The Best Approximation Method
Many important engineering problems fall into the category of being linear operators, with supporting boundary conditions. In this chapter, an inner-product and norm is used which enables the engineer to approximate such engineering problems by developing a generalized Fourier series. The resulting approximation is the “best” approximation in that a least-squares (L2) error is minimized simultaneously for fitting both the problem’s boundary conditions and satisfying the linear operator relationship (the governing equations) over the problem’s domain (both space and time). Because the numerical technique involves a well-defined inner product, error evaluation is readily available using Bessel’s inequality. Minimization of the approximation error is subsequently achieved with respect to a weighting of the inner product components, and the addition of basis functions used in the approximation.
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