# Jackson inequality

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An inequality estimating the rate of decrease of the best approximation error of a function by trigonometric or algebraic polynomials in dependence on its differentiability and finite-difference properties. Let be a -periodic continuous function on the real axis, let be the best uniform approximation error of by trigonometric polynomials of degree , i.e. and let be the modulus of continuity of (cf. Continuity, modulus of). It was shown by D. Jackson  that (*)

(where is an absolute constant), while if has an -th order continuous derivative , , then where the constant depends on only. S.N. Bernshtein  obtained inequality (*) in an independent manner for the case If is continuous or times continuously differentiable on a closed interval , and if is the best uniform approximation error of the function on by algebraic polynomials of degree , then, for one has the relation  where the constant depends on only.

The Jackson inequalities are also known as the Jackson theorems or as direct theorems in the theory of approximation of functions. They may be generalized in various directions: to approximation using an integral metric, to approximation by entire functions of finite order, to an estimate concerning the approximation using a modulus of smoothness of order , or to a function of several variables. The exact values of the constants in Jackson's inequalities have been determined in several cases.