arXiv subjects

Mitchell A. Taylor

Publications and source records attributed to Mitchell A. Taylor.

34 records · Page 2Linked to original sources

arXiv2025

Cheeger's Constant for the Gabor Transform and Ripples

We discover a new instability mechanism for short-time Fourier transform phase retrieval which yields that for any reasonable window function $ϕ$ in any dimension $d$, the local stability constant $c(f)$ defined via \begin{equation*} \inf_{|λ|=1}\|f- λg\|_{M^p(\mathbb{R}^{d})}\leq c(f)\| |V_ϕf|-|V_ϕ g|\|_\mathcal{D}, \hspace{5mm} \forall g\in M^p(\mathbb{R}^d), \end{equation*} is infinite on a dense set of vectors for all weighted fractional Sobolev norms $\mathcal{D}$, up to the sharp maximal regularity level ensuring that the problem is well-defined. This, in particular, answers an open problem of Rathmair, who asked whether exponential concentration of the Gabor transform on $\mathbb{R}^2$ guaranteed a finite local stability constant. For the specific case of Gabor phase retrieval, we further show that there is a complementary dense set where the local stability constant on $\mathbb{R}^{2d}$ is finite. Our results extend and complement a series of fundamental stability theorems for Gabor phase retrieval which have been proven over the last ten years. Of particular note is the work of Grohs and Rathmair, who showed that for sufficiently strong weighted Sobolev norms $\mathcal{D}$ on $\mathbb{R}^{2d}$, the local stability constant for Gabor phase retrieval is bounded by the inverse of the Cheeger constant of the flat metric conformally multiplied by $|V_ϕf|$. As a consequence of our analysis, we determine two dense families of functions, one of which has associated Cheeger constant zero and the other strictly positive. We also revisit the stability problem for STFT phase retrieval on bounded subsets of the time-frequency plane, for more general windows, and for restricted signal classes, extending and simplifying many influential results in the literature.

math.CA↗

arXiv2025

The Cahill-Casazza-Daubechies problem on Hölder stable phase retrieval

Phase retrieval using a frame for a finite-dimensional Hilbert space is known to always be Lipschitz stable. However, phase retrieval using a frame or a continuous frame for an infinite-dimensional Hilbert space is always unstable. In order to bridge the gap between the finite and infinite dimensional phenomena, Cahill-Casazza-Daubechies (Trans.Amer.Math.Soc. 2016) gave a construction of a family of nonlinear subsets of an infinite-dimensional Hilbert space where phase retrieval could be performed with a Hölder stability estimate. They then posed the question of whether these subsets satisfied Lipschitz stable phase retrieval. We solve this problem both by giving examples which fail Lipschitz stability and by giving examples which satisfy Lipschitz stability.

math.FA↗

arXiv2025

Sharp bounds on the failure of the hot spots conjecture

The hot spots ratio of a domain $Ω\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $Ω\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$.

math.SP↗

arXiv2025

On sharp stable recovery from clipped and folded measurements

We investigate the stability of vector recovery from random linear measurements which have been either clipped or folded. This is motivated by applications where measurement devices detect inputs outside of their effective range. As examples of our main results, we prove sharp lower bounds on the recovery constant for both the declipping and unfolding problems whenever samples are taken according to a uniform distribution on the sphere. Moreover, we show such estimates under (almost) the best possible conditions on both the number of samples and the distribution of the data. We then prove that all of the above results have suitable (effectively) sparse counterparts. In the special case that one restricts the stability analysis to vectors which belong to the unit sphere of $\mathbb{R}^n$, we show that the problem of declipping directly extends the one-bit compressed sensing results of Oymak-Recht and Plan-Vershynin.

cs.IT↗

arXiv2025

On the optimal Sobolev threshold for evolution equations with rough nonlinearities

In this article we are concerned with evolution equations of the form \begin{equation*} \partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u}) \end{equation*} where $A(D)$ is a Fourier multiplier of either dispersive or parabolic type and the nonlinear term $F$ is of limited regularity. Our objective is to develop a robust set of principles which can be used in many cases to predict the \emph{highest} Sobolev exponent $s=s(q,d)$ for which the above evolution is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ (necessarily restricting to $q=2$ for dispersive problems). We will confirm the validity of these principles for two of the most important model problems; namely, the nonlinear Schrödinger and heat equations. More precisely, we will prove that the nonlinear heat equation \begin{equation*} \partial_tu-Δu=\pm |u|^{p-1}u, \hspace{5mm} p>1, \end{equation*} is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ when $\max\{0,s_c\} 1$ was a rather longstanding open problem in the literature. As an immediate corollary of the fact that our ill-posedness threshold is dimension independent, we may conclude by taking $d\gg p$ that there are nonlinear Schrödinger equations which are ill-posed in \emph{every} Sobolev space $H_x^s(\mathbb{R}^d)$.

math.AP↗

arXiv2025

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

We study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS) $$iu_t+u_{xx}=i|u|^{2σ}u_x,$$ for small powers $σ$. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in $H^s$ when $s\in [1,4σ)$ and $σ\in (\frac{\sqrt{3}}{2},1)$. Our result when $s=1$ is particularly relevant because it corresponds to the regularity of the energy for this problem. To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and lacks the decay necessary for global smoothing type estimates. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools developed in this article are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity well-posedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold $s<4σ$ is twice as high as one might naïvely expect, given that the function $z\mapsto |z|^{2σ}$ is only $C^{1,2σ-1}$ Hölder continuous. Moreover, although we cannot prove $H^1$ well-posedness when $σ\leq \frac{\sqrt{3}}{2}$, we are able to establish $H^s$ well-posedness in the high regularity regime $s\in (2-σ,4σ)$ for the full range of $σ\in (\frac{1}{2},1)$. This considerably improves the known local results, which had only been established in either $H^2$ or in weighted Sobolev spaces.

