The Fourier series and the Fourier transform are closely related concepts in mathematics and signal processing. The Fourier series is a representation of a periodic function as a sum of sine and cosine waves, while the Fourier transform is a representation of a non-periodic function as a continuous spectrum of sine and cosine waves.

The Fourier series can be used to represent any periodic function as a sum of harmonically related sinusoids, each with a different frequency and amplitude. The Fourier transform, on the other hand, can be used to decompose any non-periodic function into its constituent sinusoids with different frequencies and amplitudes. This allows us to analyze and understand complex signals, such as audio or image data, in terms of their underlying frequency components.

The Fourier series is a special case of the Fourier transform, where the signal being analyzed is assumed to be periodic. The Fourier series is a discrete spectrum of frequencies, while the Fourier transform is a continuous spectrum of frequencies.

To compute the Fourier series, the signal is first decomposed into a series of cosine and sine waves, each with a different amplitude and frequency. The Fourier transform is computed by taking the Fourier series to the limit as the period approaches infinity. The Fourier transform expresses a non-periodic function as a continuous function of frequency, allowing us to see the frequency content of a signal over all possible frequencies.

In summary, the Fourier series and the Fourier transform are related concepts that both involve decomposing a signal into its constituent sinusoidal components. The Fourier series is used for periodic signals, while the Fourier transform is used for non-periodic signals.
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