Suppose n is a positive integer. Prove that 1 cos x cos 2x cos nx sin x sin 2x sin nx IT is an orthonormal list of vectors in C[-x, ], the vector space of contin- uous real-valued functions on [–T, 1] with inner product IT (f, 8) = | f(x)g(x) dx.

Suppose n is a positive integer. Prove that 1 cos x cos 2x cos nx sin x sin 2x sin nx IT is an orthonormal list of vectors in C[-x, ], the vector space of contin- uous real-valued functions on [–T, 1] with inner product IT (f, 8) = | f(x)g(x) dx.

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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Question

4 Suppose n is a positive integer. Prove that
1
COS X соs 2x
cos nx sin x sin 2x
sin nx
IT
is an orthonormal list of vectors in C[-n, I], the vector space of contin-
uous real-valued functions on [-x, 1] with inner product
(f. g)
=| f(x)g(x)dx.
IT
[The orthonormal list above is often used for modeling periodic phenom-
ena such as tides.]

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