In this paper, generalized Bell polynomials $(Be_n^phi)_n$ associated to a sequence of real numbers $phi=(phi_i)_{i=1}^infty$ are introduced. Bell polynomials correspond to $phi_i=0$, $ige 1$. We prove that when $phi_ige 0$, $ige 1$: (a) the zeros of the generalized Bell polynomial $Be_n^phi$ are simple, real and non positive; (b) the zeros of $Be_{n+1}^phi$ interlace the zeros of $Be_n^phi$; (c) the zeros are decreasing functions of the parameters $phi_i$. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind.
{"title":"Generalized Bell polynomials","authors":"Antonio J. Durán","doi":"arxiv-2409.11344","DOIUrl":"https://doi.org/arxiv-2409.11344","url":null,"abstract":"In this paper, generalized Bell polynomials $(Be_n^phi)_n$ associated to a\u0000sequence of real numbers $phi=(phi_i)_{i=1}^infty$ are introduced. Bell\u0000polynomials correspond to $phi_i=0$, $ige 1$. We prove that when $phi_ige\u00000$, $ige 1$: (a) the zeros of the generalized Bell polynomial $Be_n^phi$ are\u0000simple, real and non positive; (b) the zeros of $Be_{n+1}^phi$ interlace the\u0000zeros of $Be_n^phi$; (c) the zeros are decreasing functions of the parameters\u0000$phi_i$. We find a hypergeometric representation for the generalized Bell\u0000polynomials. As a consequence, it is proved that the class of all generalized\u0000Bell polynomials is actually the same class as that of all Laguerre multiple\u0000polynomials of the first kind.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254677","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 present a survey of results related to the solution of Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals $C^{alpha,r}_{beta,p}$, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of $2pi$--periodic functions $C^{alpha,r}_{beta,p}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_{p}$, $1leq pleq infty$, with generalized Poisson kernels $$ P_{alpha,r,beta}(t)=sumlimits_{k=1}^{infty}e^{-alpha k^{r}}cos big(kt-frac{betapi}{2}big), alpha>0, r>0, betain mathbb{R}.$$
{"title":"Approximation by Fourier sums on the classes of generalized Poisson integrals","authors":"Anatoly Serdyuk, Tetiana Stepaniuk","doi":"arxiv-2409.10629","DOIUrl":"https://doi.org/arxiv-2409.10629","url":null,"abstract":"We present a survey of results related to the solution of\u0000Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized\u0000Poisson integrals $C^{alpha,r}_{beta,p}$, which consists in finding of\u0000asymptotic equalities for exact upper boundaries o f uniform norms of\u0000deviations of partial Fourier sums on the classes of $2pi$--periodic functions\u0000$C^{alpha,r}_{beta,p}$, which are defined as convolutions of the functions,\u0000which belong to the unit balls pf the spaces $L_{p}$, $1leq pleq infty$,\u0000with generalized Poisson kernels $$\u0000P_{alpha,r,beta}(t)=sumlimits_{k=1}^{infty}e^{-alpha k^{r}}cos\u0000big(kt-frac{betapi}{2}big), alpha>0, r>0, betain mathbb{R}.$$","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254676","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}
Differential equations where the graph of some derivative of a function is composed of a finite number of similarity transformations of the graph of the function itself are defined. We call these self-similar differential equations (SSDEs) and prove existence and uniqueness of solution under certain conditions. While SSDEs are not ordinary differential equations, the technique for demonstrating existence and uniqueness of SSDEs parallels that for ODEs. This paper appears to be the first work on equations of this nature.
{"title":"Self-similar Differential Equations","authors":"Leon Q. Brin, Joe Fields","doi":"arxiv-2409.09943","DOIUrl":"https://doi.org/arxiv-2409.09943","url":null,"abstract":"Differential equations where the graph of some derivative of a function is\u0000composed of a finite number of similarity transformations of the graph of the\u0000function itself are defined. We call these self-similar differential equations\u0000(SSDEs) and prove existence and uniqueness of solution under certain\u0000conditions. While SSDEs are not ordinary differential equations, the technique\u0000for demonstrating existence and uniqueness of SSDEs parallels that for ODEs.\u0000This paper appears to be the first work on equations of this nature.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254678","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 use Jensen's formula to give an upper bound for the number of real zeros of polynomials with constrained coefficients. We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = sum_{j=0}^{n}{a_jz^j},, quad |a_0| = 1,, quad |a_j| leq M,, quad a_j in Bbb{C},, quad M geq 1,,$$ has at most $cn^{1/2}(1+log M)^{1/2}$ zeros in the interval $[-1,1]$. This result is sharp up to the multiplicative constant $c > 0$ and extends an earlier result of Borwein, Erd'elyi, and K'os from the case $M=1$ to the case $M geq $1. This has also been proved recently with the factor $(1+log M)$ rather than $(1+log M)^{1/2}$ in the Appendix of a recent paper by Jacob and Nazarov by using a different method. We also prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the above form has at most $(c/a)(1+log M)$ zeros in the interval $[-1+a,1-a]$ with $a in (0,1)$. Finally we correct a somewhat incorrect proof of an earlier result of Borwein and Erd'elyi by proving that there is a constant $eta > 0$ such that every polynomial $P$ of the above form with $M = 1$ has at most $eta n^{1/2}$ zeros inside any polygon with vertices on the unit circle, where the multiplicative constant $eta > 0$ depends only on the polygon.
