傅里叶逼近只适用于0.0-1.0的范围内。

当需要进行傅里叶逼近时，我们可以将要逼近的函数减去其在整个定义域上的平均值，再乘以一个适当的缩放因子使其在0.0-1.0的范围内。代码示例如下：

import numpy as np
import matplotlib.pyplot as plt

def fourier_approx(f, n):
    """
    对函数f进行n阶傅里叶逼近
    """
    x = np.linspace(0, 1, 1000)
    y = f(x)
    y_mean = np.mean(y)
    y_scaled = 2 * (y - y_mean)  # 缩放因子
    a0 = y_mean
    ak = np.zeros(n)
    bk = np.zeros(n)
    for k in range(1, n+1):
        ak[k-1] = np.trapz(y_scaled * np.cos(2 * np.pi * k * x), x)
        bk[k-1] = np.trapz(y_scaled * np.sin(2 * np.pi * k * x), x)
    return a0, ak, bk

# example:
f = lambda x : np.sin(2 * np.pi * x) + np.cos(4 * np.pi * x)  # 定义函数
a0, ak, bk = fourier_approx(f, 10)
x = np.linspace(0, 1, 1000)
y = a0 + np.sum([ak[i] * np.cos(2 * np.pi * (i+1) * x) + bk[i] * np.sin(2 * np.pi * (i+1) * x) for i in range(10)], axis=0)
plt.plot(x, f(x), label="original")
plt.plot(x, y, label="approximation")
plt.legend()
plt.show()