Can the Least Squares Method be expressed as a convolution?

[Daniel Petka]

Homework Statement

Consider a laser line position estimation by fitting using the Least Square Method (LSM) and prove (or disprove) that it can be considered as a convolution with some function and finding the center by looking for the maximum (zero‐crossing by the derivative). What is the smoothing function?

The Least Square Method (LSM) is defined as:
$$\sum_i[S(x_i)-F(x_i,;a,b,...)]^2=min,$$
where the fitting function is:
$$F(x;y_0,A,x_0,w)=y_0+A\cdot g(x-x_c,w)$$

The fit program will adjust all parameters, but we
are interested only for $x_c$.

Hint: change sums to integrals in LSM description!

Relevant Equations

fitting function: $F(x;y_0,A,x_0,w)=y_0+A\cdot g(x-x_c,w)$
convolution: $f(x)=\int S(x-y)K(y)dy$
Least Squares Method: $\sum_i[S(x_i)-F(x_i,;a,b,...)]^2=min$

I started by converting the LSM from sum to integral form:
$$f(x_c) = \sum_i[S(x_i)-F(x_i,;a,b,...)]^2 to f(x_c) = \int( S(x) - F(x-x_c)^2 dx$$

Since we are not interested in the other parameters (like offset), I assumed that they are fitted correctly and thus ignored them, turning $F(x-x_c)$ directly to $g(x-x_c)$.

Then I expanded the binomial formula as following:
$$\int S(x)^2 - 2S(x)F(x-x_c) + g(x-x_c)^2 dx$$

And used the linearity of the integral to isolate the part of the equation that doesn't depend on x_0:
$$ f(x_c) = \int S(x)^2 dx + \int 2S(x)g(x-x_c) + g(x-x_c)^2 dx$$
Hence, we have a constant q that isn't affected by the convolution:

$$ f(x_c) = q + \int 2S(x)g(x-x_c) + g(x-x_c)^2 dx$$

The middle term is a convolution og the 2 functions. My idea was to disprove that a Kernel exists, because there is a term that doesn't depend on $x_c$, but this logic doesn't make any sense after thinking about it. I am completely stuck at this point, since I can neither prove nor disprove that the kernel function exists. Any help would be highly appreciated!