Nonconforming approximation methods for function reconstruction on general polygonal meshes via orthogonal polynomials

In this work we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials and can reproduce higher degree polynomials. This setting naturally arises in applications where pointwise evaluations are inaccessible and only integral measurements over subdomains may be available. We develop a unisolvence theory and derive necessary and sufficient conditions for the associated approximation spaces to be unisolvent. Specifically, it is shown that unisolvence depends on the parity of the product of the polynomial degree $m$ and the number of polygon edges $N$. When this condition is not satisfied, we introduce an enrichment strategy involving an additional linear functional and a suitably designed enrichment function to ensure unisolvence. Numerical experiments confirm the accuracy of the proposed methods.