Editing Trigonometric interpolation (section)

===Odd number of points===
If the number of points ''N'' is odd, say ''N=2K+1'', 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} y_k\,t_k(x),</math>|{{EquationRef|5}}}}
where
:<math> t_k(x) = e^{-iKx+iKx_k} \prod_{\begin{align}m&=0 \\[-4mu] m &\ne k\end{align}}^{2K} \frac{e^{ix}-e^{ix_m}}{e^{ix_k}-e^{ix_m}}.</math>
The factor  <math>e^{-iKx+iKx_k}</math> in this formula compensates for the fact that the complex plane formulation contains also negative powers of <math>e^{ix}</math> and is therefore not a polynomial expression in <math>e^{ix}</math>. The correctness of this expression can easily be verified by observing that <math>t_k(x_k)=1</math> and that <math>t_k(x)</math> is a linear combination of the right powers of <math>e^{ix}</math>.
Upon using the identity
{{NumBlk|:|<math>e^{iz_1}-e^{iz_2}=2i\sin\tfrac12(z_1-z_2)\,e^{(z_1 + z_2)i/2},</math>|{{EquationRef|2}}}}
the coefficient <math>t_k(x)</math> can be written in the form
{{NumBlk|:|<math> t_k(x) = \prod_{\begin{align}m&=0 \\[-4mu] m &\ne k\end{align}}^{2K} \frac{\sin\tfrac12(x-x_m)}{\sin\tfrac12(x_k-x_m)}.</math>|{{EquationRef|4}}}}