S)\)-monotone coefficients in weighted spaces (Q2386961)

For an integer \(k \geq 0\) and a real sequence \(\{a_n\}_{n=0}^\infty\) denote \[ \Delta_k\, a_n = \sum_{i=0}^k (-1)^i\, {k \choose i}\, a_{n+i}\text{ and }\{\Delta\}_k\, a_n = \sum_{i=0}^k {k \choose i}\, a_{n+i}. \] A sequence \(\{a_n\}_{n=0}^\infty\) is called \((k,s)\)-monotone if \(a_n \to 0\) as \(n \to \infty\) and \(\Delta_k(\{\Delta\}_s\,a_n) \geq 0\) for all \(n \geq 0\). The paper deals with trigonometric series of the form \[ \frac{a_0}{2} + \sum_{n=1}^\infty a_n\, \cos{nx} \sim f(x) \text{ and } \frac{a_0}{2} + \sum_{n=1}^\infty a_n\, \sin{nx} \sim g(x), \] where the sequence of the coefficients \(\{a_n\}_{n=0}^\infty\) is \((k,s)\)-monotone. The author proves a number of inequalities which give two-sided estimates for the weighted \(L_p\)-norm (\(0 < p < \infty\)) of the function \(f\) or \(g\) in terms of the coefficients \(\{a_n\}_{n=0}^\infty\), in spirit of the famous Hardy-Littlewood theorem.