I want to know the answer to the following question.
If a finite (but fixed) number of Fourier coefficients (of any choice) of a Fourier series are made $0$, then will the new series be a Fourier series? Prove or give a counterexample. Now, if we define the number of coefficients turning to $0$ as $N$ and allow $N\to\infty$ then can we say anything? More precisely, what can one conclude if arbitrarily many terms of a Fourier series are made $0$?
Well, I know that the Fourier sum converges to the original function if the function is Lipshitz. If I go ahead one step and assume that the function is differentiable, then the Fourier series converges to the function. I consider the Fourier series of one such differentiable function.
After deleting (say $50$) terms of my choice from the Fourier series, the final series I obtain is still differentiable because I subtracted $50$ differentiable terms from a differentiable function. That is, if $f$ is my function and I deleted the coefficients $c_1,c_2,...,c_{50}$ then $f(x)-\sum_1^{50}c_ke^{in_kx}$ is a differentiable function which will have a Fourier series converging to the function. This Fourier series is exactly the series obtainable by removing those terms of my choice from the Fourier series of $f$.
Now if we just assumed the function $f$ is Lipschitz, then also maybe we would get the same answer.
So my answer is YES for the first part. As for the second part, I don't know.