Can you approximate a function based solely on constraints?

Question

I am having an issue now at work, I need to model a requirement to a function which I do not know. I only know the constraints of it. They are right here: $$ \begin{split} \int_0^1 \int_0^1 f(x,y) dx dy &= 1 \\ f(0,0) = f(0,1) &= 0 \\ f(1,0) &= 1 \end{split} $$

I also know that the function is defined like this $f(x,y): [0,1] \times [0,1]$. Initially, I thought it is a linear function ($f = ax + by + c$) but that violates my constraints. That being said, it might be a sigmoid, or an exponential, but I cannot figure out a way to actually solve this. Is there a mathematical or programmatic way to do this? Guessing hasn't gotten me anywhere.

Thanks in advance!

Answer 1

The problem with the simple linear solution you proposed is that it has 3 parameters but you have 4 constraints. It seems you can choose any form you like, how about $$ f(x,y) = Axy + Bx + Cy + D $$ to make life easier? Then we have

• $f(0,0) = 0 \implies D = 0$ so $f(x,y) = Axy + Bx + Cy$
• $f(0,1) = 0 \implies C = 0$ so $f(x,y) = Axy + Bx = x(Ay+B)$
• $f(1,0) = 1 \implies B = -A$ so $f(x,y) = Axy -Ax = Ax(y-1)$

And now you can impose the integration to find $A$. Can you finish?