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Chapter 6: Problem 2

Use the Gram-Schmidt process to find the first four orthogonal polynols satisfying the following: a. Interval: \((-\infty, \infty)\) Weight Function: \(e^{-x^{2}}\). b. Interval: \((0, \infty)\) Weight Function: \(e^{-x}\).

Short Answer

Expert verified
The first four orthogonal polynomials satisfying the conditions for part (a) are \(P_{0}(x) = 1\), \(P_{1}(x) = x\), \(P_{2}(x) = x^2 - 1\), and \(P_{3}(x) = x^3 - 3x\). For part (b), those are \(Q_{0}(x) = 1\), \(Q_{1}(x) = x - 1\), \(Q_{2}(x) = x^2 - 4x + 2\), and \(Q_{3}(x) = x^3 - 9x^2 + 18x - 6\).

Step by step solution

01

Part (a): Gram-Schmidt process with interval: (-∞, ∞) and weight function: \(e^{-x^{2}}\)

Consider the first four monomials \(1,x,x^{2},x^{3}\). Use the Gram Schmidt Process:1. The first polynomial \(P_{0}(x) = 1\)2. Next, use the idea of Gram Schmidt Process to find \(P_{1}(x), P_{2}(x), P_{3}(x)\). i. Orthogonalize: Subtract off the part of \(x, x^{2}, x^{3}\) that is in the direction of \(P_{0}(x)\) ii. Normalize: Divide by its norm.3. Compute them using integrals, we get: \(P_{1}(x) = x\), \(P_{2}(x) = x^2 - 1\), \(P_{3}(x) = x^3 - 3x\)
02

Part (b): Gram-Schmidt process with interval (0, ∞) and weight function \(e^{-x}\)

Again, consider the first four monomials \(1, x, x^{2}, x^{3}\). 1. The first polynomial \(Q_{0}(x) = 1\)2. Then apply Gram Schmidt Process to find \(Q_{1}(x),Q_{2}(x),Q_{3}(x)\) following the same process as in part (a)3. Compute them using integrals, we get: \(Q_{1}(x) = x - 1\), \(Q_{2}(x) = x^2 - 4x + 2\), \(Q_{3}(x) = x^3 - 9x^2 + 18x - 6\)

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