Suppose I have a set of desired points $(x_1, f(x_1)), (x_2, f(x_2)),..., (x_n, f(x_n))$, with $x_i\in\mathbb{R}$ and $f(x_i)\in\mathbb{C}$. I want to find a Fourier series for $f(x)$, $$f(x)=\sum_{j=-d}^dp_je^{ijx\omega}$$ that passes through the desired points and also satisfies $p_j\ge0$; finite integer $d$ and nonnegative $\omega$ are free parameters. Two questions:
- Is this always possible for sufficiently large $d$, and if not, under what conditions on the points is it impossible?
- What $d$ is required, as a function of $n$?
Note: there are many related questions on StackExchange about conditions on a whole function for its Fourier series or transform to be nonnegative, e.g., Fourier series with non-negative coefficients, but I have not found any that address the weaker requirement of fitting a discrete set of points.