Mean-square approximation of a function

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 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 of the function $ f( t) $ 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)