approximation theory

approximation theory A subject that is concerned with the approximation of a class of objects, say F, by a subclass, say P ⊂ F, that is in some sense simpler. For example, let F = C [a,b],

the real continuous functions on [a,b], then a subclass of practical use is P_n, i.e. polynomials of degree n. The means of measuring the closeness or accuracy of the approximation is provided by a metric or norm. This is a nonnegative function that is defined on F and measures the size of its elements. Norms of particular value in the approximation of mathematical functions (for computer subroutines, say) are the Chebyshev norm and the 2-norm (or Euclidean norm). For functions f ∈ C [a,b]

these norms are given respectively as

For approximation of data these norms take the discrete form

The 2-norm frequently incorporates a weight function (or weights). From these two norms the problems of Chebyshev approximation and least squares approximation arise. For example, with polynomial approximation we seek p_n ∈ P_n

for which ||f–p_n|| or ||f–p_n||₂

are acceptably small. Best approximation problems arise when, for example, we seek p_n ∈ P_n

for which these measures of errors are as small as possible with respect to P_n.

Other examples of norms that are particularly important are vector and matrix norms. For n-component vectors x = (x₁,x₂,…,x_n)^T

important examples are

Corresponding to a given vector norm, a subordinate matrix norm can be defined for n × n matrices A, by

For the vector norm

this reduces to the expression

where a_ij is the i,jth element of A. Vector and matrix norms are indispensible in most areas of numerical analysis.