19,609 results on '"Classical Analysis and ODEs (math.CA)"'
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2. Linearization and connection coefficients of polynomial sequences: A matrix approach
- Author
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Luis Verde-Star
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Rings and Algebras (math.RA) ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Rings and Algebras ,Geometry and Topology ,15A30, 33C45 - Abstract
For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$, called linearization coefficients, that satisfy $$ p_n(t) p_m(t)=\sum_{k=0}^{n+m} d(n,m,k) p_k(t).$$ For any pair of polynomial sequences $\{u_k(t)\}$ and $\{p_k(t)\}$ we find infinite matrices whose entries are the coefficients $e(n,m,k)$ that satisfy $$p_n(t) p_m(t)=\sum_{k=0}^{n+m} e(n,m,k) u_k(t).$$ Such results are obtained using a matrix approach. We also obtain recurrence relations for the linearization coefficients, apply the general results to general orthogonal polynomial sequences and to particular families of orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families., 14 pages
- Published
- 2023
3. On the nonoscillatory phase function for Legendre's differential equation
- Author
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Bremer, James and Rokhlin, Vladimir
- Subjects
Numerical Analysis (math.NA) ,Classical Analysis and ODEs (math.CA) - Abstract
We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic derivative of this solution, we show that Legendre's differential equation admits a nonoscillatory phase function. Moreover, we derive from our expression an asymptotic expansion useful for evaluating Legendre functions of the first and second kinds of large orders, as well as the derivative of the nonoscillatory phase function. Numerical experiments demonstrating the properties of our asymptotic expansion are presented.
- Published
- 2018
4. Irregularities of distribution for bounded sets and half‐spaces
- Author
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Brandolini, Luca, Colzani, Leonardo, Travaglini, Giancarlo, Brandolini, L, Colzani, L, and Travaglini, G
- Subjects
Mathematics - Classical Analysis and ODEs ,Settore MAT/05 - Analisi Matematica ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,irregularities of distribution, discrepancy, fractal sets ,11K38, 42B10 ,MAT/05 - ANALISI MATEMATICA - Abstract
We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.
- Published
- 2023
5. Well-posedness of the deterministic transport equation with singular velocity field perturbed along fractional Brownian paths
- Author
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Amine, Oussama, Mansouri, Abdol-Reza, and Proske, Frank
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,60H10, 49N60, 91G80 ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this article we prove path-by-path uniqueness in the sense of Davie \cite{Davie07} and Shaposhnikov \cite{Shaposhnikov16} for SDE's driven by a fractional Brownian motion with a Hurst parameter $H\in(0,\frac{1}{2})$, uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.\par Using this result, we construct weak unique regular solutions in $W_{loc}^{k,p}\left([0,1]\times\mathbb{R}^d\right)$, $p>d$ of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.\par The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons \cite{DiPernaLions89}, Ambrosio \cite{Ambrosio04} or Crippa-De Lellis \cite{CrippaDeLellis08}.\par Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from \cite{DMN92} as well as supremum concentration inequalities. \emph{keywords}: Transport equation, Compactness criterion, Singular vector fields, Regularization by noise., Strengthening of the main theorem as well as fixing of typos
- Published
- 2023
6. Univoque bases of real numbers: Simply normal bases, irregular bases and multiple rationals
- Author
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Yu Hu, Yan Huang, and Derong Kong
- Subjects
Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics - Number Theory ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Dynamical Systems (math.DS) ,Number Theory (math.NT) ,Mathematics - Dynamical Systems - Abstract
Given a positive integer $M$ and a real number $x\in(0,1]$, we call $q\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\in\{0,1,\ldots,M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$. Similarly, a base $q\in(1,M+1]$ is called a univoque irregular base of $x$ if there exists a unique sequence $(d_i)\in\{0,1,\ldots, M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$ and the sequence $(d_i)$ has no digit frequency. Let $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ be the sets of univoque simply normal bases and univoque irregular bases of $x$, respectively. In this paper we show that for any $x\in(0,1]$ both $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ have full Hausdorff dimension. Furthermore, given finitely many rationals $x_1, x_2, \ldots, x_n\in(0,1]$ so that each $x_i$ has a finite expansion in base $M+1$, we show that there exists a full Hausdorff dimensional set of $q\in(1,M+1]$ such that each $x_i$ has a unique expansion in base $q$., 25 pages, 2 figures
- Published
- 2023
7. An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments
- Author
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Bremer, James
- Subjects
Numerical Analysis (math.NA) ,Classical Analysis and ODEs (math.CA) - Abstract
We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel's equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions and representing them with piecewise bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions of orders between 0 and $1\sep,000\sep,000\sep,000$ at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has several advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calculating the logarithms of the Bessel functions rather than the Bessel functions themselves, we avoid many issues which arise from numerical overflow and underflow. Third, in the oscillatory regime, our algorithm calculates the values of a nonoscillatory phase function for Bessel's differential equation and its derivative. These quantities are useful for computing the zeros of Bessel functions, as well as for rapidly applying the Fourier-Bessel transform. The results of extensive numerical experiments demonstrating the efficacy of our algorithm are presented. A Fortran package which includes our code for evaluating the Bessel functions as well as our code for all of the numerical experiments described here is publically available.
