Algebraic polynomial of best approximation

A polynomial deviating least from a given function. More precisely, let a measurable function be in () and let be the set of algebraic polynomials of degree not exceeding . The quantity

 (*)

is called the best approximation, while a polynomial for which the infimum is attained is known as an algebraic polynomial of best approximation in . Polynomials which deviate least from a given continuous function in the uniform metric () were first encountered in the studies of P.L. Chebyshev (1852), who continued to study them in 1856 [1]. The existence of algebraic polynomials of best approximation was established by E. Borel [2]. Chebyshev proved that is an algebraic polynomial of best approximation in the uniform metric if and only if Chebyshev alternation occurs in the difference ; in this case such a polynomial is unique. If , the algebraic polynomial of best approximation is unique due to the strict convexity of the space . If , it is not unique, but it has been shown by D. Jackson [3] to be unique for continuous functions. The rate of convergence of to zero is given by Jackson's theorems (cf. Jackson theorem).

In a manner similar to (*) an algebraic polynomial of best approximation is defined for functions in a large number of unknowns, say . If the number of variables , an algebraic polynomial of best approximation in the uniform metric is, in general, not unique.

References

 [1] P.L. Chebyshev, "Questions on smallest quantities connected with the approximate representation of functions (1859)" , Collected works , 2 , Moscow-Leningrad (1947) pp. 478; 152–236 (In Russian) [2] E. Borel, "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars (1905) [3] D. Jackson, "A general class of problems in approximation" Amer. J. Math. , 46 (1924) pp. 215–234 [4] A.L. Garkavi, "The theory of approximation in normed linear spaces" Itogi Nauk. Mat. Anal. 1967 (1969) pp. 75–132 (In Russian)