Estimation of Dirichlet kernel difference in the norm of L

This paper is related to the problem of estimation of the norm of trigonometrical polynomials through their coefficients in L. It is proved that the norm of the difference of Dirichlet’s kernels in L has the precise order ln(n − m) and the lower estimate is also valid with the coefficient 4/π ². A theorem and two lemmas are presented showing that the coefficient c at ln(n − m) in an asymptotic estimate uniform with respect to m and n may be greater than 4/π ² and its value in examples depends on arithmetic properties of n and m.