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)

Comments

See also Approximation of functions, direct and inverse theorems.

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)