A theorem on the moment methods

The inner product involved in the moment methods is usually an integral, which is evaluated numerically by summing the integrand at certain discrete points. In connection with this inner product, a theorem is proved, which states that the overall number of points involved in the integration must not be smaller than the number of unknowns involved in the moment method. If these two numbers are equal, a point-matching solution is obtained, irrespective of whether one has started with Galerkin's method or the least squares method. If the number of points involved in the integration is larger than the number of the unknowns, a weighted point-matching solution is obtained.