TOPICS

Whitney-Mikhlin Extension Constants

Let be the -dimensional closed ball of radius centered at the origin. A function which is defined on is called an extension to of a function defined on if

 (1)

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,

 (2)

. Let

 (3)

then for ,

 (4)

where is a modified Bessel function of the first kind and is a modified Bessel function of the second kind. For ,

 (5)

For ,

 (6)

which is bounded by

 (7)

For odd , the recurrence relations

 (8) (9)

with

 (10) (11) (12) (13)

where e is the constant 2.71828..., give

 (14)

These can be given in closed form as

 (15) (16) (17)

The first few are

 (18) (19) (20) (21) (22) (23)

Similar formulas can be given for even in terms of , , , .

Explore with Wolfram|Alpha

More things to try:

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