# Jackson inequality

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 $f$ be a $2 \pi$- periodic continuous function on the real axis, let $E _ {n} ( f )$ be the best uniform approximation error of $f$ by trigonometric polynomials $T _ {n}$ of degree $n$, i.e.

$$E _ {n} ( f ) = \inf _ {T _ {n} } \max _ { x } | f ( x) - T _ {n} ( x) | ,$$

and let

$$\omega ( f ; \delta ) = \max _ {| t _ {1} - t _ {2} | \leq \delta } | f ( t _ {1} ) - f ( t _ {2} ) |$$

be the modulus of continuity of $f$( cf. Continuity, modulus of). It was shown by D. Jackson [1] that

$$\tag{* } E _ {n} ( f ) \leq C \omega \left ( f ; \frac{1}{n} \right )$$

(where $C$ is an absolute constant), while if $f$ has an $r$- th order continuous derivative $f ^ { ( r) }$, $r \geq 1$, then

$$E _ {n} ( f ) \leq \frac{C _ {r} }{n ^ {r} } \omega \left ( f ^ { ( r) } ; \frac{1}{n} \right ) ,$$

where the constant $C _ {r}$ depends on $r$ only. S.N. Bernshtein [3] obtained inequality (*) in an independent manner for the case

$$\omega ( f ; t ) \leq K t ^ \alpha ,\ \ 0 < \alpha < 1 .$$

If $f$ is continuous or $r$ times continuously differentiable on a closed interval $[ a , b ]$, $r = 1, 2 \dots$ and if $E _ {n} ( f ; a , b )$ is the best uniform approximation error of the function $f$ on $[ a , b ]$ by algebraic polynomials of degree $n$, then, for $n > r$ one has the relation $( f ^ { 0 } = f )$

$$E _ {n} ( f ; a , b ) \leq \frac{A _ {r} ( b - a ) ^ {r} }{n ^ {r} } \omega \left ( f ^ { ( r) } ; \frac{b - a }{n} \right ) ,$$

where the constant $A _ {r}$ depends on $r$ 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 $k$, or to a function of several variables. The exact values of the constants in Jackson's inequalities have been determined in several cases.

#### References

 [1] D. Jackson, "Ueber die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung" , Göttingen (1911) (Thesis) [2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) [3] S.N. Bernshtein, "On the best approximation of continuous functions by polynomials of a given degree (1912)" , Collected works , 1 , Moscow (1952) pp. 11–104 [4] N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian) [5] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)

Let $\omega _ {k} ( f; \delta )$ be the modulus of continuity of order $k$,

$$\omega _ {k} ( f; \delta ) = \ \sup _ { \begin{array}{c} | h | \leq t \\ x, x + kh \in [ a, b] \end{array} } \ \left | \sum _ {\nu = 0 } ^ { k } (- 1) ^ {k - \nu } \left ( \begin{array}{c} k \\ \nu \end{array} \right ) f ( x + \nu h) \ \right | .$$

Then, more generally,

$$E _ {n} ( f ) \leq C _ {k} \omega _ {k} ( f ; n ^ {-} 1 ) ,$$

where $C _ {k}$ is independent of $f$. The best possible coefficients $C _ {k}$ were determined by J. Favard. For the interval $[- 1, 1]$ the constant $C _ {1}$ is $6$. A result of S.B. Stechkin says that

$$\omega _ {k} \left ( f; { \frac{1}{n} } \right ) \leq \ \frac{C _ {k} }{n ^ {k} } \sum _ {i = 0 } ^ { n } ( i + 1) ^ {k - 1 } E _ {i} ( f ) .$$

#### References

 [a1] E.W. Cheney, "Introduction to approximation theory" , McGraw-Hill (1966) pp. Chapt. 4 [a2] G.W. Meinardus, "Approximation von Funktionen und ihre numerische Behandlung" , Springer (1964) pp. Chapt. 1, §5 [a3] T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1981)
How to Cite This Entry:
Jackson inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jackson_inequality&oldid=47452
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