In your own words explain why and how it is possible to decompose any function i
ID: 77228 • Letter: I
Question
In your own words explain why and how it is possible to decompose any function into a combination of sine and cosine functions. For the remainder of this question, consider the function T(x) = x2 + 1 on the interval [-1,1]. Since T(x) is an even* function of .x, it is sufficient to "Fourier Analyze" T(x) in the interval [0,1]. The behaviour of T(x) in [-1,0] can deduced from the resulting Fourier Series. (*NB: Make sure you know what an even function is.) On interval [0,1], expand T(x) using a Fourier cosine** series as: (**NB: Make sure you know why sine components are absent.) Compute T0 and a1 ... a5 . Plot*** the power spectrum using the harmonic coefficients obtained above. By substituting values of x from -3 to 3 using increments of 0.1 into the above series, plot*** the "harmonic extension" of T(x) on the interval [-33]. Comment on how closely your scries representation resembles the given function. It would help you make this comparison if you superimpose a plot of the original function over your series plot. ***NB: Cite software you use, if any, to produce your figure. Where appropriate describe what commands or steps you use to produce the figure.Explanation / Answer
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