An extremal nonnegative sine polynomial
In this attractive paper the authors prove five pleasant theorems. Their synopsis is given in the abstract as follows: ``For any positive integer \(n\), the sine polynomials that are nonnegative in \([0,\pi]\) and which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials \(K_n(\theta)\) are constructed in such a way that \(\{K_n (\theta)\}\) is a summability kernel. Thus, for each \(p\), \(1\leq p\leq \infty\), and for any \(2\pi\)-periodic function \(f\in L_p[-\pi,\pi]\), the sequence of convolutions \(K_n*f\) is proved to converge to \(f\) in \(L_p[-\pi,\pi]\). The pointwise and almost everywhere convergences are also consequences of our construction''.
scientific article; zbMATH DE number 2067048
scientific article