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Whitney-Mikhlin Extension Constants


Let B_n(r) be the n-dimensional 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))[[f(x)]^2+sum_(j=1)^n((partialf)/(partialx_j))^2]dx)<=chisqrt(int_(B(r))[[F(x)]^2+sum_(j=1)^n((partialF)/(partialx_j))^2]dx).
(2)

chi(1,r)=1. Let

 nu=1/2(n-2),
(3)

then for n>2,

 chi(n,r)=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))),
(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{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))])}.
(5)

For r->infty,

 chi(n,infty)=sqrt(1+(I_nu(1))/(I_(nu+1)(1))(K_nu(1))/(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)/(a_(k+1))(b_(k+1))/(b_k)).
(14)

These can be given in closed form as

a_k=(2i)/eI_(k-1/2)(1)K_(1/2)(-1)
(15)
b_k=(2i)/(epi)K_(1/2)(-1)K_(k-1/2)(1)
(16)
chi(2k+1,infty)=[I_(k/2)(1)K_((k-2)/2)(1)]^(-1/2).
(17)

The first few are

chi(3,infty)=e
(18)
chi(5,infty)=sqrt((e^2)/(e^2-7))
(19)
chi(7,infty)=sqrt(2/7)sqrt((e^2)/(37-5e^2))
(20)
chi(9,infty)=1/(sqrt(37))sqrt((e^2)/(18e^2-133))
(21)
chi(11,infty)=1/(sqrt(133))sqrt((e^2)/(2431-329e^2))
(22)
chi(13,infty)=sqrt(2/(2431))sqrt((e^2)/(3655e^2-27007)).
(23)

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


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References

Finch, S. R. "Whitney-Mikhlin Extension Constants." §3.8 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 227-229, 2003.Mikhlin, S. G. Constants in Some Inequalities of Analysis. New York: Wiley, 1986.

Referenced on Wolfram|Alpha

Whitney-Mikhlin Extension Constants

Cite this as:

Weisstein, Eric W. "Whitney-Mikhlin Extension Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Whitney-MikhlinExtensionConstants.html

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