The Chebyshev polynomials of the first kind can be otained from the recurrence relation, Tn+1(x) = 2xTn(x) − Tn−1(x) with T0(x) = 1 and T1(x) = x. a) Show that any two Chebyshev polynomials are orthogonal with respect to the weighting factor (1 − x)-1/2 in the closed interval [−1, 1]. b) Pick the first three Chebyshev polynomials and use Gram-Schmidt orthonormalization procedure to form an orthonormal set in the closed interval [−1, 1].
The Chebyshev polynomials of the first kind can be otained from the recurrence relation, Tn+1(x) = 2xTn(x) − Tn−1(x) with T0(x) = 1 and T1(x) = x. a) Show that any two Chebyshev polynomials are orthogonal with respect to the weighting factor (1 − x)-1/2 in the closed interval [−1, 1]. b) Pick the first three Chebyshev polynomials and use Gram-Schmidt orthonormalization procedure to form an orthonormal set in the closed interval [−1, 1].
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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The Chebyshev polynomials of the first kind can be otained from the recurrence relation, Tn+1(x) = 2xTn(x) −
Tn−1(x) with T0(x) = 1 and T1(x) = x.
a) Show that any two Chebyshev polynomials are orthogonal with respect to the weighting factor (1 − x)-1/2 in the closed interval [−1, 1].
b) Pick the first three Chebyshev polynomials and use Gram-Schmidt orthonormalization procedure to form an
orthonormal set in the closed interval [−1, 1].
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