  Motivated by polynomial approximations of differential forms, we study
analytical and numerical properties of a polynomial interpolation problem that
relies on function averages over interval segments. The usage of segment data
gives rise to new theoretical and practical aspects that distinguish this
problem considerably from classical nodal interpolation. We will analyse
fundamental mathematical properties of this problem as existence, uniqueness
and numerical conditioning of its solution. We will provide concrete conditions
for unisolvence, explicit Lagrange-type basis systems for its representation,
and a numerical method for its solution. To study the numerical conditioning,
we will provide concrete bounds of the Lebesgue constant in a few distinguished
cases.