math.AP↗

arXiv2025

Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations

We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.

math.AP↗

arXiv2024

Low regularity solutions for the general quasilinear ultrahyperbolic Schrödinger equation

We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schrödinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive improvement over the landmark results of Kenig, Ponce, Rolvung and Vega, as it weakens the regularity and decay assumptions to the same scale of spaces considered by Marzuola, Metcalfe, and Tataru, but removes the uniform ellipticity assumption on the metric from their result. Our method has the additional benefit of being relatively simple but also very robust. In particular, it only relies on the use of pseudodifferential calculus for classical symbols.

math.AP↗

arXiv2024

Sharp well-posedness for the free boundary MHD equations

In this article, we provide a definitive well-posedness theory for the free boundary problem in incompressible magnetohyrodynamics. Despite the clear physical interest in this system and the remarkable progress in the study of the free boundary Euler equations in recent decades, the low regularity well-posedness of the free boundary MHD equations has remained completely open. This is due, in large part, to the highly nonlinear wave-type coupling between the velocity, magnetic field and free boundary, which has forced previous works to impose restrictive geometric constraints on the data. To address this problem, we introduce a novel Eulerian approach and an entirely new functional setting, which better captures the wave equation structure of the MHD equations and permits a complete Hadamard well-posedness theory in low-regularity Sobolev spaces. In particular, we give the first proofs of existence, uniqueness and continuous dependence on the data at the sharp $s>\frac{d}{2}+1$ Sobolev regularity, in addition to a blowup criterion for smooth solutions at the same low regularity scale. Moreover, we provide a completely new method for constructing smooth solutions which, to our knowledge, gives the first proof of existence (at any regularity) in our new functional setting. All of our results hold in arbitrary dimensions and in general, not necessarily simply connected, domains. By taking the magnetic field to be zero, they also recover the corresponding sharp well-posedness theorems for the free boundary Euler equations. The methodology and tools that we employ here can likely be fruitfully implemented in other free boundary models.

math.AP↗

arXiv2023

Examples of Hölder-stable Phase Retrieval

Examples are constructed of infinite-dimensional subspaces $V\subset L^2(μ)$ with the property that for any $f,g\in V$, if $|f|$ is approximately equal to $|g|$ with respect to the $L^2$ norm, then there exists a unimodular scalar $z$ such that $f$ is approximately equal to $zg$.

math.CA↗

arXiv2021

No pure capillary solitary waves exist in 2D finite depth

We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth.

math.AP↗

arXiv2021

Free Banach lattices under convexity conditions

We prove the existence of free objects in certain subcategories of Banach lattices, including $p$-convex Banach lattices, Banach lattices with upper $p$-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (\emph{Israel J.\ Math.}~2019). In the $p$-con\-vex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear $p$-summing maps.

math.FA↗

arXiv2020

A Schauder basis for $L_2$ consisting of non-negative functions

We prove that $L_2(\mathbb{R})$ contains a Schauder basis of non-negative functions. Similarly, $L_p(\mathbb{R})$ contains a Schauder basic sequence of non-negative functions such that $L_p(\mathbb{R})$ embeds into the closed span of the sequence. We prove as well that if $X$ is a separable Banach space with the bounded approximation property, then any set in $X$ with dense span contains a quasi-basis (Schauder frame) for $X$. Furthermore, if $X$ is a separable Banach lattice with a bibasis then any set in $X$ with dense span contains a u-frame.

math.FA↗

arXiv2019

Unbounded topologies and uo-convergence in locally solid vector lattices

Suppose $X$ is a vector lattice and there is a notion of convergence $x_α \rightarrow x$ in $X$. Then we can speak of an "unbounded" version of this convergence by saying that $(x_α)$ unbounded converges to $x\in X$ if $\lvert x_α-x\rvert \wedge u\rightarrow 0$ for every $u \in X_+$. In the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo-convergence and those convergences deriving from locally solid topologies.

math.FA↗

arXiv2017

Metrizability of minimal and unbounded topologies

In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and $L_{\infty}$. Minimal topologies connect with the recent, and actively studied, subject of "unbounded convergences". In fact, a Hausdorff locally solid topology $τ$ on a vector lattice $X$ is minimal iff it is Lebesgue and the $τ$ and unbounded $τ$-topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, $uτ$, associated to $τ$ on $X$. Regarding metrizability, we prove that if $τ$ is a locally solid metrizable topology then $uτ$ is metrizable iff there is a countable set $A$ with $\overline{I(A)}^τ=X$. We prove that a minimal topology is metrizable iff $X$ has the countable sup property and a countable order basis. In line with the idea that uo-convergence generalizes convergence almost everywhere, we prove relations between minimal topologies and uo-convergence that generalize classical relations between convergence almost everywhere and convergence in measure.

math.FA↗

arXiv2017

Completeness of Unbounded Convergences

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence.

math.FA↗