On an Asymptotic Property of Dirichlet Kernels

The main object of our study is the Dirichlet kernel. The properties of this trigonometric polynomial — the sum of cosines of multiple arcs — are of undoubted interest in the theory of trigonometric series. For example, the results on the asymptotic behavior of the Lebesgue constants, which are the integral norms of the Dirichlet kernels, are well known. These results are constantly being developed and generalized as applied to various systems of functions in both one-dimensional and multidimensional situations. In this paper, we find the leading term of the asymptotics for the value of the global minimum of the Dirichlet kernel as its number tends to infinity. The leading term is the product of the said number by a negative constant, which coincides with the value of the global minimum of the sinc-function (cardinal sine). The proof uses the connection between Dirichlet kernels and Chebyshev polynomials of the second kind. As can be seen from the authors’ previous works, the result undergoes quantitative changes in the transition to lacunary sums of cosines. Our interest in such constructions is caused by the problem posed several years ago by L. E. Rossovskii and A. A. Tovsultanov on calculating the spectral radius for a special one-parameter family of functional operators. The question reduces to studying the behavior of “long” products of sines with lacunae in the arguments. It is shown that the revealed asymptotic property of Dirichlet kernels turns out to be useful in a similar “non-lacunary” problem.