{"title":"The number of real zeros of polynomials with constrained coefficients","authors":"Tamás Erdélyi","doi":"arxiv-2409.09553","DOIUrl":"https://doi.org/arxiv-2409.09553","url":null,"abstract":"We use Jensen's formula to give an upper bound for the number of real zeros\u0000of polynomials with constrained coefficients. We prove that there is an\u0000absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) =\u0000sum_{j=0}^{n}{a_jz^j},, quad |a_0| = 1,, quad |a_j| leq M,, quad a_j\u0000in Bbb{C},, quad M geq 1,,$$ has at most $cn^{1/2}(1+log M)^{1/2}$ zeros\u0000in the interval $[-1,1]$. This result is sharp up to the multiplicative\u0000constant $c > 0$ and extends an earlier result of Borwein, Erd'elyi, and K'os\u0000from the case $M=1$ to the case $M geq $1. This has also been proved recently\u0000with the factor $(1+log M)$ rather than $(1+log M)^{1/2}$ in the Appendix of\u0000a recent paper by Jacob and Nazarov by using a different method. We also prove\u0000that there is an absolute constant $c > 0$ such that every polynomial $P$ of\u0000the above form has at most $(c/a)(1+log M)$ zeros in the interval $[-1+a,1-a]$\u0000with $a in (0,1)$. Finally we correct a somewhat incorrect proof of an earlier\u0000result of Borwein and Erd'elyi by proving that there is a constant $eta > 0$\u0000such that every polynomial $P$ of the above form with $M = 1$ has at most $eta\u0000n^{1/2}$ zeros inside any polygon with vertices on the unit circle, where the\u0000multiplicative constant $eta > 0$ depends only on the polygon.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254714","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}
The purpose of this note is twofold: firstly, it intends to bring to light an apparently unknown property of the product of the extreme zeros of Laguerre polynomials, which in a very particular case leads to a twenty-year-old conjecture for Hermite polynomials posed by Gazeau, Josse-Michaux, and Moncea while developing numerical methods in quantum mechanics; and secondly to progress towards the solution of this problem as an application of a parametric eigenvalue problem.
{"title":"On the product of the extreme zeros of Laguerre polynomials","authors":"K. Castillo","doi":"arxiv-2409.09405","DOIUrl":"https://doi.org/arxiv-2409.09405","url":null,"abstract":"The purpose of this note is twofold: firstly, it intends to bring to light an\u0000apparently unknown property of the product of the extreme zeros of Laguerre\u0000polynomials, which in a very particular case leads to a twenty-year-old\u0000conjecture for Hermite polynomials posed by Gazeau, Josse-Michaux, and Moncea\u0000while developing numerical methods in quantum mechanics; and secondly to\u0000progress towards the solution of this problem as an application of a parametric\u0000eigenvalue problem.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254680","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 small cap decoupling theorem for the parabola over a general non-Archimedean local field for which $2neq 0$. We obtain polylogarithmic dependence on the scale parameter $R$ and polynomial dependence in the residue prime, except for the prime 2 for which the polynomial depends on degree. Our constants are fully explicit.
