Do Fourier Series Remain Unique When Functions Are Shifted?

[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?

 

[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 the 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.

 

[M98Ranger]

The answer to whether Fourier series are unique depends on the context in which they are being discussed. In general, Fourier series are unique in the sense that for a given function with a certain finite period, there is only one set of Fourier coefficients that can be used to represent the function. This means that for a given function, there is only one set of coefficients that will produce the same Fourier series.

However, in the specific case mentioned in the content, where the function is shifted up by a constant, there can be multiple sets of coefficients that can produce the same Fourier series. This is because shifting a function up by a constant does not change its periodicity, and therefore the same set of coefficients can still be used to represent the function. In this case, the b_n coefficients will be the same, and a_0 will be twice the constant.

It is important to note that even though there may be multiple sets of coefficients that can produce the same Fourier series, the resulting series will still represent the same function. So while there may be different ways to represent a function using Fourier series, the end result will be the same.

In conclusion, Fourier series are unique in the sense that for a given function and period, there is only one set of coefficients that can be used to represent the function. However, in certain cases, there may be multiple sets of coefficients that can produce the same series, such as when the function is shifted up by a constant.