Transforms and Fourier Series | 变换与傅里叶级数

📚 Transforms and Fourier Series | 变换与傅里叶级数

Fourier series allow us to represent any periodic function as a sum of simple sine and cosine waves. This powerful idea, introduced by Joseph Fourier in the early 19th century, is fundamental in mathematics, physics, and engineering. The transition from Fourier series to the Fourier transform extends the analysis to non‑periodic functions, opening up tools for signal processing, differential equations, and data compression.

傅里叶级数使我们能够将任何周期函数表示为简单正弦波和余弦波的和。这一强大的思想由约瑟夫·傅里叶在19世纪初提出，是数学、物理和工程学的基础。从傅里叶级数到傅里叶变换的过渡将分析扩展到非周期函数，为信号处理、微分方程和数据压缩提供了工具。

1. Introduction to Fourier Series | 傅里叶级数导论

A Fourier series expands a periodic function f(x) with period 2π into an infinite sum of trigonometric terms: f(x) = a₀/2 + Σ (aₙ cos(nx) + bₙ sin(nx)). The idea is that any sufficiently smooth periodic waveform can be decomposed into harmonics of a fundamental frequency.

傅里叶级数将周期为2π的函数f(x)展开为正弦和余弦项的无穷和：f(x) = a₀/2 + Σ (aₙ cos(nx) + bₙ sin(nx))。其思想是，任何足够平滑的周期波形都可以分解为基频的谐波。

The constant term a₀/2 represents the average value of the function over one period, while aₙ and bₙ capture the amplitudes of the cosine and sine components at integer multiples of the fundamental frequency. These coefficients are determined uniquely by the original function.

常数项a₀/2代表函数在一个周期内的平均值，而aₙ和bₙ则捕捉了在基频整数倍处余弦和正弦分量的振幅。这些系数由原函数唯一确定。

2. Periodic Functions and Orthogonality | 周期函数与正交性

The key to finding the coefficients lies in the orthogonality of sine and cosine functions over the interval [−π, π]. Specifically, ∫ cos(mx)cos(nx)dx = 0 for m≠n, ∫ sin(mx)sin(nx)dx = 0 for m≠n, and ∫ cos(mx)sin(nx)dx = 0 for all m,n. These integrals evaluate to π (or 2π for the constant case) when the indices match.

求系数的关键在于正弦和余弦函数在区间[−π, π]上的正交性。具体来说，当m≠n时，∫ cos(mx)cos(nx)dx = 0；当m≠n时，∫ sin(mx)sin(nx)dx = 0；对所有m,n，∫ cos(mx)sin(nx)dx = 0。当指标相等时，这些积分值为π（常数情况为2π）。

These orthogonal relationships allow us to isolate each coefficient by multiplying the series by the corresponding trigonometric function and integrating term by term. All terms except the desired one vanish, yielding simple formulas for aₙ and bₙ.

这些正交关系使我们能够通过将级数乘以相应的三角函数并逐项积分来分离每个系数。除了所需项外，所有项都消失了，从而得到aₙ和bₙ的简单公式。

3. Euler Formulas for Coefficients | 系数的欧拉公式

Using orthogonality, we derive the Euler formulas for a function f(x) of period 2π:

利用正交性，我们推导出周期为2π的函数f(x)的欧拉公式：

a₀ = (1/π) ∫_−π^π f(x) dx

a_n = (1/π) ∫_−π^π f(x) cos(nx) dx   (n ≥ 1)

b_n = (1/π) ∫_−π^π f(x) sin(nx) dx   (n ≥ 1)

For a function with period 2L rather than 2π, a simple change of variable gives: f(x) = a₀/2 + Σ_n=1^∞ (a_n cos(nπx/L) + b_n sin(nπx/L)), with coefficients integrated over [−L, L] and scaled by 1/L.

对于周期为2L而不是2π的函数，通过简单的变量替换可得到：f(x) = a₀/2 + Σ_n=1^∞ (a_n cos(nπx/L) + b_n sin(nπx/L))，系数在[−L, L]上积分并除以1/L。

In practice, we often compute these integrals using integration by parts, trigonometric identities, or by exploiting symmetry. The choice of the integration interval matters; any interval of length 2π (or 2L) works, such as [0, 2π], as long as it covers one full period.

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