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Lukas Liehr

Publications and source records attributed to Lukas Liehr.

At least 19 recordsLinked to original sources
arXiv2026

Grothendieck's theorem for Bessel sequences

We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence $\{ x_j \}_{j\in\mathbb{N}}$ with Bessel bound $1$ in a Hilbert space, we show that there exists functions $\{ f_j \}_{j\in\mathbb{N}}$ belonging to the unit ball of $L^\infty([0,1])$ such that for all $j,k \in \mathbb{N}$ one has $$ \langle x_j,x_k\rangle = \int_0^1 f_j(x)\overline{f_k(x)}\,dx.$$ As an application, we give an affirmative answer to an extension problem of Olevskii: if $E \subset [0,1]$ is a Lebesgue measurable set such that $[0,1]\setminus E$ has positive measure, then every Bessel sequence in $L^2(E)$ with Bessel bound $1$ extends to an orthonormal system in $L^2([0,1])$ that is bounded by the (optimal) constant $\lambda([0,1]\setminus E)^{-1/2}$ on $[0,1]\setminus E$. A formalization of our main result in Lean 4 accompanies the paper.

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arXiv2026

Banach lattices and phase retrieval: A case study for the use of AI in mathematics

The ability of large language models to assist professional mathematicians has been progressing rapidly. Earlier this year, a group of researchers in Banach lattice theory and phase retrieval began incorporating this technology into their research workflows. Facing challenges about the reliability of these models, they also decided to couple the discovery process with Lean verification. Here, we present a case study of how this has led to a more united community and a deeper understanding of our field.

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arXiv2026

Gabor Frames of Totally Positive Functions: A Complete Characterization

We prove that the set of time-frequency shifts $\{e^{2\pi i \beta l t} g(t-\alpha k) : k,l \in \mathbb{Z}\}$ with a continuous, integrable totally positive function $g$ and lattice parameters $\alpha,\beta>0$ generates a frame for $L^2(\mathbb{R})$ if and only if $\alpha\beta<1$. This fully settles the so-called frame set problem for the class of totally positive functions. As a closely related result we prove a sharp Kadets-type theorem for every shift-invariant space generated by a continuous totally positive function. The proofs are based on Fredholm theory and limit-operator theory. A formalization of our main result in Lean 4 is also provided.

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arXiv2026

Cantor measures with odd base do not admit Fourier frames

We prove that the Cantor measure with base $b$ does not admit a Fourier frame whenever $b > 1$ is an odd integer. In particular, this answers a question of Strichartz on the existence of a Fourier frame for the middle third Cantor measure. A formalization of our main result in Lean 4 is also provided.

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arXiv2026

On the existence problem of regular Gabor frames

For every dimension $d > 1$, we establish explicit criteria on lattices $\Lambda \subset \mathbb{R}^{2d}$ with density $D(\Lambda) > 1$ such that no function with a continuous Zak transform generates a Gabor frame along $\Lambda$. In particular, this gives a negative answer to the existence problem of Gabor frames with window functions in the Schwartz space, the Feichtinger algebra, and the Fourier-invariant Wiener space. Our result is based on a characterization of when a collection of quasiperiodic functions admits a common zero, which may be of independent interest. We also include a formalization of our main result in Lean 4.

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arXiv2026

Compactly supported Gabor orthonormal bases

We characterize all lattices $\Lambda \subset \mathbb{R}^2$ and all compactly supported functions $g \in L^2(\mathbb{R})$ for which the Gabor system $\left \{ e^{2\pi i s x} g(x-t) : (t,s) \in \Lambda \right \}$ forms an orthonormal basis for $L^2(\mathbb{R})$. The characterization is given in geometric terms through translation tilings and discreteness properties of lattice projections. In particular, this resolves a conjecture of Han and Wang on the non-existence of Gabor bases along specific irrational lattices. Finally, we construct Gabor bases that cannot be realized by any product set, answering a problem of Iosevich and Mayeli.

