Fluctuationlessness theorem to approximate univariate functions' matrix representations

摘要

Matrix representation of functions are required to convert an operator related problem to its algebraic counterpart over certain vectors and matrices. The problems involving operators which are purely or partially algebraic are most frequently encountered ones in applications. The algebraic operator here has a Hilbert space domain defined over square integrable univariate functions on a specified interval and its action on its argument is just multiplication by a function. We focus on univariate functions for simplicity in this very first step although the generalization to multivariance seems to be rather straightforward. The main purpose of this work is to introduce a conjecture to facilitate the numerical approximation of the matrix representation of the above algebraic operator and then to prove it to get an important theorem which seems to be capable of opening new very efficient horizons in numerical analysis and in its applications. Theorem states that the matrix representation of a univariate function is the image of the matrix representation of the independent variable under the same function for a finite Hilbert space. Illustrative numerical implementations are also given.