Ortogonalnist vid grec ὀr8os pryamij i grec gwnia kut termin yakim poznachayut perpendikulyarnist vektoriv Linijni vidrizki AB i CD ye ortogonalnimi odin do odnogo Zmist 1 Viznachennya 2 V Evklidovomu prostori 3 Ortogonalni funkciyi 4 Posilannya 5 Div takozhViznachennya RedaguvatiNehaj R R pregilbertiv prostir Elementi x R displaystyle x in R y R displaystyle y in R nazivayutsya ortogonalnimi yaksho yih skalyarnij dobutok dorivnyuye 0 tobto x y 0 displaystyle langle x y rangle 0 sho poznachayetsya x y displaystyle x perp y 1 Mnozhina vektoriv nazivayetsya ortogonalnoyu yaksho dovilna para z ciyeyi mnozhini ortogonalna Yaksho vsi vektori ciyeyi mnozhini odinichni to vona nazivayetsya mnozhinoyu ortnormovanih vektoriv Ne nulovi ortogonalni vektori linijno nezalezhni 2 Yaksho dlya sistemi vektoriv x 1 x 2 x n displaystyle x 1 x 2 ldots x n prostoru R R viznachnik Gramma dorivnyuye 0 to ci vektori linijno zalezhni V Evklidovomu prostori RedaguvatiV 2 abo 3 vimirnomu Evklidovomu prostori dva vektori ortogonalni yaksho skalyarnij dobutok cih vektoriv dorivnyuye nulyu tobto kut mizh nimi 90 abo p 2 radian Takim chinom ortogonalnist vektoriv ye uzagalnennyam perpendikulyarnosti V Evklidovih pidprostorah ortogonalnim dopovnennyam pryamoyi ye ploshina i navpaki Ortogonalni funkciyi RedaguvatiDvi dijsni funkciyi f x f x ta g x displaystyle g x ye ortogonalnimi odna shodo odnoyi u intervali a x b displaystyle a leq x leq b yaksho a b f x g x d x 0 displaystyle int a b f x g x dx 0 Analogiyeyu do ponyattya ortogonalnosti ye vektorna teoriya de u trohvimirnomu vipadku dlya vektori A B displaystyle A B ye ortogonalnimi koliA B A 1 B 1 A 2 B 2 A 3 B 3 0 displaystyle A cdot B A 1 B 1 A 2 B 2 A 3 B 3 0 U n n vimirnomu prostori vektori ortogonalni yaksho i 1 n A i B i 0 displaystyle sum i 1 n A i B i 0 U infty vimirnomu prostori u yakomu A i B i displaystyle A i B i mayut neperervnij rozpodil i i ye neperervnoyu zminnoyu f x displaystyle f x takim chinom i 1 n A i B i displaystyle sum i 1 n A i B i perehodit u A x B x d x displaystyle int A x B x dx Ponyattya funkciyi perevoditsya takim chinom u ponyattya vektora u infty vimirnomu prostori Integral a b f x g x d x f g displaystyle int a b f x g x dx f cdot g viznachaye skalyarnij dobutok u funkcionalnomu prostori U takomu prostori skalyarnij vnutrishnij dobutok viznachayetsya tak samo yak j u skinchennih vektornih prostorah vidpovidno takim samim chinom mozhna viznachiti ortogonalnist Yaksho dana pohidna neperervna na vidrizku a b displaystyle a b funkciyi f x f x i neobhidno rozklasti yiyi po naboru linijno nezalezhnih funkcij f i x displaystyle f i x dlya yakoyi isnuye a b f i x 2 d x displaystyle int a b f i x 2 dx to mozhna useredneno aproksimuvati yiyi linijnoyu sukupnistyu i 1 n c i f i x displaystyle sum i 1 n c i f i x Koeficiyenti pidibrati vazhko yaksho nabir ye ortonormovanim U procesi ortogonalizaciyi funkciyi f 1 x f 2 f n x displaystyle f 1 x f 2 f n x zaminyuyetsya