
bergman space

sankeernamaaya vishakalanam, pravartthana vishakalanam, oopparettar sidhaantham ennivayil, stephaan bergmante peril oru bergman space, sankeernamaaya thalatthinte oru domaine "dii" yill holomorphic functionukalude oru function space aanu, avaa poornnamaayum samyojippikkaavunna athirthiyil vendathra nannayi perumaarunnu. prathyekichum, "p"-norm parimithamaaya "d" leeii ellah holomorphic functionukaludeyum formula _ 1-nte spaceaanu bergman space
alavu soothravaakyathe functionte "maanadandam" ennu vilikkunnu; formula 4 aanenkil ithu oru yadhaartha maanadandamaanu. athinaal l "p" ("d") enna sthalathulla holomorphic functionukalude upavibhagamaanu ithu. bergman speysukal banach speysukalaanu, ithu esttimettinte anantharafalamaanu, ithu "dii" yude compact subsettukalil saadhuthayullathaanu
athinaal holomorphic functionukalude oru shreniyude samyojanam compact kanverjansineyum soochippikkunnu, athinaal limitt function holomorphic aanu.
oru punarnirmmaana kernel hilbert space aanenkil, athinte kernel bergman kernel nalkunnu.
prathyeka kesukalum saamaanyavalkkaranangalum.
domaine parimithamaanengil, maanadandam palappozhum nalkunnathu
ivide soothravaakyam _ 6 ennathu sankeernamaaya thalatthinte oru saadhaarana lebsg alavaanu, i.iii. . sthalam pariganikkathe, pakaramaayi upayogikkunnu.
bergman space saadhaaranayaayi sankeernna thalatthinte ooppan yoonit disk formula _ 7 ll nirvachikkappedunnu, eee saahacharyathil formula _ 8. hilbert space kesil, formula _ 9 nalkiyirikkunnuvenkil, namukk undu.
athaayath, bharamullah "l" p "(1/(" n "+ 1)) sthalathinu isometric aayi isomorphic aanu. prathyekichum polynomialukal saandramaanu. athupole, valath (allengil mukalile) sankeernamaaya pakuthi thalam aanenkil,
ivide soothravaakyam _ 12, athaayath, bharamullah "l" p "1/" t "(0, φ) space (laplace parivartthanatthiloode) isometric aayi isomorphic aanu.
bharamullah bergman space oru samana reethiyil nirvachichirikkunnu, i.iii.
aaa reethiyil thiranjedutthaal, aaa formula _ 14 oru banach space aanu (allengil oru hilbert space, iff). soothravaakyam _ 15 ullappol, oru waited bergman space formula _ 16 upayogichu nammal ardhamaakkunnathu ellah vishakalana pravarthanangaludeyum space ennaanu.
athupole thanne valath pakuthi thalatthilum (i.iii. soothravaakyam _ 18) namukkund
eee space laplace parivartthanatthiloode, space formula _ 20 lake isometric aayi isomorphic aanu,
(ivide gamma functione soochippikkunnu).
kooduthal saamaanyavalkkaranangal chilappol pariganikkunnu, udaaharanathinu formula _ 22 ennathu clossed raitt complexes haff-playin formula _ 24 leeii translation-invariyant positteev regular boral alavedukkal formula _ 23 maayi bandhappett oru waited bergman space (palappozhum seen space ennu vilikkunnu) soochippikkunnu, athaayath
kernelukal punarnirmmikkunnu.
pooyint formula 27-nte punarnirmmikkunna kernel formula _ 26 nalkunnu
athupole formula _ 29-num namukkund.
pothuve, formula _ 31 oru domaine formula _ 32 oru domaine formula _ 33-lake anuroopamaayi mapp cheiyukayaanenkil,
bharamullah saahacharyathil njangalkk
oppam