{"title":"High-low analysis and small cap decoupling over non-Archimedean local fields","authors":"Ben Johnsrude","doi":"arxiv-2409.09163","DOIUrl":"https://doi.org/arxiv-2409.09163","url":null,"abstract":"We prove a small cap decoupling theorem for the parabola over a general\u0000non-Archimedean local field for which $2neq 0$. We obtain polylogarithmic\u0000dependence on the scale parameter $R$ and polynomial dependence in the residue\u0000prime, except for the prime 2 for which the polynomial depends on degree. Our\u0000constants are fully explicit.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254715","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 relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = mbox{div}(Anablacdot) - partial_t $ on cylindrical domains $ Omega = mathcal O times mathbb R $, where the base $ mathcal O subset mathbb R^{n} $ is a $1$-sided chord arc domain (and for one result Lipschitz) in the spatial variables. In the paper we answer the question when the solvability of the $L^p$ Regularity problem for $L$ (denoted by $ (R_L)_{p} $) can be deduced from the solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$ (denoted $ (D_L^*)_{p'} $). We show that this holds if for at least of $qin(1,infty)$ the problem $ (R_L)_{q} $ is solvable. This result is a parabolic equivalent of two elliptic results of Kenig-Pipher (1993) and Shen (2006), the combination of which establishes the elliptic version of our result. For the converse, see Dindov{s}-Dyer (2019) where it is shown that $ (R_L)_{p}$ implies $ (D_L^*)_{p'}$ on parabolic $mbox{Lip}(1,1/2)$ domains.
{"title":"A relation between the Dirichlet and the Regularity problem for Parabolic equations","authors":"Martin Dindoš, Erika Sätterqvist","doi":"arxiv-2409.09197","DOIUrl":"https://doi.org/arxiv-2409.09197","url":null,"abstract":"We study the relationship between the Dirichlet and Regularity problem for\u0000parabolic operators of the form $ L = mbox{div}(Anablacdot) - partial_t $\u0000on cylindrical domains $ Omega = mathcal O times mathbb R $, where the base\u0000$ mathcal O subset mathbb R^{n} $ is a $1$-sided chord arc domain (and for\u0000one result Lipschitz) in the spatial variables. In the paper we answer the question when the solvability of the $L^p$\u0000Regularity problem for $L$ (denoted by $ (R_L)_{p} $) can be deduced from the\u0000solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$\u0000(denoted $ (D_L^*)_{p'} $). We show that this holds if for at least of $qin(1,infty)$ the problem $\u0000(R_L)_{q} $ is solvable. This result is a parabolic equivalent of two elliptic results of Kenig-Pipher\u0000(1993) and Shen (2006), the combination of which establishes the elliptic\u0000version of our result. For the converse, see Dindov{s}-Dyer (2019) where it is\u0000shown that $ (R_L)_{p}$ implies $ (D_L^*)_{p'}$ on parabolic\u0000$mbox{Lip}(1,1/2)$ domains.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254716","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}
The classical Muckenhoupt's $A_p$ condition is necessary and sufficient for the boundedness of the maximal operator $M$ on $L^p(w)$ spaces. In this paper we obtain another characterization of the $A_p$ condition. As a result, we show that some strong versions of the weighted $L^p(w)$ Coifman--Fefferman and Fefferman--Stein inequalities hold if and only if $win A_p$. We also give new examples of Banach function spaces $X$ such that $M$ is bounded on $X$ but not bounded on the associate space $X'$.
{"title":"On a non-standard characterization of the $A_p$ condition","authors":"Andrei K. Lerner","doi":"arxiv-2409.07781","DOIUrl":"https://doi.org/arxiv-2409.07781","url":null,"abstract":"The classical Muckenhoupt's $A_p$ condition is necessary and sufficient for\u0000the boundedness of the maximal operator $M$ on $L^p(w)$ spaces. In this paper\u0000we obtain another characterization of the $A_p$ condition. As a result, we show\u0000that some strong versions of the weighted $L^p(w)$ Coifman--Fefferman and\u0000Fefferman--Stein inequalities hold if and only if $win A_p$. We also give new\u0000examples of Banach function spaces $X$ such that $M$ is bounded on $X$ but not\u0000bounded on the associate space $X'$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227671","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}
Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. int_{mathbb{R}^{n}}K(x,x-y)f(y)mathrm{d}y$ and $D^{gamma}(0leqgammaleq1)$ be the fractional differentiation operator, where $K(x,z)=frac{Omega(x,z')}{|z|^{n}}$, $z'=frac{z}{|z|},~~zneq0$. Let $~T^{ast}~$and $~T^sharp~$ be the adjoint of $T$ and the pseudo-adjoint of $T$, respectively. In this paper, via the expansion of spherical harmonics and the estimates of the convolution operators $T_{m,j}$, we shall prove some boundedness results for $TD^{gamma}-D^{gamma}T$ and $(T^{ast}-T^{sharp})D^{gamma}$ under natural regularity assumptions on the exponent function on a class of generalized Herz-Morrey spaces with weight and variable exponent, which extend some known results. Moreover, various norm characterizations for the product $T_{1}T_{2}$ and the pseudo-product $T_{1}circ T_{2}$ are also established.
