Whitney inequality
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, 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. Denote the value of the best uniform approximation of a function formula_1 by algebraic polynomials formula_2 of degree formula_3 by The moduli of smoothness of order formula_5 of a function formula_1 are defined as: where formula_9 is  the finite difference of order formula_5. Theorem: [Whitney, 1957] If formula_1, then where formula_13 is a constant depending only on formula_5. The Whitney constant formula_15 is the smallest value of formula_13 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. 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. Let: where formula_20 is the Lagrange  polynomial for formula_21 at the nodes formula_22. Now fix some formula_23 and choose formula_24 for which formula_25. Then: Therefore: And since we have formula_29, (a property of moduli of smoothness) Since formula_24 can always be chosen in such a way that formula_32, this completes the proof. Whitney constants and Sendov's conjecture. It is important to have sharp estimates of the Whitney constants. It is easily shown that formula_33, and it was first proved by Burkill (1952) that formula_34, who conjectured that formula_35 for all formula_5. Whitney was also able to prove that and In 1964, Brudnyi was able to  obtain the estimate formula_39, and in 1982, Sendov proved that formula_40. Then, in 1985, Ivanov and Takev proved that formula_41, and Binev proved that formula_42. Sendov conjectured that formula_35 for all formula_5, and in 1985 was able to prove that the Whitney constants are bounded above by an absolute constant, that is, formula_45 for all formula_5. Kryakin, Gilewicz, and Shevchuk (2002) were able to show that formula_47 for formula_48, and that formula_49 for all formula_5.