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arXiv2026

Cheeger's isoperimetric problem for Gaussian mixtures

In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $\mu(x) = p\gamma(x-a) + (1-p)\gamma(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $\gamma : \mathbb{R}^n \to (0,\infty)$ denotes a Gaussian. In particular, we characterize the Cheeger sets for $\mu$ in terms of specific half-spaces perpendicular to $a-b$, thereby confirming the conjectured solution to the Cheeger problem for Gaussian mixtures. Finally, we study the regime of parameters $p,a,b$ in which $\mu$ admits a unique Cheeger set.

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arXiv2026

Nonlinear determination and phase retrieval under unimodular constraints

We study nonlinear determination problems in Hilbert spaces in which inner products are observed up to prescribed rotations in the complex plane. Given a Hilbert space $H$ and a subset $\Theta$ of the unit circle $\mathbb{T}$, we say that a system $\mathbf{G}\subseteq H$ does $\Theta$-phase retrieval ($\Theta$-PR) if for all $f,h\in H$ the condition that for every $g\in\mathbf{G}$ there exists $\theta_g\in\Theta$ with $\langle f,g\rangle=\theta_g\langle h,g\rangle$ forces $f=\theta h$ for some $\theta\in\Theta$. This framework unifies classical phase retrieval ($\Theta=\mathbb{T}$) and sign retrieval ($\Theta=\{1,-1\}$). For every countable $\Theta$ we give a complete characterization of $\Theta$-PR in terms of covers of $\mathbf{G}$ and geometric relations among vectors in the corresponding orthogonal complements, extending the complement-property characterization of Cahill, Casazza, and Daubechies. For cyclic phase sets we show that $\Theta$-PR is equivalent to the existence of specific second-order recurrence relations. We apply this to obtain a sharp lattice density criterion for $\Theta$-PR of exponential systems. For uncountable $\Theta$ we obtain a topological dichotomy in the Fourier determination setting, showing that $\Theta$-PR is characterized in terms of connectedness of $\Theta$. We further develop a M\"obius-invariant framework, proving that $\Theta$-PR is preserved under circle automorphisms and is governed by projective invariants such as the cross ratio. Finally, in $\mathbb{C}^d$ we determine sharp impossibility thresholds and prove that for countable $\Theta$ the property is generic once one passes the failure regime, yielding the minimal number of vectors required for $\Theta$-PR.

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arXiv2025

From completeness of discrete translates to phaseless sampling of the short-time Fourier transform

We study the uniqueness problem in short-time Fourier transform phase retrieval by exploring a connection to the completeness problem of discrete translates. Specifically, we prove that functions in $L^2(K)$ with $K \subseteq \mathbb{R}^d$ compact, are uniquely determined by phaseless lattice-samples of its short-time Fourier transform with window function $g$, provided that specific density properties of translates of $g$ are met. By proving completeness statements for systems of discrete translates in Banach function spaces on compact sets, we obtain new uniqueness statements for phaseless sampling on lattices beyond the known Gaussian window regime. Our results apply to a large class of window functions, which are relevant in time-frequency analysis and applications.

