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


A function $f\ge 0$ on the circle is the absolute value $f=|F|$ of an $F\in H^p$, $F\not\equiv 0$, if and only if $f\in L^p$ and $\log f\in L^1$, and obviously you can make such functions as oscillatory as you desire them to be, so there's no general principle at work here.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.