Landau’s necessary density conditions for the Hankel transform

When I started this blog (quite some time ago…) I started blogging about a then-preprint (now it’s a published paper) with Luis Daniel Abreu entitled “Landau’s necessary conditions for the Hankel transform” and (as you might have noticed) I left the quest unfinished. Well, last week I spent a few days in Vienna (very nice place!) visiting NuHAG, and Daniel and I remembered that I have to finish the series of posts about our paper. Since I have a lot of other stuff I want to blog about, I am going to write one post explaining the result (and giving some context) and in a future post I will explain the main idea of the proof.

In my first post I introduced the notion of bandlimited functions and I showed the Whittaker-Shannon-Kotel’nikov sampling theorem. This theorem guarantees that, if a function {f} is bandlimited in {[-\pi,\pi]} then it is completly determined by its values on {\mathbb{Z}}. It actually gives much more: Since it provides a stable way of reconstructing {f} from these values it allows one to sample {f} at these values and reconstruct it. Furthermore, it allows us to, given a (square summable) sequence of values in {\mathbb{Z}}, construct a bandlimited function passing through those values.

On my second post I introduced the definitions of uniqueness, sampling, and interpolation sequences of a space of functions {\mathcal{B}} with the objective of understanding the properties of {\mathbb{Z}} given by the sampling theorem. Very briefly these definitions are a sequence of points {\{t_n\}} for which, respectively, the values of {f\in\mathcal{B}} in {\{t_n\}} completely determine {f}, the values of {f\in\mathcal{B}} in {\{t_n\}} stably determine {f}, and for any square summable sequence {\{a_n\}} there exists {f\in\mathcal{B}} such that {f(t_n)=a_n}.

It is natural to question whether there exists a sequence “smaller” than {\mathbb{Z}} that is still a sampling sequence or if there exists a “larger” interpolation sequence. Before dealing with this question one needs a good definition of size of a infinite sequence. Given a sequence {\{t_n\}} we define its upper and lower density has

\displaystyle D_-(\{t_n\}) = \liminf_{r\rightarrow\infty} \inf_{a} \frac1r\#\{\{t_n\}\cap [a,a+r]\}

\displaystyle D_+(\{t_n\}) = \limsup_{r\rightarrow\infty} \sup_{a} \frac1r\#\{\{t_n\}\cap [a,a+r]\},

this is usually called Beurling density.

It is clear that {D_-(\mathbb{Z})=D_+(\mathbb{Z})=1}. One can show that if {\{t_n\}} is a sampling sequence for the space of bandlimited function in {[-\pi,\pi]} then

\displaystyle D_-(\{t_n\})\geq 1,

and that if it is an interpolation sequence we must have {D_+(\{t_n\})\leq 1}. This result can be interpreted as showing that {\mathbb{Z}} is optimal for sampling of these functions, guaranteeing that the Nyquist rate can’t be breaked.

It is easy to see that if {f} is bandlimited in {[-K\pi,K\pi]} then {\frac1K\mathbb{Z}} plays the role of sampling and interpolation sequence and the same optimality result can be adapted to this setting. A much more interesting question is to understand the density of sampling or interpolation sequences for the space of functions bandlimited in a general set {S\subset\mathbb{R}}, by general I mean it is only required to be measurable, nothing else! This is done in a very nice paper by Landau (here is a short version that is much easier to read). It turns out that there is in fact a critical density for which sampling sequences have to be more dense and interpolation sequences have to be less dense (the density in high dimensions can be defined ). This density is

\displaystyle \frac1{2\pi} |S|.

In fact, Landau also shows an analogous result in higher dimensions.

In some applications (like signals coming from some types of antennas) the function {F} on {\mathbb{R}^d} that one wants to sample (or interpolate) is radial. This means that there exists a function {f} on {\mathbb{R}^+} such that, for every {x\in\mathbb{R}^d}, {F(x) = f(|x|)}. It is not hard to show that the fourier transform of {F}, let’s call it {G}, is also radial meaning that there exists a function {g} on {\mathbb{R}^+} such that {G(x) = g(|x|)}. The Fourier transfrom induces then a transformation on functions over {\mathbb{R}^+} (that, in this case, transforms {f} into {g}), this is called the Hankel transform, and after a few normalizations looks like this:

\displaystyle \mathcal{H}_\alpha(f)(x) = \int_{\mathbb{R}^+}f(t)(xt)^{1/2}J_\alpha(xt)dt,

where {J_\alpha} is the Bessel function of order {\alpha} ({\alpha} depends on the dimension {d}).

It turns out that a version of the sampling theorem exists for functions that are bandlimited on the Hankel transform (let us call them Hankel bandlimited). It is easily shown (very much like the proof I blogged about for the Fourier case) assuming the fact that {\{x^{1/2}J_\alpha(j_{n,\alpha}x) \}_{n=0}^\infty} is an orthogonal basis for {L^2[0,1]}, where {j_{n,\alpha}} is the {n}th zero of {J_\alpha}. On this sampling theorem, the sampling (and interpolation) sequence is {\{j_{n,\alpha}\}_{n=0}^\infty}; it is known that their density is {\frac1\pi}.

Our paper solves the problem of understanding the density of sampling and interpolation sequences of the space of functions which are Hankel bandlimited on a general (measurable) subset {S} of {\mathbb{R}^+}. We were able to show an analogous result to the one of Landau, which says that if {\{t_n\}} is a sampling sequence then

\displaystyle D_-(\{t_n\})\geq \frac1\pi |S|,

and that if, on the other hand, {\{t_n\}} is a sequence of interpolation then

\displaystyle D_+(\{t_n\})\leq \frac1\pi |S|.

In the same way that sampling sequences for Fourier bandlimited functions are related to Fourier frames, the sampling sequences for Hankel bandlimited functions are related to Fourier-Bessel frames. For this reason, our result provides necessary density conditions for Fourier-Bessel frames, too.

In a future post I will briefly explain what is the main idea to show this theorem.

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