I have a list of numbers that always increase. I would like to fit a function (any function, not necessarily a polynomial). I do have a condition to this function though which is as follows-
Let us say the data I have has 'n' values. I need the expression of the function to require at most 'n' constants to define it and for x ranging from 1 to n, it must produce the values of my original data exactly. This is already possible in polynomial fits. However, this function should also have the additional property that on substituting the value (n+1) to x, the result must always be greater than the value that results in the substitution of (n) to x. Basically, on extending the function to higher input values, the result must also increase.
I was wondering if I could use a series of exponentials as a function to form the required curve, but not being very familiar to hardcore math, I am not sure if or how to do it yet. If you think it is possible, I will surely look into figuring out an algorithm.