Approximation Of Functions, Measure Of

A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function $ f $ by a function $ \phi $, the measure of approximation $ \mu (f, \phi ) $ is usually defined by the metric in a function space containing both $ f $ and $ \phi $. For example, if $ f $ and $ \phi $ are continuous functions on a segment $ [a, b] $, the uniform metric of $ C [a, b] $ is commonly used, i.e. one puts

$$ \mu (f, \phi ) = \ \max _ {a \leq t \leq b } \ | f (t) - \phi (t) | . $$

If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that $ f $ and $ \phi $ are close on $ [a, b] $ in an average sense, the integral metric of a space $ L _ {p} [a, b] $ may be used, putting

$$ \mu (f, \phi ) = \ \int\limits _ { a } ^ { b } q (t) | f (t) - \phi (t) | ^ {p} \ dt,\ p > 0, $$

where $ q (t) $ is a weight function. The case $ p = 2 $ is most often used and is most convenient from a practical point of view (cf. Mean-square approximation of a function).

The measure of approximation may take into account only values of $ f $ and $ \phi $ in discrete points $ t _ {k} $, $ k = 1 \dots n $, of $ [a, b] $, e.g.

$$ \mu (f, \phi ) = \ \max _ {1 \leq k \leq n } \ | f (t _ {k} ) - \phi (t _ {k} ) | , $$

$$ \mu (f, \phi ) = \sum _ {k = 1 } ^ { n } q _ {k} | f (t _ {k} ) - \phi (t _ {k} ) | ^ {p} , $$

where $ q _ {k} $ are certain positive coefficients.

One defines in an analogous way the measure of approximation of functions in two or more variables.

The measure of approximation of a function $ f $ by a family $ F $ of functions is usually taken to be the best approximation:

$$ E (f, F) = \ \mu (f, F) = \ \inf _ {\phi \in F } \ \mu (f, \phi ). $$

The quantity

$$ E ( \mathfrak M , F) = \ \mu ( \mathfrak M , F) = \ \sup _ {f \in \mathfrak M } \ \inf _ {\phi \in F } \ \mu (f, \phi ) $$

is usually taken as the measure of approximation of a class $ \mathfrak M $ of functions $ f $ by functions $ \phi $ from a certain fixed set $ F $. It characterizes the maximal deviation of functions in $ \mathfrak M $ from functions in $ F $ that are closest to them.

In general, when approximation in an arbitrary metric space $ X $ is considered, the measure of approximation $ \mu (x, u) $ of an element $ x $ by an element $ u $( a set $ F $) is the distance $ \rho (x, u) $( or $ \rho (x, F) $) between $ x $ and $ u $( or $ F $) in the metric of $ X $.

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The measure of approximation is also called the error measure.

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