Advances in Numerical Analysis Emphasizing Interval Data

4.4 Trigonometric Interpolation

When the function $f$ is a periodic function, we approximate it with periodic functions that have a definite periodicity. Therefore, in this section, we examine the trigonometric interpolating functions. This type of interpolation is a special mode of linear and non-recursive interpolation. The structure of a trigonometric interpolator is a combination of functions $\sin lx$ and $\cos lx$ where $l\in \mathbb{Z}$.

Suppose that for $k=0,\dots ,N-1$, $\left(x_k,f_k\right)$ s are interpolation points. We consider two modes for the trigonometric interpolating function:

$$\psi \left(x\right)=\{\begin{array}{cc}\frac{A_0}{2}+{\displaystyle \sum _{l=1}^M\left(A_l\cos lx+B_l\sin lx\right),}& N=2M+1\\ \frac{A_0}{2}+{\displaystyle \sum _{l=1}^{M-1}\left(A_l\cos lx+B_l\sin lx\right)+\frac{A_M}{2}\cos Mx,}& N=2M\end{array}$$

In both cases, $\psi \left(x\right)$ is a periodic function in terms of $x$ with periodicity of $2\pi$.

Suppose that the interpolation ...