One strategy for proving the Prime Number Theorem is to use the results of Wiener and Ikehara. In fact, he gives two different proofs using slightly different result.
Them 220 If $g$ is $W$ and $h$ is $L$ and $f$ is bounded then $P_g(f) \to P_h(f)$. In other words: $$ \left[\int g(x-t) f(t) \, dt \to l \int f(t) \, dt \right] \to \left[\int h(x-t) f(t) \, dt \to l \int f(t) \, dt \right] $$
This statement (from Hardy's Divergent Series) look oddly vacuous and not related to divergent series. As best I could tell:
- $\widehat{g}(t) \neq 0$ for all $t \in \mathbb{R}$ and $g \in L^1(\mathbb{R})$.
- $h \in L^1(\mathbb{R})$
- $ |f(x)| < M$ for all $x \in \mathbb{R}$ (i.e. bounded)
I guess the theorem is that any $L^1$ function over the real line can be spanned by translates of a single function whose Fourier transform is nowhere vanishing on $\mathbb{R}$.
Many things have bothered me about this statement:
Could this be a generalization of Karamata Tauberian theorem? That can also be phrased as showing all continuous functions are within the span of translates of single function.
How is this related to the non-vanishing of the zeta function on the line $\zeta(1 + it) \neq 0$ for $t \in \mathbb{R}$ ?
Tauberian theorems are dismissed by some analysts as a novelty. What could be the modern replacement?
For the last question I found this result in a textbook [1]:
A function $f:[0,1] \to \mathbb{R}$ is absolutely continuous iff it is differentiable a.e., if $f'$ is locally integrable and for all $a,x \in [0,1]$ we have $$f(x) = f(a) + \int_a^x f'(t) \, dt $$
Penot's textbook teaches functional analysis in analogy with convex geometry, even if it's not too visual.
Historical Aside In response to Todd Trimble's comment, the Hardy's book on divergent series was published in 1949, two years after his death.
