# Mean-square approximation of a function

An approximation of a function by a function , where the error measure is defined by the formula

where is a non-decreasing function on different from a constant.

Let

(*) |

be an orthonormal system of functions on relative to the distribution . In the case of a mean-square approximation of the function by linear combinations , the minimal error for every is given by the sums

where are the Fourier coefficients of the function with respect to the system (*); hence, the best method of approximation is linear.

#### References

[1] | V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian) |

[2] | G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) |

#### Comments

Cf. also Approximation in the mean; Approximation of functions; Approximation of functions, linear methods; Best approximation; Best approximation in the mean; Best linear method.

#### References

[a1] | E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapts. 4&6 |

[a2] | I.P. Natanson, "Constructive theory of functions" , 1–2 , F. Ungar (1964–1965) (Translated from Russian) |

**How to Cite This Entry:**

Mean-square approximation of a function. N.P. KorneichukV.P. Motornyi (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Mean-square_approximation_of_a_function&oldid=17600