Explain how Fourier series work and why a Fourier transform is useful. How can all waves just be made of different waves?
The Fourier series is a mathematical tool that can be used to represent a periodic function as a sum of sine and cosine waves. Any periodic function, such as a square wave or a sawtooth wave, can be represented as a sum of an infinite number of sine and cosine waves with different amplitudes, frequencies, and phases. This decomposition process is known as Fourier series analysis.

The basic idea behind Fourier series is that any periodic function can be approximated by adding up sine and cosine waves of different frequencies and amplitudes. By using a large enough number of these waves, the approximation becomes very accurate, and the original function can be reconstructed from the Fourier series. This is why Fourier series are often used in signal processing, such as in digital audio and image compression.

A Fourier transform is a mathematical tool that extends the concept of Fourier series to non-periodic functions. It allows us to decompose a non-periodic function into a continuous spectrum of frequencies. The Fourier transform is useful because it allows us to analyze signals in the frequency domain, which can be more intuitive and easier to understand than the time domain.

All waves can be represented as a combination of different waves with different frequencies, amplitudes, and phases. This is because waves are fundamentally composed of oscillations, and any oscillation can be represented as a sum of sine and cosine waves with different frequencies. This is the basis of Fourier analysis, which allows us to break down any complex wave into simpler sine and cosine waves. By analyzing the frequencies and amplitudes of these simpler waves, we can gain insights into the behavior and properties of the complex wave.

***TLDR***:
- Fourier series and Fourier transforms are mathematical tools that can be used to analyze and understand waves.
- Any wave can be represented as a sum of sine and cosine waves with different frequencies, amplitudes, and phases. This is known as the Fourier decomposition, and it is a powerful tool that can be used to analyze and understand waves.
- Fourier series are useful for representing periodic functions, while Fourier transforms are useful for representing non-periodic functions.
- Both Fourier series and Fourier transforms are used in a variety of applications, such as signal processing, image processing, and data compression.