“Whitney-Mikhlin Extension Constants”的概念、定义、习题解题方法及翻译-数学辞典

单词

Whitney-Mikhlin Extension Constants

释义

Whitney-Mikhlin Extension Constants

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

Let be the -D closed Ball of Radius centered at the Origin. A function which isdefined 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 fromone to the other of minimal norm. Mikhlin (1986) found the best constants

such that this condition, correspondingto 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 Functionof the Second Kind. For


	(5)

For

(6)

which is bounded by

(7)

For Odd

, the Recurrence Relations

			(8)
			(9)

with

where e is the constant 2.71828..., give

(14)

The first few are

Similar formulas can be given for even

in terms of

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.

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