- Published
- 2017
8. Quantitative Straightening of Distance Spheres
- Author
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David, Guy C., Kaczanowski, McKenna, and Pinkerton, Dallas
- Subjects
Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,28A75 ,Geometry and Topology ,Analysis - Abstract
We study "distance spheres": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is "too dense" and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be "straightened" into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$., Comment: 11 pages
- Published
- 2023
9. ON POINTWISE A.E. CONVERGENCE OF MULTILINEAR OPERATORS
- Author
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LOUKAS GRAFAKOS, DANQING HE, PETR HONZÍK, and BAE JUN PARK
- Subjects
Mathematics::Functional Analysis ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics::Classical Analysis and ODEs ,42B15, 42B25 - Abstract
In this work we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited decay. The a.e. convergence is deduced from the $L^2\times\cdots\times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.
- Published
- 2023
10. On the Minkowski content of self-similar random homogeneous iterated function systems
- Author
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Sascha Troscheit
- Subjects
self-similar sets ,random attractors ,Mathematics - Metric Geometry ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Minkowski content ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Metric Geometry (math.MG) ,Dynamical Systems (math.DS) ,Articles ,Mathematics - Dynamical Systems - Abstract
The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log comensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Z\"{a}hle that the Minkowski content exists in expectation. In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Z\"{a}hle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors., Comment: 17 pages
- Published
- 2023
11. Pathwise regularisation of singular interacting particle systems and their mean field limits
- Author
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Fabian A. Harang and Avi Mayorcas
- Subjects
Statistics and Probability ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Modeling and Simulation ,Probability (math.PR) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics - Probability - Abstract
We investigate the regularizing effect of certain perturbations by noise in singular interacting particle systems under the mean field scaling. In particular, we show that the addition of a suitably irregular path can regularise these dynamics and we recover the McKean--Vlasov limit under very broad assumptions on the interaction kernel; only requiring it to be controlled in a possibly distributional Besov space. In the particle system we include two sources of randomness, a common noise path $Z$ which regularises the dynamics and a family of idiosyncratic noises, which we only assume to converge in mean field scaling to a representative noise in the McKean--Vlasov equation., 39 pages; main update is technical changes to the proofs of the stability estimate in Lemma 4.13 and presentation of Theorem 5.1
- Published
- 2023
12. Joints tightened
- Author
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Yu, Hung-Hsun Hans and Zhao, Yufei
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) - Abstract
In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $\binom{k}{d-1}$ lines and their $d$-wise intersections to form $\binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint., 11 pages
- Published
- 2023
13. Dimension estimates on circular (s,t)-Furstenberg sets
- Author
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Liu, Jiayin
- Subjects
General Mathematics ,Mathematics::Classical Analysis and ODEs ,Mathematics::General Topology ,Metric Geometry (math.MG) ,Hausdorff dimension ,Articles ,Mathematics - Metric Geometry ,Mathematics - Classical Analysis and ODEs ,circular Furstenberg set ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,ulottuvuus ,Furstenberg set - Abstract
In this paper, we show that circular $(s,t)$-Furstenberg sets in $\mathbb R^2$ have Hausdorff dimension at least $$\max\{\frac{t}3+s,(2t+1)s-t\} \text{ for all $0, Comment: 27 pages, 9 figures; incorporated referee's comments, results unchanged
- Published
- 2023
14. On Bibasic Humbert Hypergeometric Function Φ1
- Author
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Aledamat, Ayed and Shehata, Ayman
- Subjects
General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
The main aim of this work is to derive the q-recurrence relations, q-partial derivative relations and summation formula of bibasic Humbert hypergeometric function Φ1 on two independent bases q and q1 of two variables and some developments formulae, believed to be new, by using the conception of q-calculus.
