# Fourier series, Fourier transform, Power spectrum. How to apply a Low Pass filter to this function?

Let f(t)=costf(t)=\cos{t} be a function. Using some formula/theory I can find out the associate Fourier series, the associate Fourier transform, power spectrum and line spectrum.

The used theory until now is something like this:

The associate Fourier series:
a0=12π∫π−πf(t)dta_{0}=\frac{1}{2\pi}\int^{\pi}_{-\pi}{f(t)dt}

an=∫π−πf(t)cos(nt)dta_{n}=\int^{\pi}_{-\pi}{f(t)\cos(nt)}dt

bn=∫π−πf(t)sin(nt)dtb_{n}=\int^{\pi}_{-\pi}{f(t)\sin(nt)}dt

The associate Fourier transform:

F(z)=1√2π⋅∫π−πf(t)⋅eiztdt.F(z)=\frac{1}{\sqrt{2\pi}} \cdot \int^{\pi}_{-\pi}{f(t) \cdot e^{izt}dt}.

From the tutorials/books I found out that I can apply to this function an filter – low pass.

What does it mean this low – pass and how can I see this filter from a mathematical point a view? For example, I know that if I will apply, let say, a 200Hz200\text{Hz} filter I will not see the frequencies which are higher than 200Hz200\text{Hz}

From where does the low – pass filter arise? How can I see the differences when I will apply an filter of 300Hz300\text{Hz} or a filter of 200Hz200\text{Hz}? How can I translate this in a mathematical language?

I will appreciate any help or information. It would be nice if you can indicate me some books/papers.

Thanks!

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Look for material on the convolution theorem w.r.t. Fourier transforms.
– AnonSubmitter85
4 hours ago

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