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arXiv2024

Multi-window STFT phase retrieval: lattice uniqueness

Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function $f$ from its spectrogram, i.e., the magnitudes of its short-time Fourier transform $V_gf$ with window function $g$. While it is known that for appropriate windows, any function $f \in L^2(\mathbb{R})$ can be reconstructed from the full spectrogram $|V_g f(\mathbb{R}^2)|$, in practical scenarios, the reconstruction must be achieved from discrete samples, typically taken on a lattice. It turns out that the sampled problem becomes much more subtle: recent results have demonstrated that uniqueness via lattice-sampling is unachievable, irrespective of the choice of the window function or the lattice density. In the present paper, we initiate the study of multi-window STFT phase retrieval as a way to effectively bypass the discretization barriers encountered in the single-window case. By establishing a link between multi-window Gabor systems, sampling in Fock space, and phase retrieval for finite frames, we derive conditions under which square-integrable functions can be uniquely recovered from spectrogram samples on a lattice. Specifically, we provide conditions on window functions $g_1, \dots, g_4 \in L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ is determined up to a global phase from $$\left(|V_{g_1}f(A\mathbb{Z}^2)|, \, \dots, \, |V_{g_4}f(A\mathbb{Z}^2)| \right)$$ whenever $A \in \mathrm{GL}_2(\mathbb{R})$ satisfies the density condition $|\det A|^{-1} \geq 4$. For real-valued functions, a density of $|\det A|^{-1} \geq 2$ is sufficient. Corresponding results for irregular sampling are also shown.

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arXiv2024

Phase retrieval in Fock space and perturbation of Liouville sets

We study the determination of functions in Fock space from samples of their absolute value, known as the phase retrieval problem in Fock space. An important finding in this research field asserts that phaseless sampling on lattices of arbitrary density renders the problem unsolvable. The present study establishes solvability when using irregular sampling sets of the form $A \cup B \cup C$, where $A, B,$ and $C$ constitute perturbations of a Liouville set, i.e., a set with the property that all functions in Fock space bounded on the set are constant. The sets $A, B,$ and $C$ adhere to specific geometrical conditions of closeness and noncollinearity. We show that these conditions are sufficiently generic so as to allow the perturbations to be chosen also at random. By proving that Liouville sets occupy an intermediate position between sets of stable sampling and sets of uniqueness, we obtain the first construction of uniqueness sets for the phase retrieval problem in Fock space having a finite density. The established results apply to the Gabor phase retrieval problem in subspaces of $L^2(\mathbb{R})$, where we derive additional reductions of the size of uniqueness sets: for the class of real-valued functions, uniqueness is achieved from two perturbed lattices; for the class of even real-valued functions, a single perturbation suffices, resulting in a separated set.

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arXiv2024

Translation-based completeness on compact intervals

Given a compact interval $I \subseteq \mathbb{R}$, and a function $f$ that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot - λ) : λ\in Λ\}$ are complete in $C(I)$ if and only if the series of reciprocals of $Λ$ diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299-308]. An additional characterization is obtained when $Λ$ is an arithmetic progression, and the generator $f$ constitutes a linear combination of translates of a function with sufficiently fast decay.

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arXiv2024

Arithmetic progressions and holomorphic phase retrieval

We study the determination of a holomorphic function from its absolute value. Given a parameter $θ\in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $Λ\subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{ξz}, \, ξ\in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{iθ}(\mathbb{R} + iΛ)\}$ if and only if $Λ$ is not contained in an arithmetic progression $a\mathbb{Z}+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli-type uniqueness problems. For instance, $\mathbb{Z} \times \tilde{\mathbb{Z}}$ is a uniqueness set for the Gabor phase retrieval problem in $L^2(\mathbb{R}_+)$, provided that $\tilde{\mathbb{Z}}$ is a suitable perturbation of the integers.

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arXiv2024

Phaseless sampling on square-root lattices

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions $g \in L^2(\mathbb{R}^d)$ and which sampling sets $Λ\subseteq \mathbb{R}^{2d}$ is every $f \in L^2(\mathbb{R}^d)$ uniquely determined (up to a global phase factor) by phaseless samples of the form $$ |V_gf(Λ)| = \left \{ |V_gf(λ)| : λ\in Λ\right \}, $$ where $V_gf$ denotes the short-time Fourier transform (STFT) of $f$ with respect to $g$. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if $Λ$ is a lattice, i.e $Λ= A\mathbb{Z}^{2d}, A \in \mathrm{GL}(2d,\mathbb{R})$. Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form $$ Λ= A \left ( \sqrt{\mathbb{Z}} \right )^{2d}, \ \sqrt{\mathbb{Z}} = \{ \pm \sqrt{n} : n \in \mathbb{N}_0 \}, $$ guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians.

