Double Cosine and Cosine-Sine Fourier Transforms and Generalized Lipschitz Classes in Uniform Metric

Abstract

For complex-valued functions $f \in L^1(\mathbb{R}^2_+)$, where $\mathbb{R}_+ := [0,\infty)$ we give
sufficient conditions under which the double cosine or cosine-sine Fourier transform of
$f$ belongs to a generalized Lipschitz class defined by the mixed modulus of smoothness
of orders $m,n \in \mathbb{N} = \{1,2,\cdots \}$ in uniform metric. The sharpness of these conditions
is established under some restriction for non-negative functions.