Averaging Operator

DEFINITION 11.2.21 (Averaging Operator)   The AVERAGING OPERATOR, denoted by $\mu,$ gives the average value between two central points, i.e.,

$$\mu f(x)=\frac{1}{2}\bigl[f(x+ \frac{h}{2})+f(x- \frac{h}{2})\bigr].$$

Thus $\mu \,y_i=\frac{1}{2}(y_{i+ \frac{1}{2}}+y_{i-\frac{1}{2}})$ and

$$\mu^2\,y_i=\frac{1}{2}\left[ \mu\,y_{i+ \frac{1}{2}}+ \mu\,y_{i-\frac{1}{2}}\right] =\frac{1}{4}\left[y_{i+1}+2y_i+ y_{i-1}\right].$$