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 red Nehaj R displaystyle R nbsp pregilbertiv prostir Elementi x R displaystyle x in R nbsp y R displaystyle y in R nbsp nazivayutsya ortogonalnimi yaksho yih skalyarnij dobutok dorivnyuye 0 tobto x y 0 displaystyle langle x y rangle 0 nbsp sho poznachayetsya x y displaystyle x perp y nbsp 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 nbsp prostoru R displaystyle R nbsp viznachnik Gramma dorivnyuye 0 to ci vektori linijno zalezhni V Evklidovomu prostori red V 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 red Dvi dijsni funkciyi f x displaystyle f x nbsp ta g x displaystyle g x nbsp ye ortogonalnimi odna shodo odnoyi u intervali a x b displaystyle a leq x leq b nbsp yaksho a b f x g x d x 0 displaystyle int a b f x g x dx 0 nbsp Analogiyeyu do ponyattya ortogonalnosti ye vektorna teoriya de u trohvimirnomu vipadku dva vektori A B displaystyle A B nbsp 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 nbsp U n displaystyle n nbsp vimirnomu prostori vektori ortogonalni yaksho i 1 n A i B i 0 displaystyle sum i 1 n A i B i 0 nbsp U displaystyle infty nbsp vimirnomu prostori u yakomu A i B i displaystyle A i B i nbsp mayut neperervnij rozpodil i displaystyle i nbsp ye neperervnoyu zminnoyu f x displaystyle f x nbsp takim chinom i 1 n A i B i displaystyle sum i 1 n A i B i nbsp perehodit u A x B x d x displaystyle int A x B x dx nbsp Ponyattya funkciyi perevoditsya takim chinom u ponyattya vektora u displaystyle infty nbsp 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 nbsp 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 nbsp funkciyi f x displaystyle f x nbsp i neobhidno rozklasti yiyi po naboru linijno nezalezhnih funkcij f i x displaystyle f i x nbsp dlya yakoyi isnuye a b f i x 2 d x displaystyle int a b f i x 2 dx nbsp 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 nbsp 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 nbsp zaminyuyetsya takim samim chislom chislom novih funkcij f 1 x f 2 f n x displaystyle varphi 1 x varphi 2 varphi n x nbsp 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 nbsp Takij algoritm maye nazvu procesu Grama Shmidta Na konturah takozh mozhna zastosovuvati ortogonalizaciyu V takomu vipadku a b displaystyle int a b nbsp zaminyuyetsya na C displaystyle int C nbsp Funkciya f 1 x displaystyle varphi 1 x nbsp maye viglyad a f 1 x displaystyle af 1 x nbsp de a displaystyle a nbsp otrimuyetsya z umovi a b f 1 2 d x 1 displaystyle int a b varphi 1 2 dx 1 nbsp 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 nbsp Takim chinom znahodyachi pershi n displaystyle n nbsp funkcij f 1 x f 2 f n x displaystyle varphi 1 x varphi 2 varphi n x nbsp prihodimo do funkciyi f n 1 x displaystyle varphi n 1 x nbsp yaka povinna buti linijnoyu kombinaciyeyu cih funkcij a takozh funkciyi f n 1 x displaystyle f n 1 x nbsp 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 nbsp cej viraz mozhna pomnozhiti na f i x displaystyle varphi i x nbsp 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 nbsp Umova a b f 1 2 d x 1 displaystyle int a b varphi 1 2 dx 1 nbsp daye a displaystyle a nbsp Shob poslidovno obchisliti f 1 x f 2 f n x displaystyle varphi 1 x varphi 2 varphi n x nbsp 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 nbsp 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 nbsp de D n displaystyle Delta n nbsp Viznachnik Grama dlya funkciyi f 1 f 2 f n displaystyle f 1 f 2 f n nbsp 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 nbsp Funkciyi f 1 x f 2 x f n x displaystyle f 1 x f 2 x f n x nbsp ye linijno nezalezhnimi yaksho viznachnik dorivnyuye nulyu Posilannya red Kudryavcev L D Matematicheskij analiz t 2 s 331 Kudryavcev L D s 331Div takozh red nbsp Portal Matematika Perpendikulyarnist Bazis matematika Ryad Fur ye Ryad Tejlora Ortogonalnist himiya nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Ortogonalnist matematika amp oldid 40985096
