Classical Analysis and ODEs

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New submissions for Mon, 13 May 24

[1]  arXiv:2405.06273 [pdf, ps, other]

Title: Comparison criteria for first order polynomial differential equations

Authors: G. A. Grigorian

Subjects: Classical Analysis and ODEs (math.CA)

In this paper we use the comparison method for investigation of first order polynomial differential equations. We prove two comparison criteria for these equations. The proved criteria we use to obtain some global solvability criteria for first order polynomial differential equations. On the basis of these criteria we prove some criteria for existence of a closed solution (of closed solutions) of for first order polynomial differential equations. The results obtained we compare with some known results.

[2]  arXiv:2405.06349 [pdf, other]

Title: On certain Gram matrices and their associated series

Authors: Werner Ehm

Subjects: Classical Analysis and ODEs (math.CA); Number Theory (math.NT)

We derive formulae for Gram matrices arising in the Nyman--Beurling reformulation of the Riemann hypothesis. The development naturally leads upon series of the form $S(x) = \sum_{n\ge 1} R(nx)$ and their reciprocity relations. We give integral representations of these series; and we present decompositions of the quadratic forms associated with the Gram matrices along with a discussion of the components' properties.

[3]  arXiv:2405.06423 [pdf, ps, other]

Title: Carleson Operators on Doubling Metric Measure Spaces

Authors: Lars Becker, Floris van Doorn, Asgar Jamneshan, Rajula Srivastava, Christoph Thiele

Comments: 130 pages

Subjects: Classical Analysis and ODEs (math.CA)

We prove a new generalization of a theorem of Carleson, namely bounds for a generalized Carleson operator on doubling metric measure spaces. Additionally, we explicitly reduce Carleson's classical result on pointwise convergence of Fourier series to this new theorem. Both proofs are presented in great detail, suitable as a blueprint for computer verification using the current capabilities of the software package Lean. Note that even Carleson's classical result has not yet been computer-verified.

Cross-lists for Mon, 13 May 24

[4]  arXiv:2405.06314 (cross-list from math.GT) [pdf, ps, other]

Title: Applications of the Painlevé-Kuratowski convergence: Lipschitz functions with converging Clarke subdifferentials and convergence of sets defined by converging equations

Authors: Daniel Fatuła

Subjects: Geometric Topology (math.GT); Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG)

In this note we investigate two kinds of applications of the Painlev\'e-Kuratowski convergence of closed sets in analysis that are motivated also by questions from singularity theory. First, we generalise to Lipschitz functions the classical theorem stating that given a sequence of smooth functions with locally uniformly convergent derivatives, we obtain the local uniform convergence of the functions themselves (provided they were convergent at one point). Next we turn to the study of the behaviour of the fibres of a given function. We prove some general real counterparts of the Hurwitz theorem from complex analysis stating that the local uniform convergence of holomorphic functions implies the convergence of their sets of zeros. From the point of view of singularity theory our two theorems concern the convergence of the sets when their descriptions are convergent. They are also of interest in approximation theory.

[5]  arXiv:2405.06466 (cross-list from math.DS) [pdf, other]

Title: Typical dimension and absolute continuity for classes of dynamically defined measures, Part II : exposition and extensions

Authors: Balázs Bárány, Károly Simon, Boris Solomyak, Adam Śpiewak

Subjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)

This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of [1] and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS's with place-dependent probabilities and natural (equilibrium) measures for smooth IFS's. One of the goals of this paper is to present an exposition of [1] in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by K\'aroly Simon at the Summer School "Dynamics and Fractals" in 2023 at the Banach Center, Warsaw. The main new feature, compared to [1], is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures.
[1] B. B\'ar\'any, K. Simon, B. Solomyak and A. \'Spiewak: Typical absolute continuity for classes of dynamically defined measures. Advances in Mathematics, Volume 399, 2022, 108258, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2022.108258.

[6]  arXiv:2405.06503 (cross-list from math.OC) [pdf, other]

Title: Optimal transport of measures via autonomous vector fields

Authors: Nicola De Nitti, Xavier Fernández-Real

Subjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-$1$ map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems.

Replacements for Mon, 13 May 24

[7]  arXiv:2405.01133 (replaced) [src]

Title: A missing theorem on dual spaces

Authors: David P. Blecher

Comments: Error in early lemma (Proposition 2.3) endangers Theorem 4.1. Result is valid assuming the conclusions of Proposition 2.3 for the map q in Theorem 4.1

Subjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA); Operator Algebras (math.OA)

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