# 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,[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

Denote the value of the best uniform approximation of a function ${\displaystyle f\in C([a,b])}$ by algebraic polynomials ${\displaystyle P_{n}}$ of degree ${\displaystyle \leq n}$ by

${\displaystyle E_{n}(f)_{[a,b]}:=\inf _{P_{n}}{\|f-P_{n}\|_{C([a,b])}}}$

The moduli of smoothness of order ${\displaystyle k}$ of a function ${\displaystyle f\in C([a,b])}$ are defined as:

${\displaystyle \omega _{k}(t):=\omega _{k}(t;f;[a,b]):=\sup _{h\in [0,t]}\|\Delta _{h}^{k}(f;\cdot )\|_{C([a,b-kh])}\quad {\text{ for }}\quad t\in [0,(b-a)/k],}$
${\displaystyle \omega _{k}(t):=\omega _{k}((b-a)/k)\quad {\text{ for}}\quad t>(b-a)/k,}$

where ${\displaystyle \Delta _{h}^{k}}$ is the finite difference of order ${\displaystyle k}$.

Theorem: [2] [Whitney, 1957] If ${\displaystyle f\in C([a,b])}$, then

${\displaystyle E_{k-1}(f)_{[a,b]}\leq W_{k}\omega _{k}\left({\frac {b-a}{k}};f;[a,b]\right)}$

where ${\displaystyle W_{k}}$ is a constant depending only on ${\displaystyle k}$. The Whitney constant ${\displaystyle W(k)}$ is the smallest value of ${\displaystyle W_{k}}$ 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.[3]

Let:

${\displaystyle x_{0}:=a,\quad h:={\frac {b-a}{k}},\quad x_{j}:=x+0+jh,\quad F(x)=\int _{a}^{x}f(u)\,du,}$
${\displaystyle G(x):=F(x)-L(x;F;x_{0},\ldots ,x_{k}),\quad g(x):=G'(x),}$
${\displaystyle \omega _{k}(t):=\omega _{k}(t;f;[a,b])\equiv \omega _{k}(t;g;[a,b])}$

where ${\displaystyle L(x;F;x_{0},\ldots ,x_{k})}$ is the Lagrange polynomial for ${\displaystyle F}$ at the nodes ${\displaystyle \{x_{0},\ldots ,x_{k}\}}$.

Now fix some ${\displaystyle x\in [a,b]}$ and choose ${\displaystyle \delta }$ for which ${\displaystyle (x+k\delta )\in [a,b]}$. Then:

${\displaystyle \int _{0}^{1}\Delta _{t\delta }^{k}(g;x)\,dt=(-1)^{k}g(x)+\sum _{j=1}^{k}(-1)^{k-j}{\binom {k}{j}}\int _{0}^{1}g(x+jt\delta )\,dt}$
${\displaystyle =(-1)^{k}g(x)+\sum _{j=1}^{k}{(-1)^{k-j}{\binom {k}{j}}{\frac {1}{j\delta }}(G(x+j\delta )-G(x))},}$

Therefore:

${\displaystyle |g(x)|\leq \int _{0}^{1}|\Delta _{t\delta }^{k}(g;x)|\,dt+{\frac {2}{|\delta |}}\|G\|_{C([a,b])}\sum _{j=1}^{k}{\binom {k}{j}}{\frac {1}{j}}\leq \omega _{k}(|\delta |)+{\frac {1}{|\delta |}}2^{k+1}\|G\|_{C([a,b])}}$

And since we have ${\displaystyle \|G\|_{C([a,b])}\leq h\omega _{k}(h)}$, (a property of moduli of smoothness)

${\displaystyle E_{k-1}(f)_{[a,b]}\leq \|g\|_{C([a,b])}\leq \omega _{k}(|\delta |)+{\frac {1}{|\delta |}}h2^{k+1}\omega _{k}(h).}$

Since ${\displaystyle \delta }$ can always be chosen in such a way that ${\displaystyle h\geq |\delta |\geq h/2}$, 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 ${\displaystyle W(1)=1/2}$, and it was first proved by Burkill (1952) that ${\displaystyle W(2)\leq 1}$, who conjectured that ${\displaystyle W(k)\leq 1}$ for all ${\displaystyle k}$. Whitney was also able to prove that [2]

${\displaystyle W(2)={\frac {1}{2}},\quad {\frac {8}{15}}\leq W(3)\leq 0.7\quad W(4)\leq 3.3\quad W(5)\leq 10.4}$

and

${\displaystyle W(k)\geq {\frac {1}{2}},\quad k\in \mathbb {N} }$

In 1964, Brudnyi was able to obtain the estimate ${\displaystyle W(k)=O(k^{2k})}$, and in 1982, Sendov proved that ${\displaystyle W(k)\leq (k+1)k^{k}}$. Then, in 1985, Ivanov and Takev proved that ${\displaystyle W(k)=O(k\ln k)}$, and Binev proved that ${\displaystyle W(k)=O(k)}$. Sendov conjectured that ${\displaystyle W(k)\leq 1}$ for all ${\displaystyle k}$, and in 1985 was able to prove that the Whitney constants are bounded above by an absolute constant, that is, ${\displaystyle W(k)\leq 6}$ for all ${\displaystyle k}$. Kryakin, Gilewicz, and Shevchuk (2002)[4] were able to show that ${\displaystyle W(k)\leq 2}$ for ${\displaystyle k\leq 82000}$, and that ${\displaystyle W(k)\leq 2+{\frac {1}{e^{2}}}}$ for all ${\displaystyle k}$.

## References

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.