A Proof of the HRT Conjecture for Widely Spaced Sets
Given and a finite set we demonstrate the linear independence of the set of time-frequency translates when the time coordinates of points in are far apart relative to the decay of As a corollary, we prove that for any and finite there exist infinitely many dilations such that is linearly independent. Furthermore, we prove that is linearly independent for functions like which have a singularity and are bounded away from any neighborhood of the singularity.
Consider the translation operator and the modulation operator acting on For we define the time-frequency shift The Heil-Ramanathan-Topiwala (HRT) Conjecture  states:
Suppose is nonzero and is a finite set. Then the collection of functions is linearly independent.
In this paper we will prove Conjecture 1 in cases where the distance between points in is large relative to the decay of For example, we will prove the following theorem. We denote by the space of continuous functions for which as
Let let be a finite set, and fix so that for all outside of the ball of radius around the origin. If whenever then is linearly independent.
The intuition for Theorem 1 is that if the points in are spaced far apart then in any linear dependence the tails of translates of must combine to cancel the peaks of This requires putting large coefficients on the translates of However by putting large coefficients on the translates of we make the peaks of the translates more difficult to cancel, leading to a contradiction.
Theorem 1 is most effective when drops off steeply near the origin. Given any we can engineer such a steep descent by applying a sufficiently large dilation, assuming We denote by the unitary operator which dilates a function uniformly along all the coordinate axes:
Suppose and Given there exists such that is linearly independent for all
Similarly, if has a singularity away from which it is bounded then we can find translates of which have an arbitrarily steep drop off. Thus we can prove Conjecture 1 for such functions.
Let be continuous except at a point where Assume that is bounded away from any neighborhood of Then is linearly independent for any finite
In the next section we will prove Theorems 1 and 2 along with some further variations on the same theme. The spirit of our analysis is most similar to the extension principle in , however our strategy and results are different.
2 Proofs, Examples, and Extensions
The following lemma captures the intuition for Theorem 1 described above:
Let and such that for all Furthermore, suppose that whenever are distinct points in Then the collection of functions is linearly independent.
Assume that the functions are linearly dependent, so that for some coefficients we have
Since is continuous this equality holds for all If we evaluate the LHS at the point we can rearrange to get the following inequality:
After summing all these inequalities and canceling from each side, we see that
which is a contradiction. The last equality follows since each term appears in exactly of the inner sums in the second expression. ∎
Now we can prove our first theorem:
Proof of Theorem 1.
We can apply Lemma 1 with and Since the points all lie outside the ball of radius around the origin, as required. ∎
Note that we only need to assume that the time coordinates of the points in are spaced far apart for Theorem 1 to hold. Although the specific value suspiciously appears in our hypothesis, is linearly independent if and only if is linearly independent for all so we can always translate to put the most advantageous value at the origin.
Given Theorem 1, it is straightforward to deduce Corollary 1:
Proof of Corollary 1.
Since and we can find a value such that for all outside of a ball of radius around Applying a dilation we see that whenever lies outside a ball of radius Let be the minimum distance between any two points in Then whenever we can apply Theorem 1 to show that is linearly independent. ∎
Since translations and modulations are exchanged under the Fourier transform, we get an analogous result in the frequency domain. We denote by the Fourier transform of
Let so that Let and fix so that for all outside of the ball of radius around the origin. If whenever then is linearly independent.
First note that = and Therefore we can define
and is linearly independent if and only if is linearly independent. However Theorem 1 immediately applies to show is linearly independent. ∎
We have a similar extension of Corollary 1 to the frequency domain.
