Aproximarea functionalelor liniare

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$P 0.1 Stabilişti natura şsi suma seriilor: a) Σ 1 n\ ; b) Σ arctan 1 n# + n ...$

Introducere Derivare numerică Integrare numerică Cuadraturi adaptive Cuadraturi . . . Cuadraturi . . . Formule . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 51 of 58 Go Back Full Screen Close Quit 7.1. Proprietăt¸i ale cuadraturilor gaussiene Regula de cuadratură a lui Gauss, dată de (32) ¸si (38), pe lângă faptul că este optimală (adică are grad maxim de exactitate) are ¸si unele proprietăt¸i interesante. (i) Toate nodurile sunt reale distincte ¸si situate în intervalul deschis (a, b). Aceasta este o proprietate cunoscută satisfăcută de zerourile polinoamelor ortogonale.
Introducere Derivare numerică Integrare numerică Cuadraturi adaptive Cuadraturi . . . Cuadraturi . . . Formule . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 51 of 58 Go Back Full Screen Close Quit 7.1. Proprietăt¸i ale cuadraturilor gaussiene Regula de cuadratură a lui Gauss, dată de (32) ¸si (38), pe lângă faptul că este optimală (adică are grad maxim de exactitate) are ¸si unele proprietăt¸i interesante. (i) Toate nodurile sunt reale distincte ¸si situate în intervalul deschis (a, b). Aceasta este o proprietate cunoscută satisfăcută de zerourile polinoamelor ortogonale. (ii) Tot¸i coeficient¸ii (ponderile) wk sunt pozitivi. Demonstrat¸ia se bazează pe o observat¸ie ingenioasă a lui Stieltjes 0 < b a ℓ 2 j (t)w(t)dt = n wkℓ 2 j (tk) = wj, j = 1, 2, . . . , n, k=1 prima egalitate rezultând din faptul că gradul de exactitate este d = 2n − 1.
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Aproximarea functionalelor liniare

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