From Wikipedia, the free encyclopedia
In
mathematics, the Bessel potential is a
potential (named after
Friedrich Wilhelm Bessel) similar to the
Riesz potential but with better decay properties at infinity.
If s is a complex number with positive real part then the Bessel potential of order s is the operator
where Δ is the
Laplace operator and the
fractional power is defined using Fourier transforms.
Yukawa potentials are particular cases of Bessel potentials for in the 3-dimensional space.
Representation in Fourier space
The Bessel potential acts by multiplication on the
Fourier transforms: for each
Integral representations
When , the Bessel potential on can be represented by
where the Bessel kernel is defined for by the integral formula
[1]
Here denotes the
Gamma function.
The Bessel kernel can also be represented for by
[2]
This last expression can be more succinctly written in terms of a modified
Bessel function,
[3] for which the potential gets its name:
Asymptotics
At the origin, one has as ,
[4]
In particular, when the Bessel potential behaves asymptotically as the
Riesz potential.
At infinity, one has, as ,
[5]
See also
References
- Duduchava, R. (2001) [1994],
"Bessel potential operator",
Encyclopedia of Mathematics,
EMS Press
- Grafakos, Loukas (2009), Modern Fourier analysis,
Graduate Texts in Mathematics, vol. 250 (2nd ed.), Berlin, New York:
Springer-Verlag,
doi:
10.1007/978-0-387-09434-2,
ISBN
978-0-387-09433-5,
MR
2463316,
S2CID
117771953
- Hedberg, L.I. (2001) [1994],
"Bessel potential space",
Encyclopedia of Mathematics,
EMS Press
- Solomentsev, E.D. (2001) [1994],
"Bessel potential",
Encyclopedia of Mathematics,
EMS Press
-
Stein, Elias (1970),
Singular integrals and differentiability properties of functions, Princeton, NJ:
Princeton University Press,
ISBN
0-691-08079-8