“Dirichlet space”的意思、由来-开放百科全书

词条

Dirichlet space

释义

See also
References

In mathematics, the Dirichlet space on the domain (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space , for which the Dirichlet integral, defined by

is finite (here dA denotes the area Lebesgue measure on the complex plane ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on . It is not a norm in general, since whenever f is a constant function.

For , we define

This is a semi-inner product, and clearly . We may equip with an inner product given by

where is the usual inner product on The corresponding norm is given by

Note that this definition is not unique, another common choice is to take , for some fixed .

The Dirichlet space is not an algebra, but the space is a Banach algebra, with respect to the norm

We usually have (the unit disk of the complex plane ), in that case , and if

then

and

Clearly, contains all the polynomials and, more generally, all functions , holomorphic on such that is bounded on .

The reproducing kernel of at is given by

References

{{citation|journal=New York J. Math. |volume=17a |year=2011|pages= 45–86|title=

url=http://nyjm.albany.edu/j/2011/17a-4v.pdf}}

{{cite book|first1=Omar|last1=El-Fallah|first2=Karim|last2=Kellay|first3=Javad|last3=Mashreghi|first4=Thomas|last4=Ransford|title=A primer on the Dirichlet space|date=2014|publisher=Cambridge University Press|location=Cambridge, UK|isbn=978-1-107-04752-5|url=http://cambridge.org/9781107047525}}

2 : Complex analysis|Functional analysis

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

See also

References