View source for Mean-square approximation of a function

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An approximation of a function  $  f( t) $
by a function  $  \phi ( t) $, 
where the error measure  $  \mu ( f;  \phi ) $
is defined by the formula

$$ 
\mu _  \sigma  ( f;  \phi )  =  \int\limits _ { a } ^ { b }  [ f( t) - \phi
( t)]  ^ {2}  d \sigma
( t),
$$

where  $  \sigma ( t) $
is a non-decreasing function on  $  [ a, b] $
different from a constant.

Let

$$ \tag{* }
u _ {1} ( t), u _ {2} ( t) \dots
$$

be an [[Orthonormal system|orthonormal system]] of functions on  $  [ a, b] $
relative to the distribution  $  d \sigma ( t) $. 
In the case of a mean-square approximation of the function  $  f( t) $
by linear combinations  $  \sum _ {k=} 1  ^ {n} \lambda _ {k} u _ {k} ( t) $, 
the minimal error for every  $  n = 1, 2 \dots $
is given by the sums

$$ 
\sum _ { k= } 1 ^ { n }  c _ {k} ( f  ) u _ {k} ( t),
$$

where  $  c _ {k} ( f  ) $
are the [[Fourier coefficients|Fourier coefficients]] of the function  $  f( t) $
with respect to the system (*); hence, the best method of approximation is linear.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.L. Goncharov,   "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR></table>

====Comments====
Cf. also [[Approximation in the mean|Approximation in the mean]]; [[Approximation of functions|Approximation of functions]]; [[Approximation of functions, linear methods|Approximation of functions, linear methods]]; [[Best approximation|Best approximation]]; [[Best approximation in the mean|Best approximation in the mean]]; [[Best linear method|Best linear method]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , McGraw-Hill  (1966)  pp. Chapts. 4&amp;6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.P. Natanson,   "Constructive theory of functions" , '''1–2''' , F. Ungar  (1964–1965)  (Translated from Russian)</TD></TR></table>