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函数 \( f(x) = \sin x - 3 \cos x + \frac{1}{3} \sin 5x \) 的傅里叶级数就是该函数本身。

This means: "The Fourier series of the function \( f(x) = \sin x - 3 \cos x + \frac{1}{3} \sin 5x \) is the function itself."

To explain why the given function is its own Fourier series, we need to analyze its form. The Fourier series of a periodic function \( f(x) \) can be represented as:

\[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \]

In this case, the function is already expressed in terms of sine and cosine functions with coefficients:

\[ f(x) = \sin x - 3 \cos x + \frac{1}{3} \sin 5x \]

This implies:

- The coefficient of \(\sin x\) is \(1\).
- The coefficient of \(\cos x\) is \(-3\).
- The coefficient of \(\sin 5x\) is \(\frac{1}{3}\).
- There are no other terms, implying other coefficients \(a_n\) and \(b_n\) are zero.

Since the function is already in the form of a Fourier series with only the sine and cosine terms presented, the Fourier series representation of \(f(x)\) matches the function itself.

Would you like more details or have any questions about this explanation? Here are some more questions you might consider:

1. What is the general form of a Fourier series?
2. How do we determine the coefficients \(a_n\) and \(b_n\) in a Fourier series?
3. Can every function be represented by a Fourier series?
4. How does the Fourier series of a function change with different periodicities?
5. What are some applications of Fourier series in real-world problems?

**Tip:** Understanding how to decompose a function into its Fourier series helps in analyzing the function's frequency components, which is particularly useful in signal processing and other engineering applications.

Explore the Fourier series of the function f(x) = sin(x) - 3cos(x) + 1/3sin(5x), which is its own representation. This detailed explanation covers trigonometric functions, periodic functions, and the concept of Fourier series, crucial in signal processing and engineering.

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