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Let's have the discrete periodic function $f(t)$ which has only 5 non-zero values $f(t_0) = -10, f(t_1) = -5, f(t_2) = -10, f(t_3) = -15$ and $f(t_4) = -10$, all other points within the period $[t_0,t_4]$ being zero. The obvious interpolating function which recovers these values is $f(t) = 5sin(t) - 10$. Is this a unique interpolating function or you can propose other interpolating functions, such as polynomial functions with constraints? Any concrete examples, please?

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Your $f$ only interpolates these values if $t_i=\frac{i\pi}{2}$. Also, $f$ is clearly not zero at all the other points in $[t_0,t_4]$.

Moreover note that you can, for any finite number of value pairs $(t_i,v_i)_i$, find arbitrarily many smooth functions $F$ s.t. $F(t_i)=v_i$ for all $i$. Look for example at Lagrange polynomials/Newton polynomials etc. You can always change the function values in between the prescribed points so that your function is still continuous or even smooth.

When you drop the continuity/smoothness requirement (which you do if you say that it should have only 5 non-zero values) then finding an interpolation function is completely trivial (just take the function that is 0 everywhere except on the prescribed points where it takes the prescribed values). If you necessarily want to write it down as a formula you can always use a linear combination of Kronecker deltas:

$f(t)=\sum_i y_i \delta^t_{t_i}.$

where $(t_i,y_i)_i$ is your data and $\delta^t_s$ is 1 if $t=s$ and 0 if not.

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