- Published
- 2023
15. A Meyer-Itô formula for stable processes via fractional calculus
- Author
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Alejandro Santoyo Cano and Gerónimo Uribe Bravo
- Subjects
Applied Mathematics ,Probability (math.PR) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,26A33, 60G18, 60G52 ,Analysis - Abstract
The infinitesimal generator of a one-dimensional strictly $α$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $α$ and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $α\in (0,1)$ and the recurrent potential if $α\in (1,2)$. The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $α$. One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for $|X_t - x|^γ$ with $X$ a recurrent strictly stable process of index $α$ and $γ\in (α-1,α)$, generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case., 26 b5 pages
- Published
- 2023
16. Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals
- Author
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Yakubovich, S.
- Subjects
44A15, 41A60, 33C10, 33C15 ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Analysis - Abstract
Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function $K_{i\tau}(x)$. The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [6]) whether these integrals converge for even entire functions of the exponential type in a weak sense.
- Published
- 2023
17. An Optimal Scheduled Learning Rate for a Randomized Kaczmarz Algorithm
- Author
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Marshall, Nicholas F. and Mickelin, Oscar
- Subjects
FOS: Computer and information sciences ,Computer Science - Machine Learning ,Mathematics - Classical Analysis and ODEs ,Probability (math.PR) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Numerical Analysis (math.NA) ,Mathematics - Numerical Analysis ,Mathematics - Probability ,Analysis ,Machine Learning (cs.LG) - Abstract
We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving $A x \approx b + \varepsilon$, where $A x =b$ is a consistent linear system and $\varepsilon$ has independent mean zero random entries. We derive a learning rate schedule which optimizes a bound on the expected error that is sharp in certain cases; in contrast to the exponential convergence of the standard randomized Kaczmarz algorithm, our optimized bound involves the reciprocal of the Lambert-$W$ function of an exponential., 19 pages, 7 figures
- Published
- 2023
18. Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma
- Author
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McDonald, Alex and Taylor, Krystal
- Subjects
28a75, 28a80, 42B ,Mathematics::K-Theory and Homology ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$ . We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.
- Published
- 2023
19. A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces
- Author
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Alvarado, Ryan, Hajłasz, Piotr, and Malý, Lukáš
- Subjects
Mathematics - Functional Analysis ,Mathematics - Classical Analysis and ODEs ,Poincaré inequality ,Sobolev spaces ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Primary 46E36, 30L99, Secondary 31E05, 43A85 ,analysis on metric spaces ,Articles ,uniform convexity ,Functional Analysis (math.FA) - Abstract
We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space \(X\) supports a \(p\)-Poincaré inequality, then the \(N^{1,p}(X)\) Sobolev space is reflexive and separable whenever \(p\in (1,\infty)\). We also prove separability of the space when \(p=1\). Our proof is based on a straightforward construction of an equivalent norm on \(N^{1,p}(X)\), \(p\in [1,\infty)\), that is uniformly convex when \(p\in (1,\infty)\). Finally, we explicitly construct a functional that is pointwise comparable to the minimal \(p\)-weak upper gradient, when \(p\in (1,\infty)\).
- Published
- 2023
20. On a New Measure on the Levi-Civita Field $$ \mathcal{R} $$
- Author
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Borrero, Mateo Restrepo, Srivastava, Vatsal, and Shamseddine, Khodr
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [Shamseddine-Berz-2003], a measure was defined on $\mathcal{R}$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over $\mathcal{R}$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [Shamseddine-Berz-2003]. Then we will introduce the notion of an outer measure on $\mathcal{R}$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on $\mathcal{R}$ that proves to be a better generalization of the Lebesgue measure from $\mathbb{R}$ to $\mathcal{R}$ and that leads to a family of measurable sets in $\mathcal{R}$ that strictly contains the family of measurable sets from [Shamseddine-Berz-2003], and for which most of the classic results for Lebesgue measurable sets in $\mathbb{R}$ hold.
