integralukalude vyathyasam

ganithashaasthrathil, integralukalude deferentiationte prashnam, eathu saahacharyathilaanu oru pooyintinte oru cheria ayalpakkathu anuyojyamaaya functionte sharashari moolya integral aaa ghattathil functionte moolyathe ekadeshamaayi kanakkakkunnathennu nirnnayikkuka ennathaanu. kooduthal aupachaarikamaayi, "x" ennathil "m" enna alavum "d" enna metrik alavum ulla oru space nalkiyaal, "f": "x" → r eathu functionukalaanu cheiyunnathennu oraal chodikkunnu.
ellaavarkkum (allengil kuranjathu "μ"-mikkavarum ellaam) "x" ± "x"? (ivide, lekhanathinte backie bhaagangalilennapole, "bii" aarr "(" ex ") ennathu" dii "-radius" aarr ", sentar" ex "ennivayulla" ex "leeii ooppan booal soochippikkunnu.) ithu chodikkenda oru swaabhaavika chodyamaanu, prathyekichu riman integralinte huristic nirmmaanam kanakkiledukkumbol, athil "x" nuu sameepamulla "f" nte moolyangalkku "f" ("x") oru "nalla prathinidhi" aanennu eathaandu soochippikkunnu.
integralukalude deferentiationekkurichulla sidhaanthangal.
lebsg alavu.
1910-ll henry lebsgu theliyichathupole, integralukalude deferentiationte oru phalam lebsgu deferentiation sidhaanthamaanu. "n"-dimensional euclidean space r "n-ll" n "-dimensional lebsgu alavu" λ "n" pariganikkuka. appol, eathenkilum praadeshikamaayi samyojippikkaavunna functionaaya "f": r "n" → r-nuu, oraalkk undu
λ" "n"-mikkavarum ellah pooyintukalum "x" ωr "n". ennirunnaalum, poojyamaaya "mosham" pooyintukalude alavu "f" functione aasrayichirikkunnu ennathu sradhikkendathaanu.
r "n" ll boral alakkunnu.
lebsg alavinte phalam inipparayunna phalatthinte oru prathyeka kesai maarunnu, ithu besikovich covering sidhaanthathe adisthaanamaakkiyullathaanu "m" ennathu r "n", "f" ennivayil eathenkilum praadeshikamaayi parimithamaaya boral alavaanengil r "n" → r ennathu "m" maayi bandhappett praadeshikamaayi samyojippikkaavunnathaane,
" "" ""-eathaandu ellah pooyintukalum "" x "" ωr "" n ".
gausiyan nadapadikal.
ananthamaaya alavilulla krameekaranatthil integralukalude deferentiation prashnam valare budhimuttaanu. gausiyan alavu "γ" ulla verthirikkaavunna hilbert space ("h", λ, γ) pariganikkuka. vaitti covering sidhaanthathekkurichulla lekhanathil paranjirikkunnathupole, vaitti covering sidhaantham ananthamaaya alavilulla hilbert speysukalile gausiyan alavukalkku paraajayappedunnu. david preesinte (1981,1983) rund phalangal eee pashchaathalatthil oraalkk neridendivarunna budhimuttukal kaanikkunnu
ennirunnaalum, "γ" yude koveriyansil oraalkk nalla niyanthranam undengil kurachu pratheekshayund. γ"-nte covarians oopparettar "s" aayirikkatte "h" → "h
allengil, kanakkakkavunna chila oorthoonoormal adisthaanathil ("e" i ")" i "±n" h ",
1981ll, prease jaroslav tisser kaanichathu oru sthiraangam undengil 0 <"q" <1 ennu
appol, ellah "f" ωl "l" 1 ("h", "γ"; r) num,
ivide "γ" maayi bandhappett alavil kanverjans aanu. 1988ll, tisher kaanichu,
chilarkk "α"> 5⁄2, pinne
γ"-eathaandu ellah "x" umm ellah "f" ωl "l" p "(" h "," γ "; r)," p "> 1 umm.
2007 leeii kanakkanusarichu, verthirikkaavunna hilbert space "echu" ll ananthamaaya alavilulla oru gausiyan alavu "γ" undo ennathu ippozhum oru thuranna chodyamaanu, angane ellah "eph" ωl "1 (" echu "," γ "; aarr),
γ"-eathaandu ellah "x" ω "h". ennirunnaalum, "σ" "i" valare vegathil kshayikkendivarumennathil atharam alavukalonnum nilavilillenn anumaanikkappedunnu.

