Given
- a periodic "Target Function" $F(t)$
- a set of $N$ periodic "Basis Functions" $B_i(t)$ of arbitrary shape
All functions are defined on the same interval $T$.
I am allowed to shift the basis functions individually by a constant value of $t_i$. $$ B_i'(t,t_i) = B_i(t-t_i). $$ I want to approximate $F(t)$ with the function
$$ A(t,t_1..t_N) = \sum\limits_{i=1}^N B_i'(t,t_i). $$
How can I efficiently find the values $t_1..t_n$ that minimize the difference between $A(t)$ and $F(t)$?
$$ \min\int_0^T \! (F(t)-A(t,t_1..t_N))^2 \, \mathrm{d}t. $$
PS: I'm using a discrete t axis. So the integral could be rewritten as a sum.
One method is to use normalized cross correlation with each basis function. If the normalization is with respect to the support of each basis function then you will find the best shift to fit each basis function to the function of interest (classical needle in the haystack problem). This can be accomplished in the transform domain - and exploits the Fourier transform shift theorem.
If the basis functions are intended to stack up horizontally in a mutually (temporally) orthogonal fashion (as shown in your picture) then the shift theorem in combination with the Plancherel theorem (to support the orthogonal annihilation of the cross terms) gives
$t_i = -arg(\frac{\hat{F}(\xi)}{\hat{B}_i(\xi)})/(2 \pi \xi)$ for any particular $\xi$ and $\hat{B}_i(\xi)$ is the Fourier transform of the $ith$ un-shifted basis function. This solution does minimize your data discrepancy functional $\int_0^T \! (F(t)-A(t,t_1..t_N))^2 \, \mathrm{d}t$.
Additionally - you mentioned that your functions are periodic - this means that they have a Fourier series - and that simplifies things quite a lot. Also - because your data are discrete - you can use the FFT to solve the problem. You should be able to do it using the first mode (k=1).