Closeness of Functions. Approximation in the Mean. Fourier Expansions

Abstract

It is well known that most problems of applied mechanics either cannot be solved rigorously, or the effort to obtain such solutions is too great to justify the work expended. Under these circumstances, there is no choice but to employ methods leading to approximate solutions. The question then arises concerning the evaluation of the accuracy of an approximate solution, that is, the closeness of the approximating function-vector to the exact solution-vector.