- Published
- 2023
21. The mathematical foundations of the asymptotic iteration method
- Author
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Davide Batic and Marek Nowakowski
- Subjects
High Energy Physics - Theory ,High Energy Physics - Theory (hep-th) ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,General Engineering ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,Mathematical Physics (math-ph) ,General Relativity and Quantum Cosmology ,Mathematical Physics - Abstract
We introduce a new approach to the the asymptotic iteration method (AIM) by means of which we establish the standard AIM connection with the continued fractions technique and we develop a novel termination condition in terms of the approximants. With the help of this alternative termination condition and certain properties of continuous fractions, we derive a closed formula for the asymptotic function $\alpha$ of the AIM technique in terms of an infinite series. Furthermore, we show that such a series converges pointwise to $\alpha$ which, in turn, can be interpreted as a specific term of the minimal solution of a certain recurrence relation. We also investigate some conditions ensuring the existence of a minimal solution and hence, of the function $\alpha$ itself., Comment: 15 pages
- Published
- 2023
22. On stability for generalized linear differential equations and applications to impulsive systems
- Author
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Gallegos, Claudio A. and Robledo, Gonzalo
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,34A06, 34D20 (Primary), 34A30, 34A37 (Secondary) - Abstract
In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterisation of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article., Comment: 25 pages
- Published
- 2023
23. Off-Diagonal Two Weight Bumps for Fractional Sparse Operators
- Author
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Rahm, Rob
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
In this paper, we continue some recent work on two weight boundedness of sparse operators to the "off-diagonal" setting. We use the new "entropy bumps" introduced in by Treil-Volberg ([21]) and improved by Lacey-Spencer ([9]) and the "direct comparison bumps" introduced by Rahm-Spencer ([19]) and improved by Lerner ([10]). Our results are "sharp" in the sense that they are sharp in various particular cases. A feature is that given the current machinery, the proofs are almost trivial., References added; mistakes/typos corrected
- Published
- 2023
24. EXPONENTIAL APPROXIMATION OF FUNCTIONS IN LEBESGUE SPACES WITH MUCKENHOUPT WEIGHT
- Author
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Akgün, Ramazan
- Subjects
Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Analysis - Abstract
Using a transference result, several inequalities of approximation by entire functions of exponential type in $\mathcal{C}(\mathbf{R})$, the class of bounded uniformly continuous functions defined on $\mathbf{R}:=\left( -\infty ,+\infty \right) $, are extended to the Lebesgue spaces $L^{p}\left( \mathbf{\varrho }dx\right) $ $1\leq p, 18 pages, Submitted. arXiv admin note: text overlap with arXiv:2109.02083
- Published
- 2023
25. Surprises in a Classic Boundary-Layer Problem
- Author
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William A. Clark, Mario W. Gomes, Arnaldo Rodriguez-Gonzalez, Leo C. Stein, and Steven H. Strogatz
- Subjects
Computational Mathematics ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Physics::Physics Education ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,34B16, 34E05, 34E15, 37M20, 65L11 ,Theoretical Computer Science - Abstract
We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis., 23+2 pages, 8 figures. Supplementary material available at https://github.com/duetosymmetry/surprises-in-a-classic-BVP . Version 2: Added 12 references and additional discussion. Comments welcome
- Published
- 2023
26. A domain free of the zeros of the partial theta function
- Author
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Kostov, Vladimir Petrov
- Subjects
26A06 ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
We prove that for $q\in (0,1)$, the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed domain $\{ \{ |x|\leq 3\} \cap \{${\rm Re}$x\leq 0\} \cap \{ |${\rm Im}$x|\leq 3/\sqrt{2}\} \} \subset \mathbb{C}$ and no real zeros $\geq -5$., Comment: 2 figures
- Published
- 2023
27. Additive properties of fractal sets on the parabola
- Author
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Orponen, Tuomas
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Furstenberg sets ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Fourier'n sarjat ,additive energies ,Mathematics - Combinatorics ,28A80, 11B30 ,Combinatorics (math.CO) ,Articles ,Fourier transforms ,Frostman measures - Abstract
Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 < s < 1$, then adding $K$ once more makes the sum slightly larger: $$\dim_{\mathrm{H}} (K + K + K) \geq 2s + \epsilon, $$ where $\epsilon = \epsilon(s) > 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 < s < 1$, and $\mu$ is a Borel measure on $\mathbb{P}$ satisfying $\mu(B(x,r)) \leq r^{s}$ for all $x \in \mathbb{P}$ and $r > 0$, then there exists $\epsilon = \epsilon(s) > 0$ such that $$ \|\hat{\mu}\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + \epsilon)} $$ for all sufficiently large $R \geq 1$. The proof is based on a reduction to a $\delta$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-Furstenberg set problem., Comment: 26 pages, 2 figures
- Published
- 2023
28. Estimating the Hardy constant of nonconcave Gini means
- Author
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Páles, Zsolt and Pasteczka, Paweł
- Subjects
Mathematics::Functional Analysis ,Mathematics - Classical Analysis and ODEs ,Mathematics::Complex Variables ,26D15, 26E60, 39B62 ,Applied Mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics::Spectral Theory - Abstract
The extension of the Hardy-Knopp-Carleman inequality to several classes of means was the subject of numerous papers. In the class of Gini means the Hardy property was characterized in 2015 by the second author. The precise value of the associated Hardy constant was only established for concave Gini means by the authors in 2016. The determination Hardy constant for nonconcave Gini means is still an open problem. The main goal of this paper is to establish sharper upper bounds for the Hardy constant in this case. The method is to construct a homogeneous and concave quasideviation mean which majorizes the nonconcave Gini mean and for which the Hardy constant can be computed.
