In this paper we cover a few topics on how to treat inverse problems. There are two different flows of ideas. One approach is based on Morse Lemma. The other is based on analyticity which proves that the number of solutions to the inverse problems is generically isolated for some particular class of dynamical systems.
{"title":"A note on identifiability for inverse problem based on observations","authors":"Marian Petrica, Ionel Popescu","doi":"arxiv-2408.14616","DOIUrl":"https://doi.org/arxiv-2408.14616","url":null,"abstract":"In this paper we cover a few topics on how to treat inverse problems. There\u0000are two different flows of ideas. One approach is based on Morse Lemma. The\u0000other is based on analyticity which proves that the number of solutions to the\u0000inverse problems is generically isolated for some particular class of dynamical\u0000systems.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov
Given a sequence of polynomials $Q_n$ of degree $n$, we consider the triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the only assumption that the sequence ${Q_n}$ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a unit measure $mu_0$ with compact support in the complex plane, we show that as $n, k rightarrow infty$ such that $k / n rightarrow t in(0,1)$, the Cauchy transform of the zero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood of infinity to the Cauchy transform of a measure $mu_t$. The family of measures $mu_t $, $t in(0,1)$, whose dependence on the parameter $t$ can be interpreted as a flow of the zeros under iterated differentiation, has several interesting connections with the inviscid Burgers equation, the fractional free convolution of $mu_0$, or a nonlocal diffusion equation governing the density of $mu_t$ on $mathbb R$. The main goal of this paper is to provide a streamlined and elementary proof of all these facts.
{"title":"Flow of the zeros of polynomials under iterated differentiation","authors":"Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov","doi":"arxiv-2408.13851","DOIUrl":"https://doi.org/arxiv-2408.13851","url":null,"abstract":"Given a sequence of polynomials $Q_n$ of degree $n$, we consider the\u0000triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the only\u0000assumption that the sequence ${Q_n}$ has a weak* limiting zero distribution\u0000(an empirical distribution of zeros) represented by a unit measure $mu_0$ with\u0000compact support in the complex plane, we show that as $n, k rightarrow infty$\u0000such that $k / n rightarrow t in(0,1)$, the Cauchy transform of the\u0000zero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood\u0000of infinity to the Cauchy transform of a measure $mu_t$. The family of measures $mu_t $, $t in(0,1)$, whose dependence on the\u0000parameter $t$ can be interpreted as a flow of the zeros under iterated\u0000differentiation, has several interesting connections with the inviscid Burgers\u0000equation, the fractional free convolution of $mu_0$, or a nonlocal diffusion\u0000equation governing the density of $mu_t$ on $mathbb R$. The main goal of this paper is to provide a streamlined and elementary proof\u0000of all these facts.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the setting of Carleson's convergence problem for the fractional Schr"odinger equation $i, partial_t u + (-Delta)^{a/2}u=0$ with $a > 1$ in $mathbb R^d$, which has Fourier symbol $P(xi) = |xi|^a$, it is known that the Sobolev exponent $d/(2(d+1))$ is sufficient, but it is not known whether this condition is necessary. In this article, we show that in the periodic problem in $mathbb T^d$ the exponent $d/(2(d+1))$ is necessary for all non-singular polynomial symbols $P$ regardless of the degree of $P$. Among the differential operators covered, we highlight the natural powers of the Laplacian $Delta^k$ for $k in mathbb N$.
{"title":"Counterexamples to the convergence problem for periodic dispersive equations with a polynomial symbol","authors":"Daniel Eceizabarrena, Xueying Yu","doi":"arxiv-2408.13935","DOIUrl":"https://doi.org/arxiv-2408.13935","url":null,"abstract":"In the setting of Carleson's convergence problem for the fractional\u0000Schr\"odinger equation $i, partial_t u + (-Delta)^{a/2}u=0$ with $a > 1$ in\u0000$mathbb R^d$, which has Fourier symbol $P(xi) = |xi|^a$, it is known that\u0000the Sobolev exponent $d/(2(d+1))$ is sufficient, but it is not known whether\u0000this condition is necessary. In this article, we show that in the periodic\u0000problem in $mathbb T^d$ the exponent $d/(2(d+1))$ is necessary for all\u0000non-singular polynomial symbols $P$ regardless of the degree of $P$. Among the\u0000differential operators covered, we highlight the natural powers of the\u0000Laplacian $Delta^k$ for $k in mathbb N$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we derive a pair of finite univariate biorthogonal polynomials suggested by the finite univariate orthogonal polynomials $M_{n}^{(p,q)}(x)$. The corresponding biorthogonality relation is given. Some useful relations and properties, concluding differential equation and generating function, are presented. Further, a new family of finite biorthogonal functions is obtained using Fourier transform and Parseval identity. In addition, we compute the Laplace transform and fractional calculus operators for polynomials $M_{n}(p,q,upsilon;x)$.
