# zbMATH — the first resource for mathematics

The Marr conjecture and uniqueness of wavelet transforms. (English) Zbl 1404.42062
In $${\mathbb R}^d$$, the Richter or Mexican Hat wavelet $$M$$ is the Laplacian of a Gaussian. We let $$M_\sigma(x) = \sigma^{-d} M(x/\sigma)$$. The continuous wavelet transform of a function $$f$$ at scale $$\sigma$$ is then $$f * M_\sigma$$. The question is whether $$f$$ is uniquely determined by the zeroes of its continuous wavelet transform.
The original Marr conjecture states: A locally integrable function $$f$$ of sufficiently rapid decay is uniquely determined (up to a constant multiple) by the zero sets of $$f * M_\sigma$$ for any sequence of positive scales $$\{ \sigma_j\}_{j=1}^\infty$$ tending to infinity. Some special cases of the conjecture have been proved in the past. This paper proves the conjecture in more generality, and also considers wavelets $$\psi$$ other than $$M$$.
The main result is that it is sufficient that $$f$$ is integrable and of exponential decay, and that $$\psi$$ is smooth, with all derivatives growing slower than exponentially, and satisfying the “Genericity Condition”. That condition states that the regular zero set of any derivative of order $$m$$ does not contain the regular zero set of any derivative of order $$n$$, $$m, n \geq 0$$. Here “derivative of order $$m$$” means any linear combination of partial derivatives of total order $$m$$.
The Mexican Hat wavelet in one dimension satisfies the Genericity Condition. In higher dimensions the condition can be reduced to a condition on the zeroes of Laplace-Hermite polynomials. For some values of $$m$$, $$n$$ in dimension $$d=2$$, the authors have verified the condition numerically.
Other results include counterexamples for the case when conditions are not satisfied; a counterexample to show that the conjecture does not hold for the discrete case (where the zero crossings are only known on a lattice); and a corollary that shows that the zeros of a solution to the heat equation determine the initial conditions uniquely (under some assumptions).

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: