Proof of Landau’s conditions for the Hankel transform

As promised in my last blog post, I am writing this post to briefly explain the main ideas of the proof of the main Theorem in my paper about Landau’s necessary density conditions for the Hankel transform.

Recall that, given a function {f} defined in {\mathbb{R}^+} we define its Hankel transform as

\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}.

We want to understand how dense sampling or interpolation sequences have to be. Let’s say our objective is to understand how many points a sampling sequence has to have on an interval {I}.

Let’s consider the subspace of functions very concentrated in {I} and whose Hankel-band is supported in {S}, this is obviously not a subspace but for the sake of intuition let us think of it as one. A function in this space should be determined by the values of it on the sampling points in the interval {I}, since the function should have value almost zero on the sampling points outside of this interval and the “stability” required in the sampling sequence definition suggests that these points play, in fact, no role. These ideas suggest us that the number of sampling points in {I} should be larger than the “dimension” of the subspace (and for an interpolation sequence it should be smaller). It turns out that this intuition is surprisingly accurate.

The question now is how can one “estimate” the dimension of this set of functions. The idea is to use localization operators: Let {P_I} be the operator that projects onto the space functions that take only non-zero values on the interval {I} and {B_S} the operator that projects onto the space functions whose Hankel transform take only non-zero values on the set {S}. The set of functions that are {S} Hankel-bandlimited and concentrated in {I} should be almost invariant under the localization operator {L_{S,I}=B_SP_IB_S}, it first projects on the space of {S} Hankel-bandlimited functions, then on the space of functions supported in {I} and then back to the space of {S} Hankel-bandlimited functions.

Since both {B_S} and {P_I} are projection operators, it is clear that the eigenvalues of {L_{S,I}} have to lie between {0} and {1} and that the number of eigenvalues very close to {1} count the dimension of the space of function that are almost kept invariant by {L_{S,I}}, this is precisely the quantity we want to estimate.

I don’t want to go into the details of how this is done but the way that the number of eigenvalues close to {1} is estimated is by estimating the Trace and the Norm of the operator. Note that the Trace is the sum of the eigenvalues and the Norm is the sum of the squares, this means that if {\lambda_1,\lambda_2,\dots} are the eigenvalues we have

\displaystyle Trace - Norm = \sum\lambda_i - \sum\lambda_i^2 = \sum\lambda_i(1-\lambda_i).

Note that is {Trace-Norm} is very small this implies that all eigenvalues are either very close to {0} or very close to {1} which means that {Trace = \sum\lambda_i} accurately counts how many are close to {1}.

Both the {Trace} and the {Norm} are estimated by writing {L_{S,I}} in terms of an integral operator. The {Trace} is relatively easy to estimate and can be showed to be, roughly, {\frac1\pi |S| |I|}. However, estimating the {Norm} is one of the main difficulties of proving the result, the details can be found on the paper. The estimate for the {Norm} implies that {Trace-Norm} is indeed very small. This means that the “dimension” of the subspace of functions that are concentrated in {I} and have Hankel-band supported in {S} is roughly {\frac1\pi |S| |I|}.

Although this blog post was highly informal and not rigorous at all, these ideas can be made rigorous and, in fact, the big picture of the proof is present in this post. There are other interesting steps that explain how the number of nearly {1} eigenvalues actually relates to the density of sampling and interpolation sequences but those are a bit technical (and can be found in the paper)

Another difficulty is how to deal with a general interval {I=[a,a+r]} instead of simply {[0,r]}. On the Fourier case this is trivial because the eigenvalues are translation invariant, however in the Hankel setting this is one of the main sources of difficulty, since there, the eigenvalues are not translation invariant. Dealing with general sets is what allows us to have a result in terms of this Beurling density, defined for a sequence {\{t_n\}}, as:

\displaystyle D_-(\{t_n\}) = \liminf_{r\rightarrow\infty} \inf_{a\geq 0} \frac1r\#\{\{t_n\}\cap [a,a+r]\} \qquad D_+(\{t_n\}) = \limsup_{r\rightarrow\infty} \sup_{a\geq 0} \frac1r\#\{\{t_n\}\cap [a,a+r]\},

Recall that the Theorem states that, if {\{t_n\}} is a sampling sequence for the space of functions which are Hankel bandlimited on a general (measurable) subset {S} of {\mathbb{R}^+} 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|.

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