Question

4: The relationship between the operators E and D is​

Answer 1

Answer:

The lower the Delta E, the closer the colors are to each other. A delta E of zero indicates that there is zero difference between the two colors. The higher the Delta E, the further apart the colors are and more color difference is preceived.

Answer 2

The relationship between the operators E and D is $\displaystyle \sf{E \equiv \: {e}^{hD} }$

Given :

The operators E and D

To find :

The relationship between the operators E and D

Solution :

Step 1 of 2 :

Define shift operator E

For any arbitrary function f(x) the shifting operator or shift operator E is defined as

$\displaystyle \sf{ Ef(x) = f(x + h) }$

Where h is the Step length

Step 2 of 2 :

Find relationship between E and D

Using Taylor's Theorem we get

$\displaystyle \sf f(x + h) = f(x) + hf'(x) + \frac{ {h}^{2} }{2!} f''(x) + \frac{ {h}^{3} }{3!} f'''(x) + . \: . \: .$

$\displaystyle \sf \implies Ef(x ) = f(x) + hDf(x) + \frac{ {h}^{2} }{2!} {D}^{2} f(x) + \frac{ {h}^{3} }{3!} {D}^{3} f(x) + . \: . \: .$

$\displaystyle \sf \implies Ef(x ) = \bigg[1 + hD+ \frac{ {h}^{2} {D}^{2} }{2!} + \frac{ {h}^{3} {D}^{3} }{3!} + . \: . \: .\bigg] f(x)$

$\displaystyle \sf \implies Ef(x ) = {e}^{hD} f(x)$

$\displaystyle \sf \implies E \equiv \: {e}^{hD}$