50) Find the -transform of the sequence . (Give a closed-form expression.)

51) Let be as in the previous problem.

- Find two signals and such that . (You might consider letting be a -spike.)
- If , can you find a such that ?
- Can you a solution with causal signals find and such that both and have infinitely many nonzero components and ? (You might consider -transforms for this last part.)

52) Find a simple expression for , where

.

53) Show that the -transform of the sequence

is .

(Problems 52 and 53 are from Chapter 5.1 of *Transform methods in applied mathematics*, by Lancaster and Salkauskas.)