Polynomial approximation from diffused data: unisolvence and stability

In this work, we address the problem of polynomial interpolation of non-pointwise data. More specifically, we assume that our input information comes from measurements obtained on diffuse compact domains. Although the nodal and the diffused problems are related by the mean value theorem, such an approach does not provide any concrete insights in terms of well-posedness and stability. We hence develop a different framework in which unisolvence can be again recovered from nodal results, for which a wide literature is available. To analyze the stability of the so-obtained diffused interpolation procedure, we characterize the norm of the interpolation operator in terms of a Lebesgue constant-like quantity. After analyzing some of its features, such as invariance properties and sensitivity to support overlapping, we numerically verify the theoretical findings.