Non-linear regression. Obtain B.spline coefficients using Fourier Transform?

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. $$s(\omega)=c(\omega)\cdot\beta^n(\omega)=c(\omega)\cdot \beta_0(\omega)^{n+1}$$

$$\beta_0(\omega)= (\operatorname{sinc}(\omega/2))$$ 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

• It's unclear how you would apply this approach: could you explain how it might work with actual data? – whuber Nov 26 '18 at 17:33