New Equations of Convolution Type Obtained by Replacing the Integral by Its Maximum

We study the nonlinear equation
$$ \max _{\gamma \in \mathbb R}g(\gamma )|\cos (\gamma -\alpha )| =f(\alpha ), $$
where $f(\alpha)$ is a given function and $g(\gamma)$ is the unknown function, to be found in the class of nonnegative continuous $\pi$-periodic functions. This equation arose in the context of an applied problem dealing with the construction of a hydrofoil from given pressure envelopes. Necessary and sufficient conditions for the solvability of the equation, an explicit description of the solution set, and a comparison theorem under changes of the right-hand sides are obtained. Some possible ways of generalization are indicated.