We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gr\"ochenig, Romero, and St\"ockler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.
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
We develop the structure theory of the deterministic $\sin^2$-type algorithm for complex cubic fields introduced in the companion paper, addressing the complex-signature case of Karpenkov's Problem 4. The selection rule is shown to be, exactly, the minimization of a conformal module: the hyperbolic cosine of the distan...
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...
