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Theorem on Orthogonal Polynomials The sequence of polynomials defined inductively as follows is orthogonal Pn(x) = (x − an) Pn-1(x) — bn Pn-2(x) (n ≥ 2) with po(x) = 1, p₁(x) = x − a₁, and - an = : (XPn-1, Pn-1)/(Pn-1, Pn−1) : (xPn-1, Pn-2)/(Pn-2, Pn-2)

Theorem on Orthogonal Polynomials The sequence of polynomials defined inductively as follows is orthogonal Pn(x) = (x − an) Pn-1(x) — bn Pn-2(x) (n ≥ 2) with po(x) = 1, p₁(x) = x − a₁, and - an = : (XPn-1, Pn-1)/(Pn-1, Pn−1) : (xPn-1, Pn-2)/(Pn-2, Pn-2)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 3E
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Using Theorem 5 directly, find p0, p1, p2, p3 for [a, b] = [0, 1] for w(x) = 1

THEOREM 5 Theorem on Orthogonal Polynomials The sequence of polynomials defined inductively as follows is orthogonal: Pn(x) = (x-an) Pn-1(x) — bn Pn-2(x) (n ≥ 2) with po(x) = 1, p₁(x) = x − a₁, and an = (xPn-1, Pn-1)/(Pn-1, Pn-1) bn = (xPn-1, Pn-2)/(Pn-2, Pn-2)
Transcribed Image Text:THEOREM 5 Theorem on Orthogonal Polynomials The sequence of polynomials defined inductively as follows is orthogonal: Pn(x) = (x-an) Pn-1(x) — bn Pn-2(x) (n ≥ 2) with po(x) = 1, p₁(x) = x − a₁, and an = (xPn-1, Pn-1)/(Pn-1, Pn-1) bn = (xPn-1, Pn-2)/(Pn-2, Pn-2)
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