I came up with a idea to estimate the coefficients of a B-spline fit by using the Fourier Transform but I don't know if it makes any sense to estimate them in this way.
Given that
$$s(x)=\sum_kc(k)\beta^n(x-k)$$ where $k$ is an integer and $\beta^n(x)$ is defined as a ($n+1$) times convolution :$$\beta_0*\beta_0*...*\beta_0(x)$$ where $\beta_0$ is a rectangular pulse of length 1 centred at zero.
One could hence write $s(x)=(c*\beta^n)(x)$. Applying the Fourier Transform we obtain. $s(\omega)=c(\omega)\cdot \beta^n(\omega)=c(\omega)\cdot \beta_0(\omega)^{n+1}$
From here I would like to obtain $c(\omega)$ to then apply the inverse FFT. I've seen some papers dealing with a similar methodology but without explaining in detail what are the condition in order for this to work.
Any suggestion? Thanks in advance