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208 7. Approximations and Inequalities

9. Let f(n) be a function defined on the positive integers. As in Exercises
   7 and 8, construct a table for the sequence f(l), f(2), f(3),....

(a) Verify that the nth term of the second column is .f(n+ 1) - j(n).

Call this quantity Af(n).

(b) Verify that the nth term of the third column is [f(n + 2) -

f(n + l)] - [f(n + 1) - f(n)] = f(n + 2) - V(n + 1) + f(n). Call

this quantity A(Af(n)) or A2f(n).

(c) Let us denote the nth element of the (h + 1)th column by

Akf (n).

Prove that

Akf(n) = A(Ak-‘f(n)) = &(-l)k-’ ( f ) f(n + +

i=O

10. In general, for any function f(t), we can define three operators

If(t) = f(t). identity

of = f(t + 1) I shift

Af(t) = j(t + 1) - f(t) : difference.

Any of these operators can be iterated. Thus,

Ekf@) = E(Ek-‘f(t))

Akf(t) = A(A”-‘f(t)) (h > 2).

(a) Verify that, for functions f and g and constant c,

A(cf (t)) = cAf (t)

A(f + g)(t) = Af(t) + W).

(b) Verify that E”f (t) = f(t + k).

(c) When f(t) = t2 -3t+l, verify that If(t) = t2-33t+l, Ef(t) =

t2 -t - 1, Af(t) = 2t - 2 and A2f(t) = 2.

(d) The operators I, E, A can be manipulated like numbers, with

iteration playing the role of multiplication and I playing the role

of 1. Justify the equations

A=E-I and E=I+A.

(e) Determine an expression for Ak f (t) in terms off(t), f (t + l),...

by expanding Ak = (E - I)k b inomially and applying this op-

erator to f(t). Compare with Exercise 9(c).