Exercise $3$ Data smoothing with Fourier transforms $[10$ points$]$

The attached file dow.txt contains the daily closing value for the Dow Jones Industrial Average

$($a stock market index$)$ from late $2006$ until the end of $2010.$

$($i$)$ Read in the data from dow.txt and plot them on a graph.

$($ii$)$ Calculate the coefficients of the discrete Fourier transform of the data. Smooth the data

by setting the last $90\%$ of the Fourier coefficients to zero and then calculating the inverse

Fourier transform. Remember that the numerical coefficients are mirror symmetric, so

the $90\%$ apply to either half of the coefficients, as shown in the example in the lecture

notes. Plot the result together with the original data.

$($iii$)$ Repeat the above process, but now set the last $98\%$ of the coefficients to zero before

the inverse Fourier transform. Plot the result together with the original data and the

smoothed data from $($ii$).$ Explain how you can influence the degree of smoothing.