Approximation of the Lebesgue constant of the Fourier operator by a logarithmic function

The Lebesgue constant of the classical Fourier operator is uniformly approximated by a family of logarithmic functions that depend on two parameters. The case where the corresponding residual term has non-monotonic behavior is considered. The obtained result of Lebesgue constant approximation by indicated family of functions strengthens the known results corresponding to cases of strict decrease and increase of the residual term. Various modifications of the logarithmic approximation are studied.