I’m trying to find a rigorous formulation of an impression/intuitive notion related to Hardy spaces. It seems to me that functions in Hardy spaces tend to have modulus functions which do not oscillate. This apparently generalizes a property of the exponential function. In particular:

$$exp(ikz) = cos(kz) + i*sin(kz)$$

with $sin(kz)$ the harmonic conjugate of $cos(kz)$, and

$$|exp(ikz)|^2 = cos^2(kz) + sin^2(kz) = 1.$$

For many examples of functions $F$ in Hardy spaces, one has

$$F(x) = u(x) + i v(x)$$

with $u(x)^2 + v(x)^2$ very apparently non-oscillatory. For certain specific examples, I am able to prove something. Bessel functions are a good example, I think.

The spherical Hankel function $h_n(z)$ of the first kind admits

the representation

$$h_n(z) = exp(iz) C_n int_0^infty exp(-zt) P_n(1+it) dt$$ with $C_n$ a constant depending on $n$ and $P_n$ the Legendre function of the first kind of order n. This can can be verified for integers n by direct substitution of the (finite) series expansions for each of these functions. This holds for noninteger orders as well, but proving it is a bit harder. It follows from the equivalent formula

$$h_n(z) = C_n int_1^infty exp(izt) P_n(t) dt$$

that $h_n$ is in a Hardy space of functions analytic on the upper half of the complex plane

(not $H^p$ for any $p>0$, though, since $P_n(t)$ belows up at infinity).

It can be shown by direct expansion of the relevant series that

$$|h_n(x)|^2 = 1+ int_0^infty exp(-xt) d/dt P_n(1+t^2/2) dt.$$

The derivative of $P_n(1+t^2/2)$ is positive on $(0,infty)$, so $|h_n(x)|^2$ is completely monotone on $(0,infty)$. An equivalent formula is

$$|h_n(x)|^2 = z int_0^infty exp(-xt) P_n(1+t^2/2) dt.$$

Obviously, it is too much to ask for |F(z)|^2 to be completely monotone for all F in a Hardy space. But is there a general principal here? It is not enough for F to be

in a Hardy space because there are obvious examples where |F| is highly oscillatory.

For example, take

$$F(z) = int_0^infty exp(izt) phi(t) dt$$

with $phi(t)$ a smooth function which is equal to 1 on (1,100000). The function |F(x)|^2

will oscillate rapidly on $(0,infty)$.

I have the strong impression that this must related to a well-known, standard result, but I don’t know where to look for it.