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Mean-square approximation of a function

From Encyclopedia of Mathematics
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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
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098