Estimation of Dirichlet kernel difference in the norm of $\mathrm{L}$

V. O. Tonkov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2010), pp. 12-18

Résumé

This work is related to the problem of estimation of the norm of a trigonometrical polynomials through their coefficient in $\mathrm{L}$. It is proved that the norm of the difference of Dirichlet's kernels in $\mathrm{L}$ has the precise order $\ln(n-m)$ and the lower estimate is also valid with the coefficient $4/\pi^{2}$. A theorem and two lemmas are presented showing that the coefficients $c$ at $\ln(n-m)$ in an asymptotc estimate uniform with resepect to $m$ and $n$ may be greater than $4/\pi^{2}$ and its value in examples depends on arithmetic properties of $n$ and $m$.