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Bessel potential

From Encyclopedia of Mathematics
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A potential of the form

where , are points in the Euclidean space , ; is a Borel measure on ;

and is the modified cylinder function (or Bessel function, cf. Cylinder functions) of the second kind of order or the Macdonald function of order ; is called a Bessel kernel.

The principal properties of the Bessel kernels are the same as those of the Riesz kernels (cf. Riesz potential), viz., they are positive, continuous for , can be composed

but, unlike the Riesz potentials, Bessel potentials are applicable for all , since

as .

If , where is a natural number, and the measure is absolutely continuous with square-integrable density , the Bessel potentials satisfy the identities:

and

where is the Laplace operator on . In other words, the function is a fundamental solution of the operator .

References

[1] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[2] M. Aronszajn, K.T. Smith, "Theory of Bessel potentials I" Ann. Inst. Fourier (Grenoble) , 11 (1961) pp. 385–475


Comments

The function is usually called the modified Bessel function of the third kind.

How to Cite This Entry:
Bessel potential. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bessel_potential&oldid=13961
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098