Interpolation of a collection of matrices

Suppose I have a collection of matrices, A1,A2,...,An (n~250), of size ~1000*1500, which represents the values of a function at time t=1 to 250. I would like to interpolate Ak's (at each (i-j)th entry) so that I can find the maximum for each entry over the period [1,250]. However I don't have enough memory to load all Ak's at once and then interpolate each (i-j)th entry. Is there any good algorithm to do this?

In other words, I am looking for a least-memory-consuming way to find, for each (i-j):

max {interpolant of Ak(i,j), k=1,...,250} over the period [1,250].

Thanks so much in advance.