This paper investigates the effect of basis selection on the derivation of numerical discrete schemes. In many numerical methods, especially those derived via interpolation and collocation techniques, such as Trigonometric Series, Orthogonal Functions, Polynomials and Power series, the choice of basis function is often assumed to influence the resulting discrete formulation. However, this study establishes that, irrespective of the kind of basis employed, the resulting numerical discrete scheme remains invariant provided that interpolation, collocation, and evaluation are performed at the same set of points. The invariance underscores the fundamental role of the interpolation and collocation nodes, rather than the basis itself, in determining the final scheme. The findings offer a unified perspective on scheme construction, reducing computational redundancy and strengthening the theoretical understanding of discrete approximations in numerical analysis.
Keywords: Numerical Discrete Scheme, Interpolaton, Collocation point, Basis-Invaroant, evaluation Point.
Citation: Subair, A.O. (2025). Basis-Invariant Derivation of Numerical Discrete Schemes via Interpolation, Collocation and Evaluation at Identical Points. Int J Math Expl & Comp Edu.2(3):1-4.