Is it possible to deduce a function from its fourier series ?

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SUMMARY

It is possible to deduce a function from its Fourier series, specifically recovering (1/2)(f(x-)+f(x+)). However, the Fourier series does not differentiate between functions that vary on sets of measure zero, which is often negligible in practical applications. While it is theoretically feasible to derive a specific form for a function from its Fourier series, the complexity of the function can make this process challenging, as illustrated by the example of the Fourier series for (3x^3+2x +4)e^(3x^2+9x)cos((3x^2+5x-7)sin(9x^2+4x+3).

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Is it possible to deduce a function from its Fourier series ?
 
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You can recover (1/2)(f(x-)+f(x+)) from the Fourier series of f. Among other things the Fourier series does not distinguish between functions that differ on a set of measure zero. This is not a major problem as we often either know what f does on sets of measure zero or do not care. If you mean can we find the equation of a certain form for f, generally yes in principle but it can be very difficult. Like if you were given some Fourier series of say (3x^3+2x +4)e^(3x^2+9x)cos((3x^2+5x-7)sin(9x^2+4x+3) you would have a hard time guessing that.
 

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