On the best approximation of Dirichlet kernels
The author uses the notations: \[ T_ n(x)= {a_ 0\over 2}+ \sum^ n_{k= 1} (a_ k \cos kx+ b_ k \sin kx);\quad D_ n(x)= {1\over 2}+ \sum^ n_{k=1} \cos kx, \] \[ \| T_ n\|_ L= {1\over \pi} \int^{2\pi}_ 0 | T_ n(x)| dx;\quad \| T_ n\|_ C= \max_ x | T_ n(x) |; \] \vskip2.0mm \[ C_ n= \sup\{\| T_ n\|_ C: \| T_ n\|_ L\leq 1\}; \] \[ \widetilde T_ n(x)= \sum^ n_{k= 1} (a_ k \sin kx- b_ k \cos kx); \quad \widetilde D_ n(x)= \sum^ n_{k= 1} \sin kx, \] \[ \widetilde C_ n= \sup\{\| \widetilde T_ n\|_ C: \| T_ n\|_ L\leq 1\}. \] The paper contains a proof of Stechkin's assertion that \(C_ n= cn+ o(n)\) with \(0.539\dots\leq c\leq 0.58\dots\), and of the author's result that \(\widetilde C_ n= \widetilde c\cdot n+ o(n)\) with \(0,43\dots\leq \widetilde c\leq 0,85\dots\;\). The title is motivated by the fact that \[ C_ n= \inf\biggl\{\| D_ n+ \varphi_ n\|_ C: \varphi_ n(x)= \sum^ \infty_{k= n+1} \alpha_ k \cos kx\biggr\}, \] \[ \widetilde C_ n= \inf\biggl\{\| \widetilde D_ n+ \psi_ n\|_ C: \psi_ n(x)= \sum^ \infty_{k= n+1} \beta_ k\sin kx\biggr\}. \]
scientific article; zbMATH DE number 589639