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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

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

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