Show that {1, cos(kx) | k ¤ N} is an orthogonal family on [0, π] with inner product (f,g) = f* f(x) g(x) dx. 0
Show that {1, cos(kx) | k ¤ N} is an orthogonal family on [0, π] with inner product (f,g) = f* f(x) g(x) dx. 0
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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Please show a step-by-step solution. Do not skip steps, and explain your steps. Write it on paper, preferably. Make sure the work is clear.
![5. (a) Show that {1, cos(kx) | k € N} is an orthogonal family on [0, π] with inner product
(f,g) = f* f(x) g(x) dx.
(b) Find the formulas for the coefficients in a Fourier Series in these functions, f(x) ~
S(x) = +Σk-1 ak cos(kx).
(c) Show that S(r) represents the even, 27-periodic extension of f(x) from 0<x< T
to the real line. Draw the graph of S(x) on −37 < x < 3π when f(x) = x.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cb378ba-be87-4838-bd31-84fa6b2aaf1f%2F4a3b1520-327a-4046-bf18-f1fd7226eaa4%2Fwtaudy_processed.png&w=3840&q=75)
Transcribed Image Text:5. (a) Show that {1, cos(kx) | k € N} is an orthogonal family on [0, π] with inner product
(f,g) = f* f(x) g(x) dx.
(b) Find the formulas for the coefficients in a Fourier Series in these functions, f(x) ~
S(x) = +Σk-1 ak cos(kx).
(c) Show that S(r) represents the even, 27-periodic extension of f(x) from 0<x< T
to the real line. Draw the graph of S(x) on −37 < x < 3π when f(x) = x.
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