Yan - Bin Jia Nov 29 , 2016 1

for some fixed τ , which is called the period of f . Though function approximation using orthogonal polynomials is very convenient, there is only one kind of periodic polynomial, that is, a constant. So, polynomials are not good for approximating periodic functions. In this case, trigonometric functions are quite useful. A large class of important computational problems falls under the category of “Fourier transform methods” or “spectral methods”. For some of these problems, the Fourier transform is simply an efficient computational tool for data manipulation. For other problems, the Fourier transform is itself of intrinsic interest. Fourier methods have revolutionized fields of science and engineering, from radio astronomy to medical imaging, from seismology to spectroscopy. The wide application of Fourier methods is credited principally to the existence of the fast Fourier transform (FFT). The most direct applications of the FFT are to the convolution or deconvolution of data, correlation and autocorrelation, optimal filtering, power spectrum estimation, and the computational of Fourier integrals. A physical process can be described either in the time domain, by the values of some quantity h as a function of time t, or else in the frequency domain, where the process is specified by giving its amplitude H as a function of frequency ξ with −∞ < ξ < ∞. For many purposes it is useful to think of h(t) and H(ξ) as being two different representations of the same function. These two representations are related to each other by the Fourier transform equations,