- Published
- 2023
29. Bump conditions and two-weight inequalities for commutators of fractional integrals
- Author
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Wen, Yongming and Wu, Huoxiong
- Subjects
Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,42B20, 42B25, 42B35, 47B47 - Abstract
This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given., Comment: 15 pages
- Published
- 2023
30. On the frame set of the second-order B-spline
- Author
-
A. Ganiou D. Atindehou, Christina Frederick, Yébéni B. Kouagou, and Kasso A. Okoudjou
- Subjects
Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,42C15, 42C40 - Abstract
The frame set of a function $g\in L^2(\mathbb{R})$ is the set of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the collection of time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second order $B-$spline. Our method relies on the compact support of this function to partition a subset of the putative frame set and find an explicit dual window function in each of the partition regions. Numerical evidence indicates the existence of further points belonging to the frame set., This version is a major revision and streamlined the manuscripts
- Published
- 2023
31. Modeling and Analysis of Duhem Hysteresis Operators With Butterfly Loops
- Author
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Reynier Peletier, Marco Augusto Vasquez Beltran, and Bayu Jayawardhana
- Subjects
Mathematics - Classical Analysis and ODEs ,Control and Systems Engineering ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Systems and Control (eess.SY) ,Electrical and Electronic Engineering ,Electrical Engineering and Systems Science - Systems and Control ,Computer Science Applications - Abstract
In this work we study and analyze a class of Duhem hysteresis operators that can exhibit butterfly loops. We study firstly the consistency property of such operator which corresponds to the existence of an attractive periodic solution when the operator is subject to a periodic input signal. Subsequently, we study the two defining functions of the Duhem operator such that the corresponding periodic solutions can admit a butterfly input-output phase plot. We present a number of examples where the Duhem butterfly hysteresis operators are constructed using two zero-level set curves that satisfy some mild conditions.
- Published
- 2023
32. A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials
- Author
-
Bielenova, Kateryna, Nazarenko, Hryhorii, and Vishnyakova, Anna
- Subjects
Mathematics - Complex Variables ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Complex Variables (math.CV) - Abstract
We find the constant $b_{\infty}$ ($b_{\infty} \approx 4.81058280$) such that if a complex polynomial or entire function $f(z) = \sum_{k=0}^ \omega a_k z^k, $ $\omega \in \{2, 3, 4, \ldots \} \cup \{\infty\},$ with nonzero coefficients satisfy the conditions $\left|\frac{a_k^2}{a_{k-1} a_{k+1}}\right| >b_{\infty} $ for all $k =1, 2, \ldots, \omega-1,$ then all the zeros of $f$ are simple. We show that the constant $b_{\infty}$ in the statement above is the smallest possible. We also obtain an analog of Hutchinson's theorem for polynomials or entire functions with real nonzero coefficients.
- Published
- 2023
33. Lattice points in stretched finite type domains
- Author
-
Jingwei Guo and Tao Jiang
- Subjects
Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) - Abstract
We study an optimal stretching problem, which is a variant lattice point problem, for convex domains in $\mathbb{R}^d$ ($d\geq 2$) with smooth boundary of finite type that are symmetric with respect to each coordinate hyperplane/axis. We prove that optimal domains which contain the most positive (or least nonnegative) lattice points are asymptotically balanced.
- Published
- 2023
34. Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure Spaces
- Author
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Emma-Karoliina Kurki, Department of Mathematics and Systems Analysis, Aalto-yliopisto, and Aalto University
- Subjects
Mathematics (miscellaneous) ,Muckenhoupt weights ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,reverse Hölder inequality ,42B35, 42B25, 30L99 ,annular decay ,Doubling metric space ,natural maximal function - Abstract
The natural maximal and minimal functions commute pointwise with the logarithm on $A_\infty$. We use this observation to characterize the spaces $A_1$ and $RH_\infty$ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt $A_p$ and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between $A_p$ and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of $A_1$ and $RH_\infty$, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function., 14 pages
- Published
- 2023
35. A continuum dimensional algebra of nowhere differentiable functions
- Author
-
Schlage-Puchta, Jan-Christoph
- Subjects
Mathematics - Classical Analysis and ODEs ,26A27, 26A16, 26A30 ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
We construct an algebra of dimension $2^{\aleph_0}$ consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.
