2k +1 All zeros of T+1(r) are in the interval [-1,1] and given by r = cos where k E {0,1, 2, . , n}. ... 2n + 2
2k +1 All zeros of T+1(r) are in the interval [-1,1] and given by r = cos where k E {0,1, 2, . , n}. ... 2n + 2
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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only d part
![3. The Chebyshev polynomials of the first kind are defined as
= cos (n arccos(r))
(a) Show that the Chebyshev polynomials satisfy the orthogonality condition
T„(x)Tm(x)
(b) Show that the Chebyshev polynomials of the first kind satisfy the recurrence:
Tn+1 = 2xTn – Tn-1
with To(x) = 1 and T(x) = 1.
%3D
(c) Show that T,(x) is a polynomial of degree n with leading coefficient as 2"-1 for n > 1.
2k +1
(d) All zeros of T+1(x) are in the interval [-1, 1] and given by r = cos
where k E {0,1, 2, ...,n}.
2n +2
(e) Conclude that Tn(x) alternates between +1 and -1 exactly n+1 times.
(f) Show that
1
I
2n
k%3D0
for all r € [-1, 1].
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64601ca5-1c38-465b-bbda-122f6a2674ca%2F3fe31702-e11d-4493-ba1a-91a0059a636b%2Fvm37yiq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. The Chebyshev polynomials of the first kind are defined as
= cos (n arccos(r))
(a) Show that the Chebyshev polynomials satisfy the orthogonality condition
T„(x)Tm(x)
(b) Show that the Chebyshev polynomials of the first kind satisfy the recurrence:
Tn+1 = 2xTn – Tn-1
with To(x) = 1 and T(x) = 1.
%3D
(c) Show that T,(x) is a polynomial of degree n with leading coefficient as 2"-1 for n > 1.
2k +1
(d) All zeros of T+1(x) are in the interval [-1, 1] and given by r = cos
where k E {0,1, 2, ...,n}.
2n +2
(e) Conclude that Tn(x) alternates between +1 and -1 exactly n+1 times.
(f) Show that
1
I
2n
k%3D0
for all r € [-1, 1].
1
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