The fundamental problem of approximation theory is to represent a possibly complicated function by simpler, easier to compute functions. In approximation theory it is usually assumed that the values of the function are known. This information is then used to construct an approximant. In numerical computation, information usually comes in a less explicit form. For example, the function may be the solution to a differential equation. Nevertheless, the two subjects of approximation and computation are closely related, and it is impossible to understand fully the possibilities in numerical computation without a good understanding of the elements of constructive approximation. This course will give an overview of basic classical approximation theory, i.e., best and good approximation from a finite family of functions in specific normed linear spaces (such as L1, L2, and C). We will study minimax approximation and the construction of good approximations (the exchange algorithm). We will also study orthogonal polynomials and least squares approximation. The results and techniques from approximation theory and numerical analysis will be applied in both the continuous and the discrete cases. The theory will be illustrated mainly by considering numerical approximation techniques by polynomials and splines.
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