Can Any Periodic Function Be Decomposed into Sines and Cosines?

[rakeshbs]

Fourier series is a way to express a periodic function as a sum of complex exponentials or sines and cosines.. Is there actually a proof for the fact tat a periodic function can be split up into sines and cosines or complex exponentials?

 

[HallsofIvy]

It is fairly easy to show that any integrable periodic function can be approximated arbitrarily well by a sum of sines and cosines. The Fourier series is the limit of those approximations as the "error" goes to 0- except on a set of measure 0 for dis-continuous functions. I might point out that the other way is what's hard. It can be shown that some perfectly valid Fourier series converge to non-(Riemann)-integrable functions. That was why Lebesque integration had to be invented.

 

[tacman]

Yes, there is a mathematical proof for the fact that a periodic function can be expressed as a sum of sines and cosines or complex exponentials. This proof is known as the Fourier series representation theorem and it was first introduced by French mathematician Joseph Fourier in the early 19th century.

The proof involves using the complex exponential function, which is defined as e^ix = cos(x) + i*sin(x), to represent a periodic function. By using this representation, we can express any periodic function as a combination of different frequencies and amplitudes, which can be represented by the coefficients in the Fourier series.

The key idea behind this proof is that any periodic function can be decomposed into a sum of harmonic functions, which are functions with frequencies that are integer multiples of the fundamental frequency. By using the complex exponential function, we can express these harmonic functions as sines and cosines, which are easier to work with mathematically.

Overall, the proof for the Fourier series representation theorem involves a lot of mathematical concepts and techniques such as complex analysis, trigonometry, and calculus. It is a fundamental result in mathematics and has many important applications in fields such as signal processing, engineering, and physics.