Let so that and let Then there exists a value such that is linearly independent whenever
The proof is the same as for Corollary 1, after noting that and that the Fourier transform rotates the time-frequency plane as described in the proof of Corollary 2. ∎
Consider the family of functions
The functions are in Nonetheless they decay slowly at infinity and oscillate in the tail. To the author’s knowledge, such functions are not covered by the results of , , or  which assume fast decay at infinity or ultimate positivity. Given let Then by applying Theorem 1, we can see that is linearly independent whenever For the four point set we have linearly independent whenever
Example 1 suggests that the function should satisfy Conjecture 1 in full, as it is the pointwise limit of as This is true, and is implied by Theorem 2 which we are now ready to prove.
Proof of Theorem 2.
Without loss of generality we may assume that since we can always translate to place the singularity at the origin. If we fix of size such that the minimum distance between the coordinates in is we would like to find a translate of which satisfies outside the ball of radius around If we can find such an then the argument in the proof of Lemma 1 applies to show is linearly independent. To find such an we first note that since is bounded away from we can find such that outside a ball of radius around Since we can find an less than such that and this satisfies the criteria described above. ∎
We can adapt the examples above to find functions in satisfying Conjecture 1. Consider the family of functions
The functions are in Theorem 2 applies to show that they satisfy Conjecture 1.
The previous results apply when all points in are sufficiently far apart in either time or frequency. By applying the Short Time Fourier Transform (STFT) we can demonstrate linear independence when the points in are sufficiently far apart in the time-frequency plane. For the STFT of with respect to is given by
It is easy to see  that and satisfies the identity
Suppose so that Let and fix so that for all outside of the ball of radius around the origin. If whenever then is linearly independent.
Suppose is linearly dependent so that for some coefficients we have
Then by applying the STFT with respect to we have
where denotes the time coordinate of and However we can apply Lemma 1 to with and to show that the functions must be linearly independent, which is a contradiction. ∎
Our Lemma 1 and Theorem 1 demonstrate that is linearly independent when the points of are far apart relative to the decay in However our proofs use no properties specific to the modulations and apply just as well to functions in when and Given the generality of Theorem 1 and in light of the following example, we can see that Theorem 1 alone provides only loose evidence for Conjecture 1. It would need to be combined with tools more specific to Conjecture 1 if we hoped to use it to make further progress.
In  the authors demonstrate that the function
is in for and satisfies the dependence
Nonetheless, our Theorem 1 and Corollary 1 can be applied to though Theorem 1 clearly does not rule out the dependence above.
To this end, we can try to apply metaplectic transforms in an attempt to satisfy the hypotheses of Theorem 1 in more cases. For example, for and the dilation operator satisfies
Given a point set we can apply a large dilation to move the points far apart, but we also stretch the function so that it decays very slowly away from the origin. Since the function stretches at the same rate that the points are moved apart, satisfies the hypotheses of Theorem 1 if and only if does, where Thus dilations are no help in furthering the uses of Theorem 1 for
Another useful metaplectic transform is modulation by a linear chirp. If then
Similarly, if we let then
For the purposes of applying our theorems, the operators and look promising, as compositions of them can be used to increase the distance between points in the time-frequency plane. However, as in the case of dilations, we must understand how compositions of and affect the decay of and compare this with the increase in distance between points. The author has not made substantive progress to demonstrate new examples by applying these transforms, nor has he been able to rule out their utility.
In another direction, one could try to expand the utility of Theorem 3 by leveraging the choice of window function as a free variable. One could leave and fixed but vary the window function in an attempt to satisfy the hypotheses. This leads naturally to the question:
Given can we design a window so that outside the ball of radius around the origin?
A positive answer to Question 1 would prove the HRT conjecture. We would want to design so that decreases sharply near the origin and then has a fat tail, since we know that the probability mass of cannot be too heavily concentrated near the origin due to various uncertainty principles for the STFT. Alternatively, it may be possible to develop a kind of uncertainty principle which answers Question 1 negatively.
The author would like to thank Radu Balan and Kasso Okoudjou for introducing him to the HRT Conjecture and for helpful discussions about this work. The author would also like to thank an anonymous reviewer for encouraging him to expand the results contained in earlier drafts.
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