We determine the frame set of the first Hermite function, proving a conjecture of Lyubarskii and Nes. We use a characterization of semi-regular Gabor frames due to Gr\"ochenig, Romero, and St\"ockler to transform the problem into a question about the existence of a nonzero Gaussian shift-invariant entire function $F$, with bounded coefficients, whose derivative vanishes on a lattice with spacing $\delta=ab$. By taking the Wronskian of $N$ translates of $F$, we amplify its critical points to zeros of multiplicity at least $N-1$. After rescaling, this Wronskian is again a Gaussian shift-invariant function. A Gaussian zero-density theorem then gives $(N-1)/(N\delta)\leq1$. Varying $N$ over the range allowed by the construction forces every subcritical lattice product $\delta=ab$ for which the system is not a frame to equal $(q-1)/q$ for some integer $q\geq2$.
We study the zeros of the spectrogram of complex white noise analyzed with Hermite windows of order n, extending the classical Gaussian-window correspondence with Gaussian Analytic Functions to the polyanalytic setting. Using the polyanalytic Bargmann transform, these zeros split into two families according to the sign...
Charaf Bellourak, S. Meignen, K. Polisano et al.· 0 citations
In one complex variable the Schwarzian derivative characterizes the M\"obius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors--Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in $...
We prove that every finite multivariate Gaussian mixture density has only finitely many modes. Our approach combines an algebraic formulation of the ridgeline theory of Ray and Lindsay (2005) with a transcendence-degree argument based on Ax's functional-transcendence theorem to bound the cardinality of the set of criti...
We give a complete characterization of the zero set of the Zak transform of the second-order B-spline $Q_2$. As a consequence, we obtain a new proof of the Lemvig--Nielsen obstructions to the Gabor frame property of $Q_2$, and, along the hyperbola $ab=1/q$, we show these obstructions are sharp: every point they do not...
Let a normalized continuous Parseval frame describe the phase space of a reproducing kernel Hilbert space. The frame coefficients of an orthonormal family have total mass equal to the number of states and pointwise mass at most one. This elementary exclusion principle gives explicit lower bounds for the total cost of a...
A lattice Gabor frame consists of a window function and its time-frequency shifts along a lattice. We classify the lattices that admit Schwartz-class and, equivalently, Feichtinger frame windows. Except for symplectically irrational lattices of critical covolume 1,, we determine the lattice-dependent achievable window...
Andrei Caragea, G. Pfander· 0 citations
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