Interpolation

For a comparative high polynomial degree $\vert I_{\ensuremath{\boldsymbol{N}}}\vert>M$ one expects to interpolate the given data $y_j\in \ensuremath{\mathbb{C}},\;j=0,\hdots,M-1,$ exactly. The (consistent) linear system (3.2) is under-determined. One considers the damped minimisation problem

$$\Vert\ensuremath{\boldsymbol{\hat f}}\Vert _{\ensuremath{\boldsym... ...}}} \right)^{1/2} \stackrel{\ensuremath{\boldsymbol{\hat f}}}{\rightarrow} \min \quad \ensuremath{\boldsymbol{A}} \ensuremath{\boldsymbol{\hat f}} = \ensuremath{\boldsymbol{y}},$$

which may incorporate 'damping factors' $\hat w_{\ensuremath{\boldsymbol{k}}}> 0$ , $\ensuremath{\boldsymbol{\hat W}}:= {\, \rm diag}(\hat w_{\ensuremath{\boldsymbol{k}}})_{\ensuremath{\boldsymbol{k}}\in I_{\ensuremath{\boldsymbol{N}}}}$ . A smooth solution is favoured, i.e., a decay of the Fourier coefficients $\hat f_{\ensuremath{\boldsymbol{k}}},\;\ensuremath{\boldsymbol{k}}\in I_{\ensuremath{\boldsymbol{N}}}$ , for decaying damping factors $\hat w_{\ensuremath{\boldsymbol{k}}}$ . This interpolation problem is equivalent to the damped normal equation of second kind

$$\ensuremath{\boldsymbol{A}} \ensuremath{\boldsymbol{\hat W}} \ens... ...bol{A}}^{{\vdash \hspace*{-1.72mm} \dashv}} \ensuremath{\boldsymbol{\tilde f}}.$$ (3.4)

Applying the conjugate gradient scheme to (3.4) yields the following Algorithm 8.