Equating two functions f(x), and g(x) to find parameters of first one

Question

I want to solve a mathematical problem, however I'm confused because of results that I obtained. Let we say, that we've two functions: $\ f(x)$ and $\ g(x) $. First one can be any function, describing e.g. transmitance of automatic circuit or electrical circuit. It has some fixed parameters e.g. $\ a,b,c,d$. Those parameters can be obtained by using empirical simulations. The second function i.e. $\ g(x)$ is known, analytical function such as a Chebyshev polynomial. I drew both functions, and then I observed that in some interval $\ [x_a, x_b] $ mentioned functions are the same. Now... because I want to calculate $\ a,b,c,d$ analytically, I compare functions i.e. $\ f(x) = g(x) $, and make some algebraic transformations to find mentioned parameters of $ \ f(x) $. However there is a one big problem: what should I substitute by $\ x$ value ? On this moment I use some characteristic value of $ \ x$ such as a local mimimum of $\ f(x)$, and calculate them. However, function $\ f(x)$ with new values of $\ a,b,c,d$ has the value as $\ g(x) $ only for the mentioned value of $\ x$. What should I do, when I want to calculate values of $\ a,b,c,d$ for which $\ f(x) $ will be simillar to $\ g(x)$ in mentioned range ? Regards, E.

Answer 1

First of all, Welcome to the site !.

For your problem, assuming that you know the functional form of $f(x)$ and function $g(x)$, a formal way to do it is to consider $$\Phi(a,b,c,d)=\int_{x_a}^{x_b} \Big(f(a,b,c,d,x)-g(x)\Big)^2\,dx$$ and to minimize it that is to say to solve the system of four equations $$\frac{\partial \Phi(a,b,c,d)}{\partial a}=\frac{\partial \Phi(a,b,c,d)}{\partial b}=\frac{\partial \Phi(a,b,c,d)}{\partial c}=\frac{\partial \Phi(a,b,c,d)}{\partial d}=0$$ If you think about it, it is the same as a regression using an infinite number of data points in the range $[x_a, x_b]$.