# Are fourier series unique?

1. Nov 7, 2004

### StatusX

For a given function with a certain finite period, is there only one set of fourier series coefficients $$a_n$$ and $$b_n$$? The reason I ask is, I was doing a problem where it asked for the coefficients for a certain odd function, and then it asked for the coefficients for that same function shifted up by a constant. Are all the $$b_n$$'s the same, and $$a_0$$ just twice the constant? I tried real quick using the definition to see if this came out the same, and it didn't, but I might have made a mistake. Could there be two series for the same function?

2. Nov 8, 2004

### matt grime

Given a function, the coefficients are unique, though the converse is false.

Consider the example of f(x) and f(x)+k, for some k, which is how I read your query.

Then, since the integral of ksin(nx) and kcos(mx) are zero over the interval, it follows teh non-constant terms are the same, and the constant term is then the integral of f(x)+k, which is the original constant integral plus k times the length of the interval. Note, I haven't allowed for dividing by 2pi or anything since that is a non-canonical choice, and I hope you can fill in the constants properly.