“approximation theory”的概念、定义、翻译、参考文献-科学参考

approximation theory

Mathematics

The area of numerical analysis related to finding a simpler function f(x) which has approximately the same values as F(x) over a specified interval. Analysis can then be carried out on the more accessible f(x) knowing that the difference is small.

Computer

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
$f \in C [a, b]$
these norms are given respectively as
$‖ f ‖ = max_{a \leq x \leq b} | f (x) | {‖ f ‖}_{2} = (\int_{a}^{b} f {(x)}^{2} d x)^{\frac{1}{2}}$
For approximation of data these norms take the discrete form
$‖ f ‖ = max_{i} | f (x_{i}) | {‖ f ‖}_{2} = (\underset{i}{Σ} f (x_{i})^{2})^{\frac{1}{2}}$
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
$p_{n} \in P_{n}$
for which
$‖ f - p_{n} ‖ or {‖ f - p_{n} ‖}_{2}$
are acceptably small. Best approximation problems arise when, for example, we seek
$p_{n} \in P_{n}$
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
$x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}$
important examples are
$‖ x ‖ = max_{i} | x_{i} | {‖ x ‖}_{2} = (Σ_{i = 1}^{n} x_{i}^{2})^{\frac{1}{2}}$
Corresponding to a given vector norm, a subordinate matrix norm can be defined for n × n matrices A, by
$‖ A ‖ = max_{‖ x ‖ \neq 0} \frac{‖ A x ‖}{‖ x ‖}$
For the vector norm
$‖ x ‖ = max_{i} | x_{i} |$
this reduces to the expression
$‖ A ‖ = max_{i} Σ_{j = 1}^{n} | a_{i j} |$
where a_ij is the i,jth element of A. Vector and matrix norms are indispensable in most areas of numerical analysis.

单词	approximation theory
释义	approximation theory Mathematics The area of numerical analysis related to finding a simpler function f(x) which has approximately the same values as F(x) over a specified interval. Analysis can then be carried out on the more accessible f(x) knowing that the difference is small. Computer 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 $f \in C [a, b]$ these norms are given respectively as $‖ f ‖ = max_{a \leq x \leq b} \| f (x) \| {‖ f ‖}_{2} = (\int_{a}^{b} f {(x)}^{2} d x)^{\frac{1}{2}}$ For approximation of data these norms take the discrete form $‖ f ‖ = max_{i} \| f (x_{i}) \| {‖ f ‖}_{2} = (\underset{i}{Σ} f (x_{i})^{2})^{\frac{1}{2}}$ 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 $p_{n} \in P_{n}$ for which $‖ f - p_{n} ‖ or {‖ f - p_{n} ‖}_{2}$ are acceptably small. Best approximation problems arise when, for example, we seek $p_{n} \in P_{n}$ 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 $x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}$ important examples are $‖ x ‖ = max_{i} \| x_{i} \| {‖ x ‖}_{2} = (Σ_{i = 1}^{n} x_{i}^{2})^{\frac{1}{2}}$ Corresponding to a given vector norm, a subordinate matrix norm can be defined for n × n matrices A, by $‖ A ‖ = max_{‖ x ‖ \neq 0} \frac{‖ A x ‖}{‖ x ‖}$ For the vector norm $‖ x ‖ = max_{i} \| x_{i} \|$ this reduces to the expression $‖ A ‖ = max_{i} Σ_{j = 1}^{n} \| a_{i j} \|$ where a_ij is the i,jth element of A. Vector and matrix norms are indispensable in most areas of numerical analysis.
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