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
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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,
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. Let
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then for ,
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where is a modified Bessel function of the first kind and is a modified Bessel function of the second kind. For ,
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For ,
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which is bounded by
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For odd , the recurrence relations
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with
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where e is the constant 2.71828..., give
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These can be given in closed form as
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The first few are
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Similar formulas can be given for even in terms of , , , .