Science:Math Exam Resources/Courses/MATH307/April 2012/Question 06 (a)
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Question 06 (a) 

A continuous measurement of the temperature in Vancouver, y(t), for , is decomposed as a Fourier series of the form where . What important property do the set of functions have? What is the expression for the Fourier coefficients c_{n} in terms of y(t)? 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2012/Question 06 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. For the first half of the question, the set of functions form an orthonormal basis for the space . Using Euler’s formula we can express as So the temperature function y(t) can be viewed as a wavelike function with period T decomposed into a combination of sinusoidal waves. The Fourier series represents it as a purely oscillatory components with frequency . Here the phase is 0. For the Fourier coefficients , since we have y(t) = then we an wirte as
