Structural analysis of matrix integration operators in polynomial bases

This work describes a matrix representation approach for efficient and explicit construction of generic well-conditioned matrix integration operators in terms of degree-graded and non-degree-graded polynomial bases. The main advantage of having an explicit formula for an matrix operator in a basis is that we do not need to change the basis given that conversion between bases can occasionally be unstable. We obtain and analyze the matrix integration operators and their structures for some degree-graded polynomial bases, including orthogonal bases, as well as non-degree-graded polynomial bases. The proposed framework may lead to well-structured sparse matrices which can be exploited to reduce the overall complexity in the related matrix computations including matrix–vector multiplications. It appears that applying the idea can be useful to provide an efficient method for functional integral equations.