好的，所以我一直在尝试编写一种“朴素”的方法来计算复杂形式的标准傅立叶级数的系数。我想我已经非常接近了，但是有一些奇怪的行为。这可能更像是一个数学问题而不是编程问题，但我已经在 math.stackexchange 上问过并且得到的答案为零。这是我的工作代码：
import matplotlib.pyplot as plt
import numpy as np
def coefficients(fn, dx, m, L):
"""
Calculate the complex form fourier series coefficients for the first M
waves.
:param fn: function to sample
:param dx: sampling frequency
:param m: number of waves to compute
:param L: We are solving on the interval [-L, L]
:return: an array containing M Fourier coefficients c_m
"""
N = 2*L / dx
coeffs = np.zeros(m, dtype=np.complex_)
xk = np.arange(-L, L + dx, dx)
# Calculate the coefficients for each wave
for mi in range(m):
coeffs[mi] = 1/N * sum(fn(xk)*np.exp(-1j * mi * np.pi * xk / L))
return coeffs
def fourier_graph(range, L, c_coef, function=None, plot=True, err_plot=False):
"""
Given a range to plot and an array of complex fourier series coefficients,
this function plots the representation.
:param range: the x-axis values to plot
:param c_coef: the complex fourier coefficients, calculated by coefficients()
:param plot: Default True. Plot the fourier representation
:param function: For calculating relative error, provide function definition
:param err_plot: relative error plotted. requires a function to compare solution to
:return: the fourier series values for the given range
"""
# Number of coefficients to sum over
w = len(c_coef)
# Initialize solution array
s = np.zeros(len(range))
for i, ix in enumerate(range):
for iw in np.arange(w):
s[i] += c_coef[iw] * np.exp(1j * iw * np.pi * ix / L)
# If a plot is desired:
if plot:
plt.suptitle("Fourier Series Plot")
plt.xlabel(r"$t$")
plt.ylabel(r"$f(x)$")
plt.plot(range, s, label="Fourier Series")
if err_plot:
plt.plot(range, function(range), label="Actual Solution")
plt.legend()
plt.show()
# If error plot is desired:
if err_plot:
err = abs(function(range) - s) / function(range)
plt.suptitle("Plot of Relative Error")
plt.xlabel("Steps")
plt.ylabel("Relative Error")
plt.plot(range, err)
plt.show()
return s
if __name__ == '__main__':
# Assuming the interval [-l, l] apply discrete fourier transform:
# number of waves to sum
wvs = 50
# step size for calculating c_m coefficients (trap rule)
deltax = .025 * np.pi
# length of interval for Fourier Series is 2*l
l = 2 * np.pi
c_m = coefficients(np.exp, deltax, wvs, l)
# The x range we would like to interpolate function values
x = np.arange(-l, l, .01)
sol = fourier_graph(x, l, c_m, np.exp, err_plot=True)
现在，每个系数乘以 2/N 的因子。但是，我在教授的打字笔记中推导了这个总和，其中不包括这个因子 2/N。当我自己导出表格时，我得出了一个因子为 1/N 的公式，无论我尝试什么技巧都不会取消。我在 math.stackexchange 上询问了发生了什么，但没有得到任何答案。
我注意到的是，添加 1/N 大大减少了实际解决方案和傅立叶级数之间的差异，但仍然不对。所以我尝试了 2/N 并得到了更好的结果。我真的很想弄清楚这一点，这样我就可以在尝试学习快速傅立叶变换之前为基本的傅立叶级数编写一个漂亮、干净的算法。
那么我在这里做错了什么？