Editing Trigonometric interpolation (section)

===Even number of points===
For <math>N</math> even, we define the Dirichlet kernel as
:<math>D(x,N)=\frac{1}{N} +\frac{1}{N}\cos \tfrac12 Nx + \frac{2}{N} \sum_{k=1}^{(N-1)/2}\cos(kx) = \frac{\sin\tfrac12 Nx}{N\tan\tfrac12 x}.</math>
Again, it can easily be seen that <math>D(x,N)</math> is a linear combination of the right powers of <math>e^{ix}</math>, does not contain the term <math> \sin \tfrac12 Nx </math> and satisfies
:<math>D(x_m,N)=\begin{cases}0\text{ for } m\neq0 \\1\text{ for } m=0\end{cases}.</math>
Using these properties, it follows that the coefficients <math>t_k(x)</math> in ({{EquationNote|6}}) are given by
:<math>\begin{align}
t_k(x) &= D(x-x_k,N)=\begin{cases}
\dfrac{\sin\tfrac12 N(x-x_k)}{N\tan\tfrac12 (x-x_k)}\text{ for } x\neq x_k\\[10mu]
\lim\limits_{x\to 0} \dfrac{\sin\tfrac12 Nx}{N\tan\tfrac12 x}=1 \text{ for } x= x_k.
\end{cases}\\&= \frac{\mathrm{sinc}\,\tfrac12 N(x-x_k)}{ \mathrm{sinc}\,\tfrac12 (x-x_k)}\cos\tfrac12 (x-x_k)
\end{align}</math>
Note that <math>t_k(x)</math> does not contain the <math> \sin \tfrac12 Nx </math> as well. Finally, note that the function <math> \sin \tfrac12 Nx </math> vanishes at all the points <math>x_m</math>. Multiples of this term can, therefore, always be added, but it is commonly left out.