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Approximation in the mean

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Approximation of a given function , integrable on an interval , by a function , where the quantity

is taken as the measure of approximation.

The more general case, when

where is a non-decreasing function different from a constant on , is called mean-power approximation (with exponent ) with respect to the distribution . If is absolutely continuous and , then one obtains mean-power approximation with weight , and if is a step function with jumps at points in , one has weighted mean-power approximation with respect to the system of points with measure of approximation

These concepts are extended in a natural way to the case of functions of several variables.

References

[1] V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)


Comments

References

[a1] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[a2] T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1981)
[a3] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
How to Cite This Entry:
Approximation in the mean. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_in_the_mean&oldid=15038
This article was adapted from an original article by N.P. KorneichukV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article