  Context. Several numerical problems require the interpolation of discrete
data that present various types of discontinuities. The radiative transfer is a
typical example of such a problem. This calls for high-order well-behaved
techniques to interpolate both smooth and discontinuous data. Aims. The final
aim is to propose new techniques suitable for applications in the context of
numerical radiative transfer. Methods. We have proposed and tested two
different techniques. Essentially non-oscillatory (ENO) techniques generate
several candidate interpolations based on different substencils. The smoothest
candidate interpolation is determined from a measure for the local smoothness,
thereby enabling the essential non-oscillatory property. Weighted ENO (WENO)
techniques use a convex combination of all candidate substencils to obtain
high-order accuracy in smooth regions while keeping the essentially
non-oscillatory property. In particular, we have outlined and tested a novel
well-performing fourth-order WENO interpolation technique for both uniform and
nonuniform grids. Results. Numerical tests prove that the fourth-order WENO
interpolation guarantees fourth-order accuracy in smooth regions of the
interpolated functions. In the presence of discontinuities, the fourth-order
WENO interpolation enables the non-oscillatory property, avoiding oscillations.
Unlike B\'ezier and monotonic high-order Hermite interpolations, it does not
degenerate to a linear interpolation near smooth extrema of the interpolated
function. Conclusions. The novel fourth-order WENO interpolation guarantees
high accuracy in smooth regions, while effectively handling discontinuities.
This interpolation technique might be particularly suitable for several
problems, including a number of radiative transfer applications such as
multidimensional problems, multigrid methods, and formal solutions.
