A Comparison of the Discrete Fourier Transform and Wavelet Decompositions of a Function

Abstract

Scientists like analyzing complex phenomena. The most common way is to break them into simple pieces. Since the properties of those simple pieces are well understood, the combination of these properties can help them understand the characteristics of the original phenomena. In 1822, the French mathematician Joseph Fourier introduced Fourier analysis in his essay "Analytic Theory of Heat." Fourier analysis is a method to decompose a function into the sum of trigonometric sine and cosine waves of various frequencies and amplitudes. The familiar and well-understood trigonometric functions are easy to analyze. By combining information about a function's sine and cosine components, we can deduce the properties of the function itself.(1) Another method of approximating a function is the wavelet theory, which was introduced by geophysicist Jean Morlet and mathematical physicist Alexander Grossmann in the early 1980s in France. Instead of working with the infinitely undulating sine and cosine waves, wavelet analysis relies on translations and dilations of suitably chosen "mother wavelets. "(2) Then we can express a function as linear combinations of all these wavelets. Fourier analysis is especially suited to analyzing periodic phenomena, since periodicity is the most prominent property of sines and cosines. Wavelet theory, on the other hand, is good at approximating a function over a finite and localized interval. In my paper, I used the computer program Mathematica to decompose a smooth polynomial function into a set of Fourier coefficients and a set of Haar wavelets. After rounding the data to within a certain relative error, I reconstructed the functions and compared them graphically with the original one. To understand how that works requires some introduction to the Fourier transform and wavelet theory .