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arXiv2023

Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality

We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^\infty(φ)$ from spectrogram measurements $|\mathcal{G} f(X)|$ where $\mathcal{G}$ is the Gabor transform and $X \subseteq \mathbb{R}^2$. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, Appl. Comput. Harmon. Anal. 62 (2023), pp. 173-193] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^\infty(φ)$, such as Paley-Wiener spaces.

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arXiv2023

Non-uniqueness theory in sampled STFT phase retrieval

The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important applications, including coherent diffraction imaging and audio processing. It is a classical result that for suitable windows any function can, in principle, be uniquely recovered up to a global phase factor from its spectrogram. However, for most practical applications only discrete samples - typically from a lattice - of the spectrogram are available. This raises the question of whether lattice samples of the spectrogram contain sufficient information for determining a function $f\in L^2(\mathbb{R}^d)$ up to a global phase factor. In the present paper, we answer this question in the negative by providing general non-identifiability results which lead to a non-uniqueness theory for the sampled STFT phase retrieval problem. Precisely, given any dimension $d$, any window function $g$ and any (symplectic or separable) lattice $\mathcal{L} \subseteq \mathbb{R}^d$, we construct pairs of functions $f,h\in L^2(\mathbb{R}^d)$ that do not agree up to a global phase factor, but whose spectrograms agree on $\mathcal{L}$. Our techniques are sufficiently flexible to produce counterexamples to unique recoverability under even more stringent assumptions; for example, if the window function is real-valued, the functions $f,h$ can even be chosen to satisfy $|f|=|h|$. Our results thus reveal the non-existence of a critical sampling density in the absence of phase information, a property which is in stark contrast to uniqueness results in time-frequency analysis.

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arXiv2022

Injectivity of Gabor phase retrieval from lattice measurements

We establish novel uniqueness results for the Gabor phase retrieval problem: if $\mathcal{G} : L^2(\mathbb{R}) \to L^2(\mathbb{R}^2)$ denotes the Gabor transform then every $f \in L^4[-\tfrac{c}{2},\tfrac{c}{2}]$ is determined up to a global phase by the values $|\mathcal{G}f(x,ω)|$ where $(x,ω)$ are points on the lattice $b^{-1}\mathbb{Z} \times (2c)^{-1}\mathbb{Z}$ and $b>0$ is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Gröchenig, Romero and Stöckler, we prove analogous uniqueness results for functions in shift-invariant spaces with Gaussian generator. Generalizations to nonuniform sampling are also presented. Finally, we compare our results to the situation where the considered signals are assumed to be real-valued.

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arXiv2022

On foundational discretization barriers in STFT phase retrieval

We prove that there exists no window function $g \in L^2(\mathbb{R})$ and no lattice $\mathcal{L} \subset \mathbb{R}^2$ such that every $f \in L^2(\mathbb{R})$ is determined up to a global phase by spectrogram samples $|V_gf(\mathcal{L})|$ where $V_gf$ denotes the short-time Fourier transform of $f$ with respect to $g$. Consequently, the forward operator $f \mapsto |V_gf(\mathcal{L})|$ mapping a square-integrable function to its spectrogram samples on a lattice is never injective on the quotient space $L^2(\mathbb{R}) / {\sim}$ with $f \sim h$ identifying two functions which agree up to a multiplicative constant of modulus one. We will further elaborate this result and point out that under mild conditions on the lattice $\mathcal{L}$, functions which produce identical spectrogram samples but do not agree up to a unimodular constant can be chosen to be real-valued. The derived results highlight that in the discretization of the STFT phase retrieval problem from lattice measurements, a prior restriction of the underlying signal space to a proper subspace of $L^2(\mathbb{R})$ is inevitable.

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