Master Theorem on oscillating function

Question

Consider a recurrence of the form

$T(n) = a T(n/b) + f(n)$

Master theorem's regularity condition excludes some cases (for example, when $f(n)$ is oscillating).

Suppose, however, that $f(n)$ is always less than or equal to a function $g(n)$ that does not violate the regularity condition, so that the master theorem is applicable if $g(n)$ is used instead of $f(n)$. Consider then the following recurrence:

$T'(n) = a T'(n/b) + g(n)$

and assume that the master theorem gives the solution $T'(n)=\Theta(g(n))$.

My doubt is this: can I then safely conclude that $T(n)=O(g(n))$?

In other words, can I be sure that the solution of a recurrence with a term replaced by its upper bound is an upper bound of the solution of the original occurrence? After all, the recurrence establishes a relationship between $T(n)$ and $T(n/b)$ but says nothing about what happens between $n/b$ and $n$ (and $f(n)$ here is oscillating...).

Answer 1

Yes, you can. You are defining the recurrence $T'$, so define $T'(1) = T(1)$. Then, since $g(n)$ satisfies the regularity conditions, the Master theorem tells you that $T'(n) = \Theta(g(n))$. Note that the Master theorem doesn't depend on the choice of $T'(1)$; it is a constant, and you get the same asymptotic bound for $T'(n)$ regardless of which constant you choose.

Now, you can prove by induction that $T(n) \le T'(n)$ holds for all $n$. This is a simple induction. The base case holds because you defined $T'$ that way. The inductive step holds because $f(n) \le g(n)$ for all $n$.

Once you've proven that $T(n) \le T'(n)$ for all $n$, you are entitled to conclude that $T(n) = O(T'(n))$ and thus $T(n) = O(g(n))$.