# Representation for a function that, when added/multiplied/composed with another function of the same form, yields a new function of the same form

I apologize for the possibly unclear wording of the title. I'm not well versed in math terminology.
I'm after a concrete representation of a function, eg $y(x) = Ax^p$ (where $A$ and $p$ are constant), where multiplying $y_1(x)$ by $y_2(x)$, $(A_1x^{p_1})(A_2x^{p_2})$, results in a function of the same form*. In this case, $y_3(x) = y_1(x)*y_2(x) = A_1A_2x^{p_1+p_2}$, which, after combining constants, could be represented in the original form: $y_3(x) = y_1(x)*y_2(x) = A_3x^{p_3}$

However, I want this to also be true when adding two such functions, and composing them as in $y_3(x) = y_1(y_2(x))$. The example function can be composed, but does not appear to be addable.

The constraints upon $y(x)$ is that the function must be able to pass through 3 provided points, which are guaranteed to either be all increasing or all decreasing (thus $y(x)$ will be either purely increasing or purely decreasing on its domain, though an operation involving $y(x)$ may result in a function that is not purely increasing/decreasing).

*any operation involving two functions $y_1(x)$ and $y_2(x)$ must return a piecewise function composed solely of pieces in the form of $y(x)$. In the examples, the resulting functions can be considered to be piecewise functions with only one piece.

If no such functional representation exists, I'd like to find a suitable alternative that can approximate the addition/multiplication/composition operations, perhaps through the use of many pieces.

Additional context for those who may find use in it:
Initially, I have a set of points in 2-dimensional space (numbered 1, 2, 3, 4, 5, ...) which I am to connect in a specific way - form a function that connects points 1, 2 and 3 smoothly and with no points of inflection or local extrema between point 1 and 3, then connect points 3, 4, and 5 smoothly & with no inflection/extrema between point 3 and 5 (though the curve could easily have a cusp at point 3), then connect points 5, 6, 7 smoothly, and so on.

In general, a polynomial satisfies the majority of my requirements, but as I perform many operations on these polynomials, their degree becomes very high and it becomes too computationally expensive to evaluate the polynomial accurately.

So, I've decided that the most obvious thing to do is represent my curve as a piecewise function of pieces that connect 3 points in the defined way. Thus my initial function of pieces in the form of $y(x)$ is exact and not an approximation.

The difficulty comes in that I have many of these piecewise functions that I need to multiply, add, or compose with eachother. Addition/multiplication operations can be reduced to first splitting pieces such that each critical value of x where the functions change from one piece to the other is aligned, then performing the operation on the individual corresponding pieces of each function and concatenating the resulting pieces to form a new piecewise function. Composition can be achieved in a similar piece-by-piece way. In order to preserve the same form, each operation on each piece, $y(x)$, must result in a sequence of pieces each in the form $y_n(x)$. I mentioned approximation earlier because I am not sure that an exact solution is possible.

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Polynomials have the property you are describing. The product of two polynomials is a polynomial, the sum of two polynomials is a polynomial, the composition of one polynomial into another is a polynomial. I think that is the simplest example of such thing you will find that is non trivial. – Spencer Oct 12 '13 at 22:41
just to see if I understand what you are trying to do. You have ahead of time some function f(x) which you want to approximate with simpler functions because the operations you want to perform will be easier on them. Is that right? – Spencer Oct 12 '13 at 23:01
Thanks. You can now delete all these obsolete comments... Hopefully someone will come up with an answer! – dfeuer Oct 13 '13 at 2:24