Answered: Ex.1 Let f(x) be a 27-periodic and odd… | bartleby

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.5: Applications Of Inner Product Spaces

Problem 89E

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could you explain how to get that answer, as well as disprove the other options

Ex.1 Let f(x) be a 27-periodic and odd function, coinciding with the function (x) = x²
in the interval [0, 7), and such that ƒ(k) = 0, Vk = Z. The Fourier series of f is:
(A) not converging in quadratic norm to f(x) on [-π,π]
pointwise converging to f(x) on R
∞
2π2
(C) in the form
3
+Σak cos(kx), ak #0
k=1
∞
2π2
(D) in the form
3
+Σ bk sin(kx), bk 0
k=1

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Exercise 4. Consider the following function f(x) = 2|x| if -л≤ x ≤π, extended to the entire real line as a periodic function with period 2. (a) (5 points) Sketch a graph of ƒ on the real line. (b) (30 points) Calculate the Fourier series of f. (c) (5 points) For which values of x = R does the value of the Fourier series of f coincide with f(x)?
Suppose that f(t) is periodic with period [-n, 7) and has the following real Fourier coefficients: a1 = 1, b1 an = 4, a2 = 3, az = 3, 2, b, = -3, bz = 0, (A) Write the beginning of the real Fourier series of f(t) (through frequency 3): f(t) = (B) Give the real Fourier coefficients for the following functions: (i) The derivative f' (t) , a1 = a2 = -6 az = b1 = , b2 = -6 b3 = (ii) The function f(t) – 2 , a1 = , a2 = 3 , az = 3 b, = , b2 = , b3 = (iii) The antiderivative of (f(t) – 2) (with C 0) an = , a1 = , a2 3/2 , аз — b, = b2 = 3/2 , b3 = (iv) The function f(t) + 3 sin(3t) +3 cos(2t) an = , a1 = 1 , a2 = 6 , az = 3 b, = 2 , b2 = -3 bz = 3 (iv) The function f(2t) an = , a1 = , a2 = , аз — b1 = b2 = bz = 0
Calculate the first few terms of the Fourier series of ao ao 2 and L is the half-period of f. a1 a2 Enter simplified expressions for the coefficients ao, a₁, a2, a3, b₁, b2 and b3, where the Fourier series is given by a3 || = f(x): || || = f(x): 2. sin(x), -2.sin(x), 0 < x < 1/1/2 − < x < 0 ' Σ (am m=1 • COS m. π X L b₁ b₂ b3 || ƒ (x + π) = f(x) || + bm sin m• π• xX L
Assume that there is a Fourier series converging to the function f defined by х, —2 <х < 0, f(x) = {-x, 0sx< 2 f(x): the first coefficient in this Fourier series ao=

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