“Bessel potential”的意思、由来-开放百科全书

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Bessel potential

释义

Representation in Fourier space
Integral representations
Asymptotics
See also
References

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]

Asymptotics

At the origin, one has as ,^[3]

In particular, when the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as , ^[4]

References

1. ^{{cite book|last1=Stein|first1=Elias|title=Singular integrals and differentiability properties of functions|date=1970|publisher=Princeton University Press|isbn=0-691-08079-8|at=Chapter V eq. (26)}}
2. ^{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|at=385–475, (4,2)}}
3. ^{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|at=385–475, (4,3)}}
4. ^{{cite journal|last1=N. Aronszajn|last2=K. T. Smith|title=Theory of Bessel potentials I|journal=Ann. Inst. Fourier|date=1961|volume=11|pages=385–475}}

{{eom|id=B/b110420|title=Bessel potential operator|first=R. |last=Duduchava}}
{{Citation | last1=Grafakos | first1=Loukas | title=Modern Fourier analysis | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-09433-5 | doi=10.1007/978-0-387-09434-2 | mr=2463316 | year=2009 | volume=250}}
{{eom|id=B/b120170|title=Bessel potential space|first=L.I. |last= Hedberg}}
{{eom|id=B/b015870|first=E.D.|last= Solomentsev}}
{{citation |first=Elias |last=Stein |authorlink=Elias Stein |title=Singular integrals and differentiability properties of functions |publisher=Princeton University Press |location=Princeton, NJ |year=1970 |isbn=0-691-08079-8}}

4 : Fractional calculus|Partial differential equations|Potential theory|Singular integrals

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Representation in Fourier space

Integral representations

Asymptotics

See also

References