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Classical Analysis and ODEs

New submissions

[ total of 7 entries: 1-7 ]
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New submissions for Fri, 12 Apr 24

[1]  arXiv:2404.07375 [pdf, ps, other]
Title: Subcritical Fourier uncertainty principles
Subjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)

It is well known that if a function $f$ satisfies $$ \norm{f(x) e^{\pi \alpha |x|^2}}_p + \norm{\widehat{f}(\xi) e^{\pi \alpha |\xi|^2}}_q<\infty \eqno{(*)}$$ with $\alpha=1$ and $1\le p,q<\infty$, then $f\equiv 0.$
We prove that if $f$ satisfies $(*)$ with some $0<\alpha<1$ and $1\le p,q\leq \infty$, then $$ |f(y)|\le C
(1+|y|)^{\frac{d}{p}}
e^{- \pi \alpha |y|^2}, \quad y\in \mathbb{R}^d, $$ with $ C=C(\alpha,d,p,q)$ and this bound is sharp for $p\neq 1$. We also study a related uncertainty principle for functions satisfying $\;\;\displaystyle\norm{f(x)|x|^m}_p+ \norm{\widehat{f}(\xi)|\xi|^n}_q <\infty.$

[2]  arXiv:2404.07566 [pdf, ps, other]
Title: Nevai's condition for measures with unbounded supports
Comments: 19 pages
Subjects: Classical Analysis and ODEs (math.CA); Probability (math.PR); Spectral Theory (math.SP)

We study Nevai's condition from the theory of orthogonal polynomials on the real line. We prove that a large class of measures with unbounded Jacobi parameters satisfies Nevai's condition locally uniformly on the support of the measure away from a finite explicit set. This allows us to give applications to relative uniform and weak asymptotics of Christoffel-Darboux kernel on the diagonal and to limit theorems of unconventionally normalized global linear statistics of orthogonal polynomial ensembles.

[3]  arXiv:2404.07971 [pdf, ps, other]
Title: The Newman algorithm for constructing polynomials with restricted coefficients and many real roots
Comments: 19 pages
Subjects: Classical Analysis and ODEs (math.CA)

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with at least $c\sqrt{n}$ distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood ($E_k=\{-1,1\}$) and Newman ($E_k=\{0,(-1)^k\}$) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence $E_k$ is periodic.

Cross-lists for Fri, 12 Apr 24

[4]  arXiv:2404.07832 (cross-list from math.NT) [pdf, other]
Title: Zeros of $L$-Functions in Low-Lying Intervals and de Branges spaces
Comments: 26 pages, 1 figure
Subjects: Number Theory (math.NT); Classical Analysis and ODEs (math.CA)

We consider a variant of a problem first introduced by Hughes and Rudnick (2003) and generalized by Bernard (2015) concerning conditional bounds for small first zeros in a family of $L$-functions. Here we seek to estimate the size of the smallest intervals centered at a low-lying height for which we can guarantee the existence of a zero in a family of $L$-functions. This leads us to consider an extremal problem in analysis which we address by applying the framework of de Branges spaces, introduced in this context by Carneiro, Chirre, and Milinovich (2022).

Replacements for Fri, 12 Apr 24

[5]  arXiv:2304.08358 (replaced) [pdf, ps, other]
Title: Integral representation of functions on the circle
Authors: Giuliano Basso
Comments: To appear in Real Analysis Exchange
Subjects: Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG)
[6]  arXiv:2203.16523 (replaced) [pdf, other]
Title: Shifted Witten classes and topological recursion
Comments: v1: 33 pages; v2: 36 pages, incorporated referee comments, final version
Journal-ref: Trans. Amer. Math. Soc. 377 (2024), 1069-1110
Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
[7]  arXiv:2301.01992 (replaced) [pdf, other]
Title: Monotonicity of the period and positive periodic solutions of a quasilinear equation
Subjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
[ total of 7 entries: 1-7 ]
[ showing up to 2000 entries per page: fewer | more ]

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