The Whitney-Mikhlin extension constants are the least operator norms of extension operators with the following support condition. For a positive
integer 
and 
, let 
be the closed ball of radius 
centered at the origin in 
.
An extension 
of a real-valued function 
on 
satisfies
 |
(1)
|
with 
in the Sobolev space 
and 
in 
. Given 2 Banach spaces
of functions defined on 
and 
, find the extension operator from one to the other of minimal
norm. Mikhlin (1986) found the best constants 
such that this condition, corresponding to the Sobolev 
integral norm, is satisfied,
![sqrt(int_(B(r))[[F(x)]^2+sum_(j=1)^n((partialF)/(partialx_j))^2]dx)<=chisqrt(int_(B(1))[[f(x)]^2+sum_(j=1)^n((partialf)/(partialx_j))^2]dx).](/images/equations/Whitney-MikhlinExtensionConstants/NumberedEquation2.svg) |
(2)
|
For 
,
 |
(3)
|
Let
 |
(4)
|
then for 
,
 |
(5)
|
where 
is a modified
Bessel function of the first kind and 
is a modified
Bessel function of the second kind. For 
,
![chi(2,r)=max{sqrt(1+(I_nu(1))/(I_(nu+1)(1))(I_nu(r)K_(nu+1)(1)+K_nu(r)I_(nu+1)(1))/(I_nu(r)K_nu(1)-K_nu(r)I_nu(1))),sqrt(1+(I_1(1))/(I_1(1)+I_2(1))[1+(I_1(r)K_0(1)+K_1(r)I_0(1))/(I_1(r)K_1(1)-K_1(r)I_1(1))])}.](/images/equations/Whitney-MikhlinExtensionConstants/NumberedEquation6.svg) |
(6)
|
For 
,
 |
(7)
|
which is bounded by
 |
(8)
|
For odd 
, the recurrence relations
 |  |  |
(9)
|
 |  |  |
(10)
|
with
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
where e is the constant 2.71828..., give
 |
(15)
|
These can be given in closed form as
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  | ![[I_(k+1/2)(1)K_(k-1/2)(1)]^(-1/2).](/images/equations/Whitney-MikhlinExtensionConstants/Inline50.svg) |
(18)
|
The first few are
 |  |  |
(19)
|
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
 |  |  |
(24)
|
Similar formulas can be given for even 
in terms of 
, 
, 
, 
.
