Non-orthogonal interpolation on closed interval and convergence

Abstract

Building upon the concept of discretely orthogonal bases, this paper develops a generalized interpolation framework, with the classical Lagrange interpolation method serving as a special case. Specifically, for an arbitrary number of specific non-equidistant interpolation nodes, this paper constructs corresponding discretely orthogonal polynomial bases, whose associated orthogonal matrices coincide with the well-known Discrete Cosine Transforms (DCTs). Using these polynomial bases, we show that when interpolation nodes are chosen as extended Chebyshev nodes, the interpolation of continuous functions converge in the square-integrable sense. Furthermore, we prove that the resulting interpolation functions based on extended Chebyshev nodes exhibit uniform convergence in the Hölder continuity class. These results not only provide a rigorous theoretical foundation for polynomial-based signal representation in digital conditioning of sensors, but also suggest a viable candidate for spectral-type approach for numerical schemes for partial differential equations (PDEs).