For integers $a_0,a_1,ldots,a_n$ with $|a_0a_n|=1$ and either $alpha =u$ with $1leq u leq 50$ or $alpha=u+ frac{1}{2}$ with $1 leq u leq 45$, we prove that $psi_n^{(alpha)}(x;a_0,a_1,cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},alpha=89/2$ are necessary. The above result with $0leqalpha leq 10$ is due to Filaseta, Finch and Leidy and with $alpha in {-1/2,1/2}$ due to Schur.
对于整数$a_0,a_1,ldots,a_n$与$|a_0a_n|=1$, $alpha =u$与$1leq u leq 50$或$alpha=u+ frac{1}{2}$与$1 leq u leq 45$，我们证明了$psi_n^{(alpha)}(x;a_0,a_1,cdots,a_n)$除了一个显式有限对集$(u,n)$外是不可约的。此外，除了$n=2^{12},alpha=89/2$之外的所有例外都是必要的。上面的结果与$0leqalpha leq 10$是由于Filaseta、Finch和Leidy，与$alpha in