On bounds for the characteristic functions of some degenerate multidimensional distributions

We discuss an application of an inequality for the modulus of the characteristic function of a system of monomials in random variables to the convergence of the density of the corresponding system of the sample mixed moments. We also consider the behavior of constants in the inequality for the characteristic function of a trigonometric analogue of the above-mentioned system when the random variables are independent and uniformly distributed. Both inequalities were derived earlier by the from a multidimensional analogue of Vinogradov's inequality for a trigonometric integral. As a byproduct the lower bound for the spectrum of A sub k A sub k ' is obtained, where A sub k is the matrix of coefficients of the first k+1 Chebyshev polynomials of first kind.