Suppose that {Pn (x)}n20 is a system of orthogonal polynomials on the in- terval [a, b] with positive integrable weight p(x); namely, deg p, = n and Pn(x) Pm(x)p(x)dx = 0 for any n + m. Show that each pn(x) possesses n simple roots in the in- terval (a, b) and that there exists an exactly one root of pn-1(x) between any consecutive roots of p„(x). Note that Po(x) is a non-zero constant.
Suppose that {Pn (x)}n20 is a system of orthogonal polynomials on the in- terval [a, b] with positive integrable weight p(x); namely, deg p, = n and Pn(x) Pm(x)p(x)dx = 0 for any n + m. Show that each pn(x) possesses n simple roots in the in- terval (a, b) and that there exists an exactly one root of pn-1(x) between any consecutive roots of p„(x). Note that Po(x) is a non-zero constant.
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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![Suppose that {pn(x)}n>0 is a system of orthogonal polynomials on the in-
terval [a, b] with positive integrable weight p(x); namely, deg p, = n and
Pa(x) Pm(x)p(x)dx = 0
for any n + m. Show that each p„(x) possesses n simple roots in the in-
terval (a, b) and that there exists an exactly one root of pn-1(x) between any
consecutive roots of p„(x). Note that po(x) is a non-zero constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25bf4691-7e3a-4c14-be0e-69758f750a69%2F7d47f16e-5471-4a2c-a099-a457eb4eab6c%2F76azjrl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that {pn(x)}n>0 is a system of orthogonal polynomials on the in-
terval [a, b] with positive integrable weight p(x); namely, deg p, = n and
Pa(x) Pm(x)p(x)dx = 0
for any n + m. Show that each p„(x) possesses n simple roots in the in-
terval (a, b) and that there exists an exactly one root of pn-1(x) between any
consecutive roots of p„(x). Note that po(x) is a non-zero constant.
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