Editing Trigonometric interpolation (section)

===Odd number of points===
Further simplification by using ({{EquationNote|4}}) would be an obvious approach, but is obviously involved. A much simpler approach is to consider the [[Dirichlet kernel]]
:<math>D(x,N)=\frac{1}{N} +\frac{2}{N} \sum_{k=1}^{(N-1)/2}\cos(kx) = \frac{\sin\tfrac12 Nx}{N\sin\tfrac12 x},</math>
where <math>N>0</math> is odd. It can easily be seen that <math>D(x,N)</math> is a linear combination of the right powers of <math>e^{ix}</math> and satisfies
:<math>D(x_m,N)=\begin{cases}0\text{ for } m\neq0 \\1\text{ for } m=0\end{cases}.</math>
Since these two properties uniquely define the coefficients <math>t_k(x)</math> in ({{EquationNote|5}}), it follows that
:<math>\begin{align}
t_k(x) &= D(x-x_k,N)=\begin{cases}
\dfrac{\sin\tfrac12 N(x-x_k)}{N\sin\tfrac12 (x-x_k)} \text{ for } x\neq x_k\\[10mu]
\lim\limits_{x\to 0} \dfrac{\sin\tfrac12 Nx}{N\sin\tfrac12 x}=1 \text{ for } x= x_k
\end{cases}\\&= \frac{\mathrm{sinc}\,\tfrac12 N(x-x_k)}{\mathrm{sinc}\,\tfrac12 (x-x_k)}.
\end{align}</math>
Here, the [[sinc]]-function prevents any singularities and is defined by
:<math> \mathrm{sinc}\,x=\frac{\sin x}{x}.</math>