How do I Fit a Resonance Curve?

Are you looking for something like polynomial regression? The general idea is, if you have measured pairs of (x, y(x)) and you are looking for find a fit of the form:

$$y = alpha_0 + alpha_1 x + alpha_2 x^2 ...$$

You can write this in matrix form as:

$$begin{bmatrix} y_1 y_2 y_3 vdots y_n end{bmatrix} = begin{bmatrix} 1 & x_1 & x_1^2 & cdots 1 & x_2 & x_2^2 & cdots 1 & x_3 & x_3^2 & cdots vdots & vdots & vdots & vdots 1 & x_n &x_n^2 & cdots end{bmatrix} begin{bmatrix} alpha_0 alpha_1 alpha_2 vdots alpha_m end{bmatrix}$$

This can now be solved for your coefficients, $alpha_i$. That being said, and as was hinted at in your comments, I've never actually done this, and have instead used non-linear fitting functions provided by libraries.

More information on polynomial regression on the wikipedia page.

Edit: As you say in the comments, this method is only applicable if you can write your function that you wish to fit in polynomial form, which I don't think you can do for your example. In which case you are best off referring to the other answers to this question.

Don't try using any general-purpose curve fitting algorithm for this.

The form of your function looks like a frequency response function, with the two unknown parameters $omega_0$ and $gamma$ - i.e. the resonant frequency, and the damping parameter. The function you specified omits an important feature if this is measured data, namely the relative phase between the "force" driving the oscillation and the response.

If you didn't measure the phase at each frequency, repeat the experiment, because that is critical information.

When you have the amplitude and phase data, there are curve fitting techniques devised specifically for this problem of "system identification" in experimental modal analysis. A simple one is the so-called "circle fitting" method. If you make a Nyquist plot of your measured data (i.e. plot imaginary part of the response against the real part), the section of the curve near the resonance is a circle, and you can fit a circle to the measured data and find the parameters from it.

In practice, a simplistic approach assuming the system only has one resonance often doesn't work well, because the response of a real system near resonance also includes the off-resonance response to all the other vibration modes. If the resonant frequencies are well separated and lightly damped, it is possible to correct for this while fitting "one mode at a time". If this is not the case, you need methods that can identify several resonances simultaneously from one response function.

Rather than re-invent the wheel, use existing code. The signal processing toolbox in MATLAB would be a good starting point - for example https://uk.mathworks.com/help/signal/ref/modalfit.html

If we put:

$$Y = frac{omega^2}{u(omega)^2}$$

and

$$X = omega^2$$

the equation becomes:

$$Y =frac{X^2}{C^2} +frac{(gamma^2 - 2 omega_0^2)}{C^2} X + frac{omega_0^4}{C^2}$$

You can then extract the coefficients using polynomial fitting. To get the least-squares fit right, you have to compute the errors in $Y$ and $X$ for each data point from the measurement errors in $omega$ and $u(omega)$.