Above and beyond what is stated in the assignment, there is some assumed knowledge for this set. I'll include links here to various wikipedia articles and so on. However, your TA is very happy to workshop this stuff in class!
What does a fourier transform do? All transforms that switch between a time or spatial domain and a frequency or spectral domain essentially multiply the data by all frequencies, then perform a sum. As most data possesses no strong frequencies, the sum returns an average of zero. If, however, there is a frequency present in the data, then multiplying by that frequency will lead to non-random addition, and that frequency will be extracted by the sum (or integral, as the case may be). I find visualising the fourier transform as a stretchy slinky that's pulled through a time series to be quite illuminating, though possibly I may be unique in this regard…
Your ears perform a fourier transform (sort of). When you hear sounds, your inner ear nerves separate the sound into its frequency components automatically, although they cannot generally determine phase information. Sonolocation works in other ways! It is possible to 'hear' phase information by using a reference frequency, such as a tuning fork. This enables musicians to tune instruments very precisely. Playing them precisely is another matter, of course. Most people are able to distinguish frequencies within about 10% of a musical semi-tone, which is in turn a 12th of an octave, or doubling of pitch. This ammounts to a half a percent increase or decrease in frequency. Some trained musicians are more than 10 times as sensitive.
Using mathematica, I performed fourier analysis on a piece of music that I wanted to transcribe, to help me hear the chords. It can be seen on youtube here. This was achieved by performing the fft on 1/10s chunks of the music. Such an approach has poor response for lower frequencies, however, due to the opposite of the Nisquist sampling limit. For my next song, I performed a wavelet transform (very inefficient), which used a gaussian enveloped kernel to perform sampling at different, frequency dependent rates. The result can be seen here. No comments will be brokered concerning my musical taste, or lack thereof!
Musical tones are generally the opposite of pure tones, however, so the utility of such visualisations for transcription is undermined by the fact that every note also has its family of overtones (multiples of the fundamental frequency) present. It makes for pretty pictures, however!
Fourier transforms are often used in analysis for the solving of difficult differential equations. The advantage of the fourier transform is that it can turn derivatives into multiplicative factors, and thus differential equations into algebraic equations. The standard transform for solving tricky engineering problems is the Laplace transform, which is a Wick rotation of the fourier transform, and includes boundary conditions in a more natural way.
Analytical fourier transforms often depend upon a great deal of skill and knowledge in performing integrals, including via complex contour methods. A lot of this difficulty is bypassed by taking a numerical approach, however awareness of convergence criteria and other technicalities is essential for avoiding silly mistakes, or understanding what might have gone wrong.
If you have forgotten what orthogonal families of complex exponentials, sines, and cosines look like, do a mathematica notebook of them! For the continuous Fourier transform, the set of orthogonal spectral basis functions is continuous, because the integral of any two is the delta function. Discrete Fourier transforms, such as numerical ones, are by definition on non-infinite domains, and an ordered set of orthogonal basis functions is necessarily NOT continuous, but rather normally involves the resonant harmonics of the volume involved. I like to visualise them in terms of modes of guitar strings or organ pipes, or twists of a slinky.
There are MANY different conventions for the Fourier Transform, and for normalisations of fft. fftw3 (or fastest fourier transform in the west) is a freely available documented package that works with many programming languages, and thus forms the industry standard. Confusingly, mathematica, though it uses fftw3 often puts a different normalisation on top. The differing conventions make sense in the context in which they are used. The trick is to be aware of them, and to not get stuck.
The Fourier Transform is technically a complex to complex transform. In Part 1, Question 3, you show that a real valued function transform has coefficients which are complex conjugates for the negative part - an important way that fftw3 and other algorithms save memory! There are an infinite variety of other transforms out there. For example, the Fourier sine transform uses sin as an orthonormal basis set in spectral space instead of the complex exponentials. Similarly, the Fourier cosine transform uses cosines as a basis set. Both are excellent real to real (r2r) transforms, uniquely suited to certain sorts of problems. Each comes in 8 different varieties, relating to assumptions on boundaries, normalisations, and so on. Mathematica and fftw3 support all 8 variants for each transform, and details can be found on wikipedia.
Wavelet transforms are another near-infinite class of transforms. Unlike fourier transforms, wavelet transforms partially mix in spectral information, but still preserve some temporal information. Like fourier transforms, they are a linear (lossless) transformation. The archetypal form is the Haar Wavelet transform, though since their definitions are rather easy to dream up, there are as many varieties as there are problems. Wavelet transforms are often used in image compression, sound analysis, music transcription, code breaking, and many other areas.
All transforms are generalisable to arbitrary numbers of dimensions.