- Published
- 2023
36. Estimating the Hardy constant of nonconcave homogenus quasideviation means
- Author
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Páles, Zsolt and Pasteczka, Paweł
- Subjects
Mathematics - Classical Analysis and ODEs ,26D15, 26E60, 39B62 ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
In this paper, we consider homogeneous quasideviation means generated by real functions (defined on $(0,\infty)$) which are concave around the point $1$ and possess certain upper estimates near $0$ and $\infty$. It turns out that their concave envelopes can be completely determined. Using this description, we establish sufficient conditions for the Hardy property of the homogeneous quasideviation mean and we also furnish an upper estimates for its Hardy constant.
- Published
- 2023
37. A Sharp Mizohata-Takeuchi Type Estimate for the Cone in $\mathbb R^3$
- Author
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Ortiz, Alex
- Subjects
Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics - Abstract
We prove an analog of the Mizohata-Takeuchi conjecture for the cone in $\mathbb R^3$ and the 1-dimensional weights., 30 pages, 2 figures
- Published
- 2023
38. Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains
- Author
-
Zhu, Chenfeng, Yang, Dachun, and Yuan, Wen
- Subjects
Mathematics - Functional Analysis ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Primary 46E35, Secondary 35A23, 42B25, 26D10 ,Functional Analysis (math.FA) ,Analysis of PDEs (math.AP) - Abstract
Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some mild assumptions, and $\{\rho_\nu\}_{\nu\in(0,\nu_0)}$ with $\nu_0\in(0,\infty)$ a $\nu_0$-radial decreasing approximation of the identity on $\mathbb{R}^n$. In this article, the authors establish two extension theorems, respectively, on the inhomogeneous ball Banach Sobolev space $W^{m,X}(\Omega)$ and the homogeneous ball Banach Sobolev space $\dot{W}^{m,X}(\Omega)$ for any $m\in\mathbb{N}$. On the other hand, the authors prove that, for any $f\in\dot{W}^{1,X}(\Omega)$, $$ \lim_{\nu\to0^+} \left\|\left[\int_\Omega\frac{|f(\cdot)-f(y)|^p}{ |\cdot-y|^p}\rho_\nu(|\cdot-y|)\,dy \right]^\frac{1}{p}\right\|_{X(\Omega)}^p =\frac{2\pi^{\frac{n-1}{2}}\Gamma (\frac{p+1}{2})}{\Gamma(\frac{p+n}{2})} \left\|\,\left|\nabla f\right|\,\right\|_{X(\Omega)}^p, $$ where $\Gamma$ is the Gamma function and $p\in[1,\infty)$ is related to $X(\mathbb{R}^n)$. Using this asymptotics, the authors further establish a characterization of $W^{1,X}(\Omega)$ in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation, two extension theorems on weighted Sobolev spaces, and some recently found profound properties of $W^{1,X}(\mathbb{R}^n)$ to overcome those difficulties caused by that the norm of $X(\mathbb{R}^n)$ has no explicit expression and that $X(\mathbb{R}^n)$ might be neither the reflection invariance nor the translation invariance. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, all of which are new., Comment: arXiv admin note: substantial text overlap with arXiv:2307.10528, arXiv:2304.00949
- Published
- 2023
39. An upper bound for the Nevanlinna matrix of an indeterminate moment sequence
- Author
-
Pruckner, Raphael, Reiffenstein, Jakob, and Woracek, Harald
- Subjects
Mathematics - Spectral Theory ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Spectral Theory (math.SP) ,44A60, 47B36, 34L20, 30D15, 37J99 - Abstract
The solutions of an indeterminate Hamburger moment problem can be parameterised using the Nevanlinna matrix of the problem. The entries of this matrix are entire functions of minimal exponential type, and any growth less than that can occur. An indeterminate moment problem can be considered as a canonical system in limit circle case by rewriting the three-term recurrence of the problem to a first order vector-valued recurrence. We give a bound for the growth of the Nevanlinna matrix in terms of the parameters of this canonical system. In most situations this bound can be evaluated explicitly. It is sharp in the sense that for well-behaved parameters it coincides with the actual growth of the Nevanlinna matrix up to multiplicative constants.