takim samim chislom chislom novih funkcij f 1 x f 2 f n x displaystyle varphi 1 x varphi 2 varphi n x yaki ye linijnimi kombinaciyami poperednih funkcij tobtof n x c 1 n f 1 x c 2 n f 2 x c n n f n x displaystyle varphi n x c 1 n f 1 x c 2 n f 2 x c n n f n x Takij algoritm maye nazvu procesu Grama Shmidta Na konturah takozh mozhna zastosovuvati ortogonalizaciyu V takomu vipadku a b displaystyle int a b zaminyuyetsya na C displaystyle int C Funkciya f 1 x displaystyle varphi 1 x maye viglyad a f 1 x displaystyle af 1 x de a a otrimuyetsya z umovi a b f 1 2 d x 1 displaystyle int a b varphi 1 2 dx 1 Mayemof 1 x f 1 x a b f 1 x 2 d x 1 2 displaystyle varphi 1 x f 1 x int a b f 1 x 2 dx 1 2 Takim chinom znahodyachi pershi n n funkcij f 1 x f 2 f n x displaystyle varphi 1 x varphi 2 varphi n x prihodimo do funkciyi f n 1 x displaystyle varphi n 1 x yaka povinna buti linijnoyu kombinaciyeyu cih funkcij a takozh funkciyi f n 1 x displaystyle f n 1 x Vidpovidno f n 1 x a 1 f 1 x a 2 f 2 x a n f n x f n i x displaystyle varphi n 1 x a 1 varphi 1 x a 2 varphi 2 x a n varphi n x f n i x cej viraz mozhna pomnozhiti na f i x displaystyle varphi i x j prointegruvati otrimanij viraza i a a b f n 1 x f i x d x 0 displaystyle a i a int a b f n 1 x varphi i x dx 0 Umova a b f 1 2 d x 1 displaystyle int a b varphi 1 2 dx 1 daye a displaystyle a Shob poslidovno obchisliti f 1 x f 2 f n x displaystyle varphi 1 x varphi 2 varphi n x mozhna zastosuvati rivnyannyaf n 1 x f n 1 x i 1 n a b f n 1 f i d x f i x a b f n 1 x i 1 n a b f n 1 f i d x f i x 2 d x 1 2 displaystyle varphi n 1 x frac f n 1 x sum i 1 n int a b f n 1 varphi i dx varphi i x int a b f n 1 x sum i 1 n int a b f n 1 varphi i dx varphi i x 2 dx 1 2 Abo cherez viznachniki mozhna zapisatif n x f 1 f 1 f 1 f 2 f 1 f n 1 f 1 x f 2 f 1 f 2 f 1 f 2 f n 1 f 2 x f n f 1 f n f 2 f n f n 1 f n x D n 1 D n 1 2 displaystyle varphi n x frac begin vmatrix f 1 cdot f 1 amp f 1 cdot f 2 amp amp f 1 cdot f n 1 amp f 1 x f 2 cdot f 1 amp f 2 cdot f 1 amp amp f 2 cdot f n 1 amp f 2 x amp amp amp amp f n cdot f 1 amp f n cdot f 2 amp amp f n cdot f n 1 amp f n x end vmatrix Delta n 1 cdot Delta n 1 2 de D n displaystyle Delta n Viznachnik Grama dlya funkciyi f 1 f 2 f n displaystyle f 1 f 2 f n D n f 1 f 1 f 1 f 2 f 1 f n f 2 f 1 f 2 f 1 f 2 f n f n f 1 f n f 2 f n f n displaystyle Delta n begin vmatrix f 1 cdot f 1 amp f 1 cdot f 2 amp amp f 1 cdot f n f 2 cdot f 1 amp f 2 cdot f 1 amp amp f 2 cdot f n amp amp amp f n cdot f 1 amp f n cdot f 2 amp amp f n cdot f n end vmatrix Funkciyi f 1 x f 2 x f n x displaystyle f 1 x f 2 x f n x ye linijno nezalezhnimi yaksho viznachnik dorivnyuye nulyu Posilannya Redaguvati Kudryavcev L D Matematicheskij analiz t 2 s 331 Kudryavcev L D s 331Div takozh Redaguvati Portal Matematika Perpendikulyarnist Bazis matematika Ryad Fur ye Ryad Tejlora Ortogonalnist himiya Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Ortogonalnist amp oldid 39696843
