Editing Trigonometric interpolation (section)

===Even number of points===
If the number of points ''N'' is even, say ''N=2K'', applying the [[Lagrange polynomial|Lagrange formula for polynomial interpolation]] to the polynomial formulation in the complex plane yields that the solution can be written in the form
{{NumBlk|:|<math> p(x) = \sum_{k=0}^{2K-1} y_k\,t_k(x),</math>|{{EquationRef|6}}}}
where
{{NumBlk|:|<math> t_k(x) = e^{-iKx+iKx_k} \frac{e^{ix}-e^{i\alpha_k}}{e^{ix_k}-e^{i\alpha_k}} \prod_{\begin{align}m&=0 \\[-4mu] m &\ne k\end{align}}^{2K-1} \frac{e^{ix}-e^{ix_m}}{e^{ix_k}-e^{ix_m}}.</math>|{{EquationRef|3}}}}
Here, the constants <math>\alpha_k</math> can be chosen freely. This is caused by the fact that the interpolating function ({{EquationNote|1}}) contains an odd number of unknown constants. A common choice is to require that the highest frequency is of the form a constant times <math>\cos(Kx)</math>, i.e. the <math>\sin(Kx)</math> term vanishes, but in general the phase of the highest frequency can be chosen to be <math>\varphi_K</math>. To get an expression for <math>\alpha_k</math>, we obtain by using ({{EquationNote|2}}) that ({{EquationNote|3}}) can be written on the form
:<math> t_k(x) = \frac{\cos\tfrac12\Biggl(2Kx-\alpha_k+\displaystyle\sum\limits_{m=0,\,m \ne k}^{2K-1} x_m\Biggr)+\sum\limits_{m=-(K-1)}^{K-1}c_k e^{imx}}{2^N\sin\tfrac12(x_k-\alpha_k)\displaystyle\prod\limits_{m=0,\,m \ne k}^{2K-1}\sin\tfrac12(x_k-x_m)}.</math>
This yields
:<math>\alpha_k=\sum_{\begin{align}m&=0 \\[-4mu] m &\ne k\end{align}}^{2K-1} x_m - 2 \varphi_K</math>
and
:<math> t_k(x) = \frac{\sin\tfrac12(x-\alpha_k)}{\sin\tfrac12(x_k-\alpha_k)}\prod_{\begin{align}m&=0 \\[-4mu] m &\ne k\end{align}}^{2K-1} \frac{\sin\tfrac12(x-x_m)}{\sin\tfrac12(x_k-x_m)}.</math>

Note that care must be taken in order to avoid infinities caused by zeros in the denominators.