  In this paper, we provide explicit upper and lower bounds for the argument of
the Riemann zeta-function and its antiderivatives in the critical strip under
the assumption of the Riemann hypothesis. This extends the previously known
bounds for these quantities on the critical line (and sharpens the error terms
in such estimates). Our tools come not only from number theory, but also from
Fourier analysis and approximation theory. An important element in our strategy
is the ability to solve a Fourier optimization problem with constraints,
namely, the problem of majorizing certain real-valued even functions by
bandlimited functions, optimizing the $L^1(\mathbb{R})-$error. Deriving
explicit formulae for the Fourier transforms of such optimal approximations
plays a crucial role in our approach.