- Published
- 2023
40. Painlev\'e-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions
- Author
-
Barhoumi, Ahmad, Lisovyy, Oleg, Miller, Peter D., and Prokhorov, Andrei
- Subjects
Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics - Classical Analysis and ODEs ,Mathematics - Complex Variables ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Complex Variables (math.CV) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Primary 34M55, Secondary 34E05, 34M50, 34M56, 33E17 ,Mathematical Physics - Abstract
The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by\[ \dfrac{\mathrm{d}^2u}{\mathrm{d}x^2}=\dfrac{1}{u}\left( \dfrac{\mathrm{d}u}{\mathrm{d}x} \right)^2-\dfrac{1}{x} \dfrac{\mathrm{d}u}{\mathrm{d}x}+\dfrac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \quad \alpha,\beta \in \mathbb{C}.\] Starting from a generic initial solution $u_0(x)$ corresponding to parameters $\alpha, \beta$, denoted as the triple $(u_0(x),\alpha,\beta)$, we apply an explicit B\"acklund transformation to generate a family of solutions $(u_n(x),\alpha+4n,\beta+4n)$ indexed by $n\in \mathbb{N}$. We study the large $n$ behavior of the solutions $(u_n(x),\alpha+4n,\beta+4n)$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate Painlev\'e-III equation, known as Painlev\'e-III($D_8$), \[\dfrac{\mathrm{d}^2U}{\mathrm{d}z^2}=\dfrac{1}{U}\left(\dfrac{\mathrm{d}U}{\mathrm{d}z}\right)^2-\dfrac{1}{z} \dfrac{\mathrm{d}U}{\mathrm{d}z}+\dfrac{4U^2 + 4}{z}.\] A notable application of our result is to rational solutions of Painlev\'e-III($D_6$), which are constructed using the seed solution $(1,4m,-4m)$ where $m\in \mathbb{C} \setminus (\mathbb{Z} + \frac{1}{2})$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well-defined, and by its monodromy data in general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlev\'e-III, both $D_6$ and $D_8$ at $z = 0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z = 0$., Comment: 69 pages, 12 figures
- Published
- 2023
41. Approximation of functions on a compact set by solutions of elliptic equations. Quantitative results
- Author
-
Rozenblum, Grigori and Shirokov, Nikolai
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,41A30 (primary) 35J15, 41A25 (secondary) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
We establish that a generalized H\"{o}lder continuous function on an $(m-2)$-Ahlfors regular compact set in $\mathbb{R}^m$ can be approximated by solutions of an elliptic equation, with the rate of approximation determined by the continuity modulus of the function.
- Published
- 2023
42. Weighted estimates for Hardy-Littlewood maximal functions on Harmonic $NA$ groups
- Author
-
Ganguly, Pritam, Rana, Tapendu, and Sarkar, Jayanta
- Subjects
Mathematics - Functional Analysis ,Primary: 43A80, Secondary: 43A90, 43A15, 42B25 ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Functional Analysis (math.FA) - Abstract
Our aim in this article is to study the weighted boundedness of the centered Hardy-Littlewood maximal operator in Harmonic $NA$ groups. Following Ombrosi et al. \cite{ORR}, we define a suitable notion of $A_p$ weights, and for such weights, we prove the weighted $L^p$-boundedness of the maximal operator. Furthermore, as an endpoint case, we prove a variant of the Fefferman-Stein inequality, from which vector-valued maximal inequality has been established. We also provide various examples of weights to substantiate many aspects of our results. In particular, we have shown certain spherical functions of the Harmonic $NA$ group constitute examples of $A_p$ weights. The purely exponential volume growth property of the Harmonic $NA$ group has played a crucial role in our proofs., 32 pages
- Published
- 2023
43. On some operator-valued Fourier pseudo-multipliers associated to Grushin operators
- Author
-
Bagchi, Sayan, Basak, Riju, Garg, Rahul, and Ghosh, Abhishek
- Subjects
Mathematics - Functional Analysis ,Mathematics - Analysis of PDEs ,58J40, 43A85, 42B25 ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Analysis of PDEs (math.AP) ,Functional Analysis (math.FA) - Abstract
This is a continuation of our work [BBGG23, BBGG22] where we have initiated the study of sparse domination and quantitative weighted estimates for Grushin pseudo-multipliers. In this article, we further extend this analysis to study analogous estimates for a family of operator-valued Fourier pseudo-multipliers associated to Grushin operators $G = - \Delta_{x^{\prime}} - |x^{\prime}|^2 \Delta_{x^{\prime \prime}}$ on $\mathbb{R}^{n_1+n_2}.$, Comment: This work consists of results extracted from arXiv:2112.06634v1. arXiv admin note: substantial text overlap with arXiv:2112.06634
- Published
- 2023
44. Estimating the $k$th coefficient of $(f(z))^n$ when $k$ is not too large
- Author
-
De Angelis, Valerio
- Subjects
Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,41A60 - Abstract
