Fourier and Wavelet Representations of Functions
Nicholas G. Roland · Furman University Scholar Exchange (Furman University) · 2000
Abstract. Representations of functions are compared using the traditional technique of Fourier series with a more modern technique using wavelets. Un-der certain conditions, a function can be represented with a sum of sine and cosine functions. Such a representation is called a Fourier series. This classi-cal method is used in applications such as storage of sound waves and visual images on a computer. One problem with this sum is that it is infinite. In use, only a finite number of terms can be used. More accuracy requires more terms in the series, but more terms require more time to compute and more space to store. A new type of sum called a wavelet series was first introduced in the 1980’s. With these new series the same accuracy often takes fewer terms. Since wavelet representations can be more accurate and take less computer time, they are often more useful. 1. Background in Fourier Series Jean Baptise Joseph Fourier (1768–1830) was the inventor of Fourier series in the late 1700’s. Fourier was a mathematical physicist who developed a way to express a function by combining an infinite number of sine and cosine terms. For example a tuning fork produces a sound wave when it is stuck. The sound wave that we hear is a pure tone with one frequency and can be represented by a single sine function. When we hear a piano key struck, we do not hear just one frequency but rather a mixture of frequencies. This mixture contains a fundamental and then a number of overtones (harmonics) of frequency 2, 3, 4,... multiplied by the fundamental frequency. The combination of the fundamental and the harmonics can be represented by a sum of sine functions. The series which describes this combination is called a Fourier series. Expanding a function in a Fourier series breaks the function down into its various harmonics. More generally, a 2l-periodic function f(x), can be represented by its Fourier series, f(x) ∼ 1 2 a0 + n=1 an cos( npix