Whitney-Mikhlin Extension Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $B_n(r)$ be the $n$-D closed Ball of Radius $r>1$ centered at the Origin. A function which is defined on $B(r)$ is called an extension to $B(r)$ of a function $f$ defined on $B(1)$ if

$$F(x)=f(x)\forall\ x\in B(1).$$ (1)

Given 2 Banach Spaces of functions defined on $B(1)$ and $B(r)$, find the extension operator from one to the other of minimal norm. Mikhlin (1986) found the best constants $\chi$ such that this condition, corresponding to the Sobolev $W(1,2)$ integral norm, is satisfied,

$\sqrt{\int_{B(1)} \left[{[f(x)]^2+\sum_{j=1}^n\left({\partial f\over\partial x_j}\right)^2}\right]\,dx}$
$\leq \chi \sqrt{\int_{B(r)} \left[{[F(x)]^2+\sum_{j=1}^n\left({\partial F\over\partial x_j}\right)^2}\right]\,dx}\,.\quad$	(2)

$\chi(1,r)=1$. Let

$$\nu={\textstyle{1\over 2}}(n-2),$$ (3)

then for $n>2$,

$$\chi(n,r)=\sqrt{1+{I_\nu(1)\over I_{\nu+1}(1)}{I_\nu(r)K_{\n... ...K_\nu(r)I_{\nu+1}(1)\over I_\nu(r)K_\nu(1)-K_\nu(r)I_\nu(1)}},$$ (4)

where $I_\nu(z)$ is a Modified Bessel Function of the First Kind and $K_\nu(z)$ is a Modified Bessel Function of the Second Kind. For $n=2$,

$\chi(2,r)=\max\left\{{\sqrt{1+{I_\nu(1)\over I_{\nu+1}(1)}{I_\nu(r)K_{\nu+1}(1)+K_\nu(r)I_{\nu+1}(1)\over I_\nu(r)K_\nu(1)-K_\nu(r)I_\nu(1)}},}\right.$
$\left.{\sqrt{1+{I_1(1)\over I_1(1)+I_2(1)}\left[{1+{I_1(r)K_0(1)+K_1(r)I_0(1)\over I_1(r)K_1(1)-K_1(r)I_1(1)}}\right]}}\right\}.\quad$	(5)

For $r\to\infty$,

$$\chi(n,\infty)=\sqrt{1+{I_\nu(1)\over I_{\nu+1}(1)}{K_\nu(1)\over K_\nu(1)}},$$ (6)

which is bounded by

$$n-1<\chi(n,\infty)<\sqrt{(n-1)^2+4}.$$ (7)

For Odd $n$, the Recurrence Relations

$$a_{k+1} = a_{k-1}-(2k-1)a_k$$ (8)

$$b_{k+1} = b_{k-1}+(2k-1)b_k$$ (9)

with

$$a_0 = e+e^{-1}$$ (10)

$$a_1 = e-e^{-1}$$ (11)

$$b_0 = e^{-1}$$ (12)

$$b_1 = e^{-1}$$ (13)

where e is the constant 2.71828..., give

$$\chi(2k+1,\infty)=\sqrt{1+{a_k\over a_{k+1}}{b_{k+1}\over b_k}}.$$ (14)

The first few are

$$\chi(3,\infty) = e$$ (15)

$$\chi(5,\infty) = \sqrt{e^2\over e^2-7}$$ (16)

$$\chi(7,\infty) = \sqrt{2\over 7}\sqrt{e^2\over 37-5e^2}$$ (17)

$$\chi(9,\infty) = {1\over\sqrt{37}}\sqrt{e^2\over 18e^2-133}$$ (18)

$$\chi(11,\infty) = {1\over\sqrt{133}}\sqrt{e^2\over 2431-329e^2}$$ (19)

$$\chi(13,\infty) = \sqrt{2\over 2431}\sqrt{e^2\over 3655e^2-27007}.$$ (20)

Similar formulas can be given for even $n$ in terms of $I_0(1)$, $I_1(1)$, $K_0(1)$, $K_1(1)$.

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/mkhln/mkhln.html

Mikhlin, S. G. Constants in Some Inequalities of Analysis. New York: Wiley, 1986.