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 2norm (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 2norm 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 ncomponent vectors x = (x_{1},x_{2},…,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.
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