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- Bergman_kernel abstract "In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cn.In detail, let L2(D) be the Hilbert space of square integrable functions on D, and let L2,h(D) denote the subspace consisting of holomorphic functions in D: that is,where H(D) is the space of holomorphic functions in D. Then L2,h(D) is a Hilbert space: it is a closed linear subspace of L2(D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D</dl>for every compact subset K of D. Thus convergence of a sequence of holomorphic functions in L2(D) implies also compact convergence, and so the limit function is also holomorphic.Another consequence of is that, for each z ∈ D, the evaluationis a continuous linear functional on L2,h(D). By the Riesz representation theorem, this functional can be represented as the inner product with an element of L2,h(D), which is to say thatThe Bergman kernel K is defined byThe kernel K(z,ζ) holomorphic in z and antiholomorphic in ζ, and satisfiesOne key observation about this picture is that L2,h(D) may be identified with the space of holomophic (n,0)-norms on D, via multiplication by . Since the inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated Bergman metric are therefore automatically invariant under the automorphism group of the domain.".
- Bergman_kernel wikiPageID "3205923".
- Bergman_kernel wikiPageRevisionID "604300543".
- Bergman_kernel first "E.M.".
- Bergman_kernel hasPhotoCollection Bergman_kernel.
- Bergman_kernel id "B/b015560".
- Bergman_kernel last "Chirka".
- Bergman_kernel title "Bergman kernel function".
- Bergman_kernel subject Category:Several_complex_variables.
- Bergman_kernel type PhysicalEntity100001930.
- Bergman_kernel type SeveralComplexVariables.
- Bergman_kernel type Thing100002452.
- Bergman_kernel type Variable109468959.
- Bergman_kernel comment "In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cn.In detail, let L2(D) be the Hilbert space of square integrable functions on D, and let L2,h(D) denote the subspace consisting of holomorphic functions in D: that is,where H(D) is the space of holomorphic functions in D.".
- Bergman_kernel label "Bergman kernel".
- Bergman_kernel label "Noyau de Bergman".
- Bergman_kernel label "Núcleo de Bergman".
- Bergman_kernel sameAs Núcleo_de_Bergman.
- Bergman_kernel sameAs Noyau_de_Bergman.
- Bergman_kernel sameAs m.06wc99h.
- Bergman_kernel sameAs Q3345664.
- Bergman_kernel sameAs Q3345664.
- Bergman_kernel sameAs Bergman_kernel.
- Bergman_kernel wasDerivedFrom Bergman_kernel?oldid=604300543.
- Bergman_kernel isPrimaryTopicOf Bergman_kernel.