實用調和分析新法

<正> 實用調和分析的目標是從給定的一段曲線y=f(x),0≤x≤2π定出這個函數的頑里哀係數

A NEW METHOD OF PRACTICAL HARMONIC ANALYSIS

The ordinary method of practical harmonic analysis is essentially an approximation of the following definite integrals by a fixed number (2n) of rectangles. As the function cos kx or sin kx oscillates between -1 and 1 2k times in the interval (0, 2π), it requires about 8k subintervals to get a reasonablly good approximation of the definite integral for a_k or b_k. It is obvious that as k increases, the number of subintervals has to be proportionally increased in order to insure the same accuracy. This explains why the ordinary method of practical harmonic analysis gives good approximations only for the first few coefficients.In this paper, the author approximates the curve of the function f(x) itself by a continuous curve consisting of 2n straight line-segments or n parabolic arcs. The Fourier coefficients of the approximating curve is then computed and used as approximations of the Fourier coefficients of the original function. Computation involved is almost as easy as the ordinary method. Approximation is found to be much closer than the ordinary method and more uniform for all k.The relation between approximations by the straight line-segment method A_k and B_k and those by the ordinary method α_k and β_k is found to be A_k/αk=B_k/β_k=sin~2 kπ/2n/(kπ/2n)~2.Two examples are given at the end of the paper.