Polynomial Interpolation of Function Averages on Interval Segments

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 analyze fundamental mathematical properties of this problem as existence, uniqueness, and numerical conditioning of its solution. In a few selected scenarios, we will provide concrete conditions for unisolvence and explicit Lagrange-type basis systems for its representation. To study the numerical conditioning, we will provide respective concrete bounds for the Lebesgue constant.