We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{\delta }$, where $0, Comment: 18 pages
- Published
- 2023
45. Stability of the Faber-Krahn inequality for the Short-time Fourier Transform
- Author
-
Gómez, Jaime, Guerra, André, Ramos, João P. G., and Tilli, Paolo
- Subjects
Mathematics - Functional Analysis ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Analysis of PDEs (math.AP) ,Functional Analysis (math.FA) - Abstract
We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $\delta(f;\Omega)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$ fails to be optimally concentrated on an arbitrary set $\Omega\subset \mathbb R^2$ of positive, finite measure. We then show that an optimal power of the deficit $\delta(f;\Omega)$ controls both the $L^2$-distance of $f$ to an appropriate class of Gaussians and the distance of $\Omega$ to a ball, through the Fraenkel asymmetry of $\Omega$. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context., Comment: 46 pages
- Published
- 2023
46. Pointwise convergence to initial data for some evolution equations on symmetric spaces
- Author
-
Bruno, Tommaso and Papageorgiou, Effie
- Subjects
Mathematics - Functional Analysis ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,22E30, 35B40, 26A33, 58J47, 35C15 ,Analysis of PDEs (math.AP) ,Functional Analysis (math.FA) - Abstract
Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the fractional heat equation, and the Caffarelli--Silvestre extension problem associated with $\mathscr{L}$, and in each of these cases we characterize the weights $v$ on $\mathbb{X}$ for which the solution converges pointwise a.e. to the initial data when the latter is in $L^{p}(v)$, $1\leq p < \infty$. As a tool, we also establish vector-valued weak type $(1,1)$ and $L^{p}$ estimates ($1, Comment: 25 pages
- Published
- 2023
47. Recurrence coefficients for orthogonal polynomials with a logarithmic weight function
- Author
-
Deift, Percy and Piorkowski, Mateusz
- Subjects
Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,42C05, 34M50, 45E05, 45M05 - Abstract
We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \big(\frac{2}{1-x}\big) dx$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by A. Magnus and extends earlier results by T. O. Conway and one of the authors. The proof relies on the Riemann-Hilbert method. The main difficulty in applying the method to the problem at hand is the lack of an appropriate local parametrix near the logarithmic singularity at $x = +1$.
- Published
- 2023
48. Infinite elliptic hypergeometric series: convergence and difference equations
- Author
-
Krotkov, D. I. and Spiridonov, V. P.
- Subjects
Mathematics - Number Theory ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) - Abstract
We derive finite difference equations of infinite order for theta hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion on the convergence of $q$-hypergeometric series for $|q|=1, \, q^n\neq 1$, to the elliptic level and prove convergence of the infinite ${}_{r+1}V_r$ very-well poised elliptic hypergeometric series for restricted values of $q$., 24 pp
- Published
- 2023
49. Algebraic Modification of the Method of Undetermined Coefficients For Solving Nonhomogeneous Linear Difference Equations
- Author
-
Lomonosov, Timofey
- Subjects
Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,39A06 - Abstract
In this paper, an algebraic modification of the method of undetermined coefficients for solving nonhomogeneous linear stationary difference equations for quasipolynomial right-hand sides is proposed. Although the classical method of undetermined coefficients is well-known in both differential equations and difference equations case, its application in the difference equations case is severely limited. For example, it is hard to apply for rather complex expressions that can arise in case of complex quasipolynomials and resonance. The novelty of the research is the proposition of an algebraic modification to the method. That modification eliminates major drawbacks of the primary method and also allows to modify the superposition principle to apply the method to the entire difference equation at once without dividing the problem into several less complicated ones. The superposition principle in matrix form is formulated.
- Published
- 2023
50. Existence of optimal flat ribbons
- Author
-
Blatt, Simon and Raffaelli, Matteo
- Subjects
Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,49Q10, 53A05 (Primary) 49J45, 74B20, 74K20 (Secondary) - Abstract
We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in $\mathbb{R}^{3}$ can be extended to an infinitely narrow flat ribbon having minimal bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish., 11 pages, no figures
- Published
- 2023
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