{"title":"Finite biorthogonal polynomials suggested by the finite orthogonal polynomials $M_{n}^{(p,q)}(x)$","authors":"Esra Güldoğan Lekesiz","doi":"arxiv-2408.15010","DOIUrl":"https://doi.org/arxiv-2408.15010","url":null,"abstract":"In this paper, we derive a pair of finite univariate biorthogonal polynomials\u0000suggested by the finite univariate orthogonal polynomials $M_{n}^{(p,q)}(x)$.\u0000The corresponding biorthogonality relation is given. Some useful relations and\u0000properties, concluding differential equation and generating function, are\u0000presented. Further, a new family of finite biorthogonal functions is obtained\u0000using Fourier transform and Parseval identity. In addition, we compute the\u0000Laplace transform and fractional calculus operators for polynomials\u0000$M_{n}(p,q,upsilon;x)$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function $pp$ for a fractional maximal operator $M_alpha$ or a non-degenerate fractional singular integral operator $T_alpha$, $0 leq alpha < n$, to satisfy weak $(pp,qq)$ inequalities or strong $(pp,qq)$ inequalities, with $qq$ being defined pointwise almost everywhere by % [ frac{1}{p(x)} - frac{1}{q(x)} = frac{alpha}{n}. ] % We first prove preliminary results linking fractional averaging operators and the $K_0^alpha$ condition, a qualitative condition on $pp$ related to the norms of characteristic functions of cubes, and show some useful implications of the $K_0^alpha$ condition. We then show that if $M_alpha$ satisfies weak $(pp,qq)$ inequalities, then $pp in K_0^alpha(R^n)$. We use this to prove that if $M_alpha$ satisfies strong $(pp,qq)$ inequalities, then $p_->1$. Finally, we prove a powerful pointwise estimate for $T_alpha$ that relates $T_alpha$ to $M_alpha$ along a carefully chosen family of cubes. This allows us to prove necessary conditions for fractional singular integral operators similar to those for fractional maximal operators.
{"title":"Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces","authors":"David Cruz-Uribe, Troy Roberts","doi":"arxiv-2408.12745","DOIUrl":"https://doi.org/arxiv-2408.12745","url":null,"abstract":"In this paper we prove necessary conditions for the boundedness of fractional\u0000operators on the variable Lebesgue spaces. More precisely, we find necessary\u0000conditions on an exponent function $pp$ for a fractional maximal operator\u0000$M_alpha$ or a non-degenerate fractional singular integral operator\u0000$T_alpha$, $0 leq alpha < n$, to satisfy weak $(pp,qq)$ inequalities or\u0000strong $(pp,qq)$ inequalities, with $qq$ being defined pointwise almost\u0000everywhere by % [ frac{1}{p(x)} - frac{1}{q(x)} = frac{alpha}{n}. ] % We first prove preliminary results linking fractional averaging operators and\u0000the $K_0^alpha$ condition, a qualitative condition on $pp$ related to the\u0000norms of characteristic functions of cubes, and show some useful implications\u0000of the $K_0^alpha$ condition. We then show that if $M_alpha$ satisfies weak\u0000$(pp,qq)$ inequalities, then $pp in K_0^alpha(R^n)$. We use this to prove\u0000that if $M_alpha$ satisfies strong $(pp,qq)$ inequalities, then $p_->1$.\u0000Finally, we prove a powerful pointwise estimate for $T_alpha$ that relates\u0000$T_alpha$ to $M_alpha$ along a carefully chosen family of cubes. This allows\u0000us to prove necessary conditions for fractional singular integral operators\u0000similar to those for fractional maximal operators.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive optimal asymptotic and non-asymptotic lower bounds on the Widom factors for weighted Chebyshev and orthogonal polynomials on compact subsets of the real line. In the Chebyshev case we extend the optimal non-asymptotic lower bound previously known only in a handful of examples to regular compact sets and a large class weights. Using the non-asymptotic lower bound, we extend Widom's asymptotic lower bound for weights bounded away from zero to a large class of weights with zeros including weights with strong zeros and infinitely many zeros. As an application of the asymptotic lower bound we extend Bernstein's 1931 asymptotics result for weighted Chebyshev polynomials on an interval to arbitrary Riemann integrable weights with finitely many zeros and to some continuous weights with infinitely many zeros. In the case of orthogonal polynomials, we derive optimal asymptotic and non-asymptotic lower bound on arbitrary regular compact sets for a large class of weights in the non-asymptotic case and for arbitrary SzegH{o} class weights in the asymptotic case, extending previously known bounds on finite gap and Parreau--Widom sets.
