I'll write everything down from the start.

First, the notation:

, where

Then the theorem:

If is summable for every , then

, where denotes Fourier transform of f.

Now, the problem. :(

We're supposed to prove this using induction on , and I just can't do it. The definition of is somewhat confusing, and I would b really grateful if someone could help me out, just for the basis, and I will try to work from there.

Thank you!