I have a function that evolves over time. I know the function in the initial and terminal states.
$f(t,x_1,x_2,...,x_n) = a_1x_1 + a_2x_2 + ... +a_nx_n$
$f(T,x_1,x_2,...,x_n) = b_1x_2 + b_2x_2 + ... + b_nx_n$
I want to approximate the function at some $\tau$, $t < \tau < T$.
There is probably some way to interpolate the function just like drawing straight line between two points in space?
The first thing that came into my mind is to do the following
$f(\tau,x_1,x_2,...x_n) = \sum c_ix_i$
where $c_i = \frac{\tau}{T-t}a_i + (1-\frac{\tau}{T-t})b_i$
Is it the best I can do?
Your coefficient $c_i$ should be egal to $a_i$ when $\tau=t$ and should be equal to $b_i$ when $\tau=T$. Then, they should be $$c_i=\left(\frac{T-\tau}{T-t}\right)a_i+\left(\frac{\tau-t}{T-t}\right)b_i$$ If $\tau=t$ then $$c_i= \left(\frac{T-t}{T-t}\right)a_i+\left(\frac{t-t}{T-t}\right)b_i=1a_i+0b_i$$ If $\tau=T$ then $$c_i= \left(\frac{T-T}{T-t}\right)a_i+\left(\frac{T-t}{T-t}\right)b_i=0a_i+1b_i$$