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Suppose the first few Fourier coeficients of some function f in C[0,2x] are ao, a1, a2, and b,, b2, b3. Which of the following trigonometric polynomials is closer to f? Defend your answer. ao +a, cost+a, cos 2t + b, sint g(t) = ao h(t) = +a, cost+a, cos 2t +b, sint+ b, sin 2t Choose the correct answer below. O A. The function h(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and h(t) is the orthogonal projection of f onto that space. O B. The functions g(t) and h(t) are not members of the same space, and they are both orthogonal projections. It cannot be determined which function is closer to f. OC. The function g(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and g(t) is an orthogonal projection that requires fewer terms to approximate f. O D. The function h(t) is a member of a higher order space than g(t) so it must be a closer approximation of f.

Suppose the first few Fourier coeficients of some function f in C[0,2x] are ao, a1, a2, and b,, b2, b3. Which of the following trigonometric polynomials is closer to f? Defend your answer. ao +a, cost+a, cos 2t + b, sint g(t) = ao h(t) = +a, cost+a, cos 2t +b, sint+ b, sin 2t Choose the correct answer below. O A. The function h(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and h(t) is the orthogonal projection of f onto that space. O B. The functions g(t) and h(t) are not members of the same space, and they are both orthogonal projections. It cannot be determined which function is closer to f. OC. The function g(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and g(t) is an orthogonal projection that requires fewer terms to approximate f. O D. The function h(t) is a member of a higher order space than g(t) so it must be a closer approximation of f.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 72E
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Suppose the first few Fourier coefficients of some function f in C[0,2n] are an, a1, a2, and b,, b,, b3. Which of the following trigonometric polynomials is closer to f? Defend your answer. ao g(t) = + a, cost+a, cos 2t + b, sint 2 h(t) = + a, cost+a, cos 2t + b, sint+ b, sin 2t 2 ..... Choose the correct answer below. O A. The function h(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and h(t) is the orthogonal projection of f onto that space. O B. The functions g(t) and h(t) are not members of the same space, and they are both orthogonal projections. It cannot be determined which function is closer to f. OC. The function g(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and g(t) is an orthogonal projection that requires fewer terms to approximate f. O D. The function h(t) is a member of a higher order space than g(t) so it must be a closer approximation of f.
Transcribed Image Text:Suppose the first few Fourier coefficients of some function f in C[0,2n] are an, a1, a2, and b,, b,, b3. Which of the following trigonometric polynomials is closer to f? Defend your answer. ao g(t) = + a, cost+a, cos 2t + b, sint 2 h(t) = + a, cost+a, cos 2t + b, sint+ b, sin 2t 2 ..... Choose the correct answer below. O A. The function h(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and h(t) is the orthogonal projection of f onto that space. O B. The functions g(t) and h(t) are not members of the same space, and they are both orthogonal projections. It cannot be determined which function is closer to f. OC. The function g(t) is a better approximation because both functions are members of the space of trigonometric polynomials of order 2, and g(t) is an orthogonal projection that requires fewer terms to approximate f. O D. The function h(t) is a member of a higher order space than g(t) so it must be a closer approximation of f.
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