Need to find function related to Knoedel numbers that satisfies these conditions

Question

I need to find the continuous function $f(x)$ that satisfies $f(0)=0$ and:

$$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ $$\frac{f(\sin(\pi/3))^2}{\sin^4(\pi/3)}=39$$ $$\frac{f(\sin(\pi/2))^2}{\sin^4(\pi/2)}=27$$

The RHS integers above have an interesting interpretation in terms of Knoedel numbers. It is also known that $f(x)$ must satisfy:

$$\frac{f(\sin(\pi/6))^2}{\sin^6(\pi/6)}=540$$ $$\frac{f(\sin(\pi/4))^2}{\sin^6(\pi/4)}=126$$ $$\frac{f(\sin(\pi/3))^2}{\sin^6(\pi/3)}=52$$ $$\frac{f(\sin(\pi/2))^2}{\sin^6(\pi/2)}=27$$

And

$$\frac{f(\sin(\pi/6))^2}{\sin^8(\pi/6)}=2160$$ $$\frac{f(\sin(\pi/4))^2}{\sin^8(\pi/4)}=252$$ $$\frac{f(\sin(\pi/3))^2}{\sin^8(\pi/3)}=69.3333$$ $$\frac{f(\sin(\pi/2))^2}{\sin^8(\pi/2)}=27$$

I guess it's also kind of interesting that the digits in the RHS integers always add up to 9 except for the $f(\pi/3)$ case.

Answer 1

$$f(b(t))=\sqrt{36b(t)^2-9b(t)^4} \:\:\:\:; \:\: b(t)=\sin{t}, \:\:\: 0 < t <\pi/2$$

Answer 2

Since you are given $f()$ only at a finite number of points, what's wrong with a piecewise linear or polynomial interpolation?

Answer 3

This is a comment, but it is quite long, so I am writing it as an answer, hope you don't mind.

OK, you asked this question yesterday and you didn't specify what type of function you want. Again, there is not enough information in this question. So, what are all these people asking as more information, let me try to explain. For example, the factorial function $n!$. Similarly, I can ask a question as follows:

Find me a function defined on all positive reals such that $f(n+1) = n!$ for all positive integers $n$.

There are infinitely many such functions, and one cannot give a definitive answer. So, you have to add extra conditions. For example, let's say we want $f(x+1) = x f(x),\,x>0$ and $f(1)=1$. This is much more restrictive, but again, there are many such functions. If we add one more condition, i.e. if we require $f$ to be log-convex, then we get only one function, namely the Gamma function (This is known as the Bohr–Mollerup theorem). Similarly, you should specify what extra conditions you want. I am not familiar with those Knoedel numbers, but perhaps you have an intuition as to how to extend their definitions to reals.