Is Summation a linear operator Say I have the following recursive function, where superscript does not denote a power, but instead denotes a point in time. And I wonder if it's linear for the points in space (denoted by subscript i). $T$ means the tempurature, where $t$ means the actual time. (So timesteps in the recursion don't have to be equal)
$$f(t,T_i) = \sum_{n=1}^{m} \frac{T^n_i-T^{n-1}_i}{\sqrt{t^m-t^n} + \sqrt{t^m-t^{n-1}}}$$
In other words does the following hold true (once again superscript is NOT power):
$$f(t, T_0+T_1) \equiv f(t, T_0) + f(t, T_1)$$
I guess the following can then be said:
$$\sum_{n=1}^{m} \frac{(T^n_0+T^n_1)-(T^{n-1}_0+T^{n-1}_1)}{\sqrt{t^m-t^n} + \sqrt{t^m-t^{n-1}}} = \\ \sum_{n=1}^{m} \frac{T^n_0-T^{n-1}_0}{\sqrt{t^m-t^n} + \sqrt{t^m-t^{n-1}}} + \sum_{n=1}^{m} \frac{T^n_1-T^{n-1}_1}{\sqrt{t^m-t^n} + \sqrt{t^m-t^{n-1}}}$$
Is that correct? - and so does it prove the linearity? I guess this boils down to the question: is summation a linear operator.
EDIT:
Also am I correct in removing teh $i$ subscript of the time, considering that the time (vector) for all positional elements is equal?

Write $T_i=(T_i^0,\dots,T_i^m)\in\mathbb{R}^{m+1}$ (I'm assuming "temperature" is a real variable...).  Your map, for fixed $t$, is equivalent to $$f(T_i,t)=[\begin{array}{ccc}r_1(t)&\cdots&r_m(t)\end{array}]MT_i,$$ where $M$ is the $m\times (m+1)$ matrix defined by the equation $$M_{ij} = 
\cases{-1 & i=j\\
1 & j=i+1\\
0 & else},$$ and the $r_n(t)$ are the denominators in your sum, for each summand $n$.  With this representation, it is clearly linear.
Comment: Just wanted to make sure you were aware: your function is not defined recursively.
