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1.F.3 One of my favorite bases of P<4 is X = {1,x,2x² – 1,4x3 – 3r, 8x“ – 822 + 1}. Find the coordinates of 1, x, x², x³, and xª with respect to X. Aside: These polynomials are called Chebyshev polynomials (of the first kind). You may remember your trig identity cos(26) = 2 cos²(8) – 1. If x = cos(0) then cos(20) = 2x² – 1. Similarly, cos(30) = 4.x3 – 3x and cos(40) = 8x4 – 8x² +1. So, when you found the coordinates of x4 above, you secretly found a way to write cos (0) as a linear combination of {1, cos(0), cos(20), cos(30), cos(40)}.

1.F.3 One of my favorite bases of P<4 is X = {1,x,2x² – 1,4x3 – 3r, 8x“ – 822 + 1}. Find the coordinates of 1, x, x², x³, and xª with respect to X. Aside: These polynomials are called Chebyshev polynomials (of the first kind). You may remember your trig identity cos(26) = 2 cos²(8) – 1. If x = cos(0) then cos(20) = 2x² – 1. Similarly, cos(30) = 4.x3 – 3x and cos(40) = 8x4 – 8x² +1. So, when you found the coordinates of x4 above, you secretly found a way to write cos (0) as a linear combination of {1, cos(0), cos(20), cos(30), cos(40)}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1.F.3 One of my favorite bases of P<4 is X = {1,x,2x² – 1,4x3 – 3r, 8x“ – 822 + 1}. Find the coordinates of 1, x, x², x³, and xª with respect to X. Aside: These polynomials are called Chebyshev polynomials (of the first kind). You may remember your trig identity cos(26) = 2 cos²(8) – 1. If x = cos(0) then cos(20) = 2x² – 1. Similarly, cos(30) = 4.x3 – 3x and cos(40) = 8x4 – 8x² +1. So, when you found the coordinates of x4 above, you secretly found a way to write cos (0) as a linear combination of {1, cos(0), cos(20), cos(30), cos(40)}.
Transcribed Image Text:1.F.3 One of my favorite bases of P<4 is X = {1,x,2x² – 1,4x3 – 3r, 8x“ – 822 + 1}. Find the coordinates of 1, x, x², x³, and xª with respect to X. Aside: These polynomials are called Chebyshev polynomials (of the first kind). You may remember your trig identity cos(26) = 2 cos²(8) – 1. If x = cos(0) then cos(20) = 2x² – 1. Similarly, cos(30) = 4.x3 – 3x and cos(40) = 8x4 – 8x² +1. So, when you found the coordinates of x4 above, you secretly found a way to write cos (0) as a linear combination of {1, cos(0), cos(20), cos(30), cos(40)}.
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