Extremal functions of integral functionals in $H^\omega[a,b]$

In this paper we give a solution of the discrete and continuous versions of the problem
$$ \int_a^bh(t)\psi(t)\,dt\to\sup, \quad h\in H^\omega[a,b]\colon\quad h(a)=E_1, \quad h(b)=E_2, $$
where $H^\omega[a,b]$ is the class of absolutely continuous functions on $[a,b]$ with common majorizing modulus of continuity $\omega$. We also discuss applications of the results obtained to mathematical economics (the Kantorovich–Monge mass transfer problem), approximation theory and numerical differentiation (Chebyshev $\omega$-polynomials and splines), the constructive theory of functions (inequalities for $\omega$-rearrangements), graph theory (graphs of rearrangements), and optimal control theory (the theory of total control and the Fel'dbaum–Bushaw problem).