In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitney in 1957,[1] and is an important tool in the field of approximation theory for obtaining upper estimates on the errors of best approximation.

Statement of the theorem

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Denote the value of the best uniform approximation of a function   by algebraic polynomials   of degree   by

 

The moduli of smoothness of order   of a function   are defined as:

 
 

where   is the finite difference of order  .

Theorem: [2] [Whitney, 1957] If  , then

 

where   is a constant depending only on  . The Whitney constant   is the smallest value of   for which the above inequality holds. The theorem is particularly useful when applied on intervals of small length, leading to good estimates on the error of spline approximation.

Proof

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The original proof given by Whitney follows an analytic argument which utilizes the properties of moduli of smoothness. However, it can also be proved in a much shorter way using Peetre's K-functionals.[3]

Let:

 
 
 

where   is the Lagrange polynomial for   at the nodes  .

Now fix some   and choose   for which  . Then:

 
 

Therefore:

 

And since we have  , (a property of moduli of smoothness)

 

Since   can always be chosen in such a way that  , this completes the proof.

Whitney constants and Sendov's conjecture

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It is important to have sharp estimates of the Whitney constants. It is easily shown that  , and it was first proved by Burkill (1952) that  , who conjectured that   for all  . Whitney was also able to prove that [2]

 

and

 

In 1964, Brudnyi was able to obtain the estimate  , and in 1982, Sendov proved that  . Then, in 1985, Ivanov and Takev proved that  , and Binev proved that  . Sendov conjectured that   for all  , and in 1985 was able to prove that the Whitney constants are bounded above by an absolute constant, that is,   for all  . Kryakin, Gilewicz, and Shevchuk (2002)[4] were able to show that   for  , and that   for all  .

References

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  1. ^ Hassler, Whitney (1957). "On Functions with Bounded nth Differences". J. Math. Pures Appl. 36 (IX): 67–95.
  2. ^ a b Dzyadyk, Vladislav K.; Shevchuk, Igor A. (2008). "3.6". Theory of Uniform Approximation of Functions by Polynomials (1st ed.). Berlin, Germany: Walter de Gruyter. pp. 231–233. ISBN 978-3-11-020147-5.
  3. ^ Devore, R. A. K.; Lorentz, G. G. (4 November 1993). "6, Theorem 4.2". Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1st ed.). Berlin, Germany: Springer-Verlag. ISBN 978-3540506270.
  4. ^ Gilewicz, J.; Kryakin, Yu. V.; Shevchuk, I. A. (2002). "Boundedness by 3 of the Whitney Interpolation Constant". Journal of Approximation Theory. 119 (2): 271–290. doi:10.1006/jath.2002.3732.