Relations between Difference operators

1. We note that
   $$Ef(x)=f(x+h) \;\; = [f(x+h)-f(x)]+f(x) \;\; = \Delta f(x)+f(x)= (\Delta +1)f(x).$$
   Thus,
   $$\framebox[1in][c]{ $E \equiv 1 + \Delta$ } \;\; {\mbox{ or }} \;\; \Delta \equiv E-1.$$

2. Further, $\nabla (E(f(x)) = \nabla(f(x+h)) = f(x+h) - f(x).$ Thus,
   $$(1-\nabla) Ef(x)= E(f(x)) - \nabla(E(f(x)) = f(x+h) - [f(x+h)-f(x)] = f(x).$$
   Thus $E \equiv 1 + \Delta,$ gives us
   $$(1-\nabla)(1+\Delta) f(x)=f(x)\; {\mbox{ for all }} \; x.$$
   So we write,
   $$(1+\Delta)^{-1}=1-\nabla \,\,\,{\mbox{ or }} \;\; \framebox[1.5in][c]{$ \nabla=1-(1+\Delta)^{-1},$} \;\; {\mbox{ and }}$$
   $$(1-\nabla)^{-1}=1+\Delta =E.$$
   Similarly,
   $$\Delta=(1-\nabla )^{-1}-1.$$

3. Let us denote by $E^{\frac{1}{2}}f(x)=f(x+ \frac{h}{2}).$ Then, we see that
   $$\delta f(x)=f(x+ \frac{h}{2})-f(x-\frac{h}{2}) = E^{\frac{1}{2}}f(x)- E^{-\frac{1}{2}}f(x).$$
   Thus,
   $$\;\; \framebox[1.4in][c]{$\delta = E^{\frac{1}{2}}-E^{-\frac{1}{2}}.$}$$
   Recall,
   $$\delta ^2 f(x)=f(x+h)-2f(x)+f(x-h) = [f(x+h)+2f(x)+f(x-h)]-4f(x) = 4(\mu^2-1)f(x).$$
   So, we have,
   $$\framebox[1in][c]{$\mu^2\equiv \frac{\delta^2}{4}+1$} \;\; {\mbox{ or }} \;\; \framebox[1in][c]{$\mu \equiv \sqrt{1+\frac{\delta^2}{4}}$}.$$
   That is, the action of $${\sqrt{1+\frac{\delta^2}{4}}}$$ is same as that of $\mu.$
4. We further note that,
   

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

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

   
   and
   

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

   $$= \frac{1}{2}\left[f(x+h)-f(x-h)\right].$$

   
   Thus,
   $$\Delta f(x) = \left[\frac{1}{2}\delta^2+\delta \mu\right]f(x),$$
   i.e.,
   $$\Delta \equiv \frac{1}{2} \delta ^2 +\delta \mu \equiv \frac{1}{2} \delta ^2 +\delta\sqrt{1+\frac{\delta^2}{4}}.$$

   In view of the above discussion, we have the following table showing the relations between various difference operators:

   	E	$\Delta$	$\nabla$	$\delta$
   E	E	$\Delta+1$	$(1-\nabla)^{-1}$	$\frac{1}{2} \delta ^2 +\delta\sqrt{1+\frac{\delta^2}{4}}+1$
   $\Delta$	$E-1$	$\Delta$	$(1-\nabla)^{-1}-1$	$\frac{1}{2} \delta ^2 +\delta\sqrt{1+\frac{1}{4}\delta^2}$
   $\nabla$	$1-E^{-1}$	$1-(1+\nabla)^{-1}$	$\nabla$	$-\frac{1}{2} \delta ^2 +\delta\sqrt{1+\frac{1}{4}\delta^2}$
   $\delta$	$E^{1/2}-E^{-1/2}$	$\Delta(1+\Delta)^{-1/2}$	$\nabla(1-\nabla)^{-1/2}$	$\delta$

EXERCISE 11.3.1  

1. Verify the validity of the above table.
2. Obtain the relations between the averaging operator and other difference operators.
3. Find $\Delta^2 y_2, \; \nabla^2 y_2, \; \delta^2 y_2$ and $\mu^2 y_2$ for the following tabular values:

   $i$	0	1	2	3	4
   $x_i$	93.0	96.5	100.0	103.5	107.0
   $y_i$	11.3	12.5	14.0	15.2	16.0