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.
Denote the value of the best uniform approximation of a function
by algebraic polynomials
by The moduli of smoothness of order
are defined as: where
is the finite difference of order
Theorem: [2] [Whitney, 1957] If
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.
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.
is the Lagrange polynomial for
at the nodes
Now fix some
, (a property of moduli of smoothness) Since
, this completes the proof.
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
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
, and in 1985 was able to prove that the Whitney constants are bounded above by an absolute constant, that is,
Kryakin, Gilewicz, and Shevchuk (2002)[4] were able to show that