{"title":"Weighted bounds for a class of singular integral operators in variable exponent Herz-Morrey spaces","authors":"Yanqi Yang, Qi Wu","doi":"arxiv-2409.07152","DOIUrl":"https://doi.org/arxiv-2409.07152","url":null,"abstract":"Let T be the singular integral operator with variable kernel defined by\u0000$Tf(x)= p.v. int_{mathbb{R}^{n}}K(x,x-y)f(y)mathrm{d}y$ and\u0000$D^{gamma}(0leqgammaleq1)$ be the fractional differentiation operator,\u0000where $K(x,z)=frac{Omega(x,z')}{|z|^{n}}$, $z'=frac{z}{|z|},~~zneq0$. Let\u0000$~T^{ast}~$and $~T^sharp~$ be the adjoint of $T$ and the pseudo-adjoint of\u0000$T$, respectively. In this paper, via the expansion of spherical harmonics and\u0000the estimates of the convolution operators $T_{m,j}$, we shall prove some\u0000boundedness results for $TD^{gamma}-D^{gamma}T$ and\u0000$(T^{ast}-T^{sharp})D^{gamma}$ under natural regularity assumptions on the\u0000exponent function on a class of generalized Herz-Morrey spaces with weight and\u0000variable exponent, which extend some known results. Moreover, various norm\u0000characterizations for the product $T_{1}T_{2}$ and the pseudo-product\u0000$T_{1}circ T_{2}$ are also established.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226649","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}
Philippe JamingIMB, Karim KellayIMB, Chadi SabaIMB, Yunlei WangIMB
In this paper we show that, if an increasing sequence $Lambda=(lambda_k)_{kinmathbb{Z}}$ has gaps going to infinity $lambda_{k+1}-lambda_kto +infty$ when $ktopminfty$, then for every $T>0$ and every sequence $(a_k)_{kinmathbb{Z}}$ and every $Ngeq 1$, $$ Asum_{k=0}^Nfrac{|a_k|}{1+k}leqfrac{1}{T}int_{-T/2}^{T/2} left|sum_{k=0}^N a_k e^{2ipilambda_k t}right|,mbox{d}t$$ further, if $sum_{kinmathbb{Z}}dfrac{1}{1+|lambda_k|}<+infty$,$$ Bmax_{|k|leq N}|a_k|leqfrac{1}{T}int_{-T/2}^{T/2} left|sum_{k=-N}^N a_k e^{2ipilambda_k t}right|,mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $Lambda$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by Ingham for $Tgeq 1$ under the condition that $lambda_{k+1}-lambda_kgeq 1$. The main novelty is that if those gaps go to infinity, then $T$ can be taken arbitrarily small. The result is new even when the $lambda_k$'s are integers where it extends a result of McGehee, Pigno and Smith. The results are then applied to observability of Schr"odinger equations with moving sensors.
{"title":"On L1-norms for non-harmonic trigonometric polynomials with sparse frequencies","authors":"Philippe JamingIMB, Karim KellayIMB, Chadi SabaIMB, Yunlei WangIMB","doi":"arxiv-2409.07093","DOIUrl":"https://doi.org/arxiv-2409.07093","url":null,"abstract":"In this paper we show that, if an increasing sequence\u0000$Lambda=(lambda_k)_{kinmathbb{Z}}$ has gaps going to infinity\u0000$lambda_{k+1}-lambda_kto +infty$ when $ktopminfty$, then for every $T>0$\u0000and every sequence $(a_k)_{kinmathbb{Z}}$ and every $Ngeq 1$, $$\u0000Asum_{k=0}^Nfrac{|a_k|}{1+k}leqfrac{1}{T}int_{-T/2}^{T/2}\u0000left|sum_{k=0}^N a_k e^{2ipilambda_k t}right|,mbox{d}t$$ further, if\u0000$sum_{kinmathbb{Z}}dfrac{1}{1+|lambda_k|}<+infty$,$$ Bmax_{|k|leq\u0000N}|a_k|leqfrac{1}{T}int_{-T/2}^{T/2} left|sum_{k=-N}^N a_k\u0000e^{2ipilambda_k t}right|,mbox{d}t $$ where $A,B$ are constants that depend\u0000on $T$ and $Lambda$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by\u0000Ingham for $Tgeq 1$ under the condition that $lambda_{k+1}-lambda_kgeq 1$.\u0000The main novelty is that if those gaps go to infinity, then $T$ can be taken\u0000arbitrarily small. The result is new even when the $lambda_k$'s are integers\u0000where it extends a result of McGehee, Pigno and Smith. The results are then\u0000applied to observability of Schr\"odinger equations with moving sensors.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226650","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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