{"title":"Lower Bounds for Weighted Chebyshev and Orthogonal Polynomials","authors":"Gökalp Alpan, Maxim Zinchenko","doi":"arxiv-2408.11496","DOIUrl":"https://doi.org/arxiv-2408.11496","url":null,"abstract":"We derive optimal asymptotic and non-asymptotic lower bounds on the Widom\u0000factors for weighted Chebyshev and orthogonal polynomials on compact subsets of\u0000the real line. In the Chebyshev case we extend the optimal non-asymptotic lower\u0000bound previously known only in a handful of examples to regular compact sets\u0000and a large class weights. Using the non-asymptotic lower bound, we extend\u0000Widom's asymptotic lower bound for weights bounded away from zero to a large\u0000class of weights with zeros including weights with strong zeros and infinitely\u0000many zeros. As an application of the asymptotic lower bound we extend\u0000Bernstein's 1931 asymptotics result for weighted Chebyshev polynomials on an\u0000interval to arbitrary Riemann integrable weights with finitely many zeros and\u0000to some continuous weights with infinitely many zeros. In the case of\u0000orthogonal polynomials, we derive optimal asymptotic and non-asymptotic lower\u0000bound on arbitrary regular compact sets for a large class of weights in the\u0000non-asymptotic case and for arbitrary SzegH{o} class weights in the asymptotic\u0000case, extending previously known bounds on finite gap and Parreau--Widom sets.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a restricted projection theorem for a certain one dimensional family of projections from $mathbb R^n$ to $mathbb R^k$. The family we consider here arises naturally in the study of quantitative equidistribution problems in homogeneous dynamics.
{"title":"Projection Theorems in the Presence of Expansions","authors":"K. W. Ohm","doi":"arxiv-2408.11159","DOIUrl":"https://doi.org/arxiv-2408.11159","url":null,"abstract":"We prove a restricted projection theorem for a certain one dimensional family\u0000of projections from $mathbb R^n$ to $mathbb R^k$. The family we consider here\u0000arises naturally in the study of quantitative equidistribution problems in\u0000homogeneous dynamics.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the formation of breathers in multi-dimensional lattices with nonlocal coupling that decays algebraically. By variational methods, the exact relationship between various parameters (dimension, nonlinearity, nonlocal parameter $alpha$) that defines positive excitation thresholds is characterized. At the anti-continuum regime, there exists a family of unique ground states that determines excitation thresholds. We not only characterize the sharp spatial decay of ground states, which varies continuously in $alpha$, but also identify the time decay of dispersive waves, which undergoes a discontinuous transition in $alpha$.
{"title":"Nonlinear excitations in multi-dimensional nonlocal lattices","authors":"Brian Choi","doi":"arxiv-2408.11177","DOIUrl":"https://doi.org/arxiv-2408.11177","url":null,"abstract":"We study the formation of breathers in multi-dimensional lattices with\u0000nonlocal coupling that decays algebraically. By variational methods, the exact\u0000relationship between various parameters (dimension, nonlinearity, nonlocal\u0000parameter $alpha$) that defines positive excitation thresholds is\u0000characterized. At the anti-continuum regime, there exists a family of unique\u0000ground states that determines excitation thresholds. We not only characterize\u0000the sharp spatial decay of ground states, which varies continuously in\u0000$alpha$, but also identify the time decay of dispersive waves, which undergoes\u0000a discontinuous transition in $alpha$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a family of Fourier integral operators by allowing their symbols to satisfy a multi-parameter differential inequality on R^N. We show that these operators of order -(N-1)/2 are bounded from classical, atom decomposable H^1-Hardy space to L^1(R^N). Consequently, we obtain a sharp L^p-regularity result due to Seeger, Sogge and Stein.
{"title":"Regularity of Fourier integrals on product spaces","authors":"Chaoqiang Tan, Zipeng Wang","doi":"arxiv-2408.09691","DOIUrl":"https://doi.org/arxiv-2408.09691","url":null,"abstract":"We study a family of Fourier integral operators by allowing their symbols to\u0000satisfy a multi-parameter differential inequality on R^N. We show that these\u0000operators of order -(N-1)/2 are bounded from classical, atom decomposable\u0000H^1-Hardy space to L^1(R^N). Consequently, we obtain a sharp L^p-regularity\u0000result due to Seeger, Sogge and Stein.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Endpoint regularity of general Fourier integral operators","authors":"Xiangrong Zhu, Wenjuan Li","doi":"arxiv-2408.15280","DOIUrl":"https://doi.org/arxiv-2408.15280","url":null,"abstract":"Let $ngeq 1,0<rho<1, max{rho,1-rho}leq deltaleq 1$ and\u0000$$m_1=rho-n+(n-1)min{frac 12,rho}+frac {1-delta}{2}.$$ If the amplitude\u0000$a$ belongs to the H\"{o}rmander class $S^{m_1}_{rho,delta}$ and $phiin\u0000Phi^{2}$ satisfies the strong non-degeneracy condition, then we prove that the\u0000following Fourier integral operator $T_{phi,a}$ defined by begin{align*}\u0000T_{phi,a}f(x)=int_{mathbb{R}^{n}}e^{iphi(x,xi)}a(x,xi)widehat{f}(xi)dxi,\u0000end{align*} is bounded from the local Hardy space $h^1(mathbb{R}^n)$ to\u0000$L^1(mathbb{R}^n)$. As a corollary, we can also obtain the corresponding\u0000$L^p(mathbb{R}^n)$-boundedness when $1<p<2$. These theorems are rigorous improvements on the recent works of Staubach and\u0000his collaborators. When $0leq rholeq 1,deltaleq max{rho,1-rho}$, by\u0000using some similar techniques in this note, we can get the corresponding\u0000theorems which coincide with the known results.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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