Let $D = \left\{0, t_1, t_2, \ldots, t_n\right\} \times [0,1]$ and $$ f: D\to [0,1], $$ be a function of time and a one-dimensional space.
There is no analytical formula for $f$, but $f(t_i, \cdot)$ is Lipschitz.
The function $f$ can be computed with arbitrary precision at a finite number of points $\{x_{t_i, j}\}_{j=0}^m$, $(t_i, j) \in D$. Alternatively, the Fourier coefficients $$ \hat f(t_i,k) = \int_0^1 f(t_i,x) e^{2\pi i kx} dx $$ can be computed for each time slice $f(t_i,\cdot)$, up to a finite order $K$.

I know that the time slices $f(t_i,\cdot)$ look like this

time slice

What will be a good choice of basis functions to approximate as accurately as possible $f$?
Is the approximation better with a finite sample $\{f(x_{t_i, j})\}_{j=0}^m$, or with a truncated Fourier series?
Can the particular basis functions used give bound on the approximation error?

I think a truncated Fourier series is the way to good as the error is of order $O(1/K)$.
But I'm not too familiar with gibbs phenomenom, so I don't know if the graph of $f$ discard Fourier series at once.

Also, I know Chebyshev polynomial are for non periodic function.
Would they perform better?

Numerically, the easiest thing is to use a polynomial of some order, and use a collocation methods on the finite sample $\{f(x_{t_i, j})\}_{j=0}^m$.
Is this viable?
I think the main problem of this approach is that there is no bound on the approximation error.

My function $f$ is the solution of a global optimization problem, which is Lipschitz itself $$ |V\left(f_1\right) - V\left( f_2\right) | \le L \|f_1 - f_2\|_{L^2(D)}. $$ I know $f$ lives in a compact subset of $L^2(D)$, so by constructing a fine enough grid, $f$ can be approximate arbitrarily well.


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