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.

翻译：受微分形式多项式近似的启发，我们研究了一个依赖于区间段上函数平均的多项式插值问题的解析与数值性质。段数据的使用带来了新的理论与实践特性，使得这一问题与经典节点插值有显著区别。我们将分析该问题解的存在性、唯一性及数值适定性等基本数学性质，给出唯一可解性的具体条件、用于表达式的显式拉格朗日型基函数系统以及求解该问题的数值方法。为研究数值适定性，我们将提供若干典型情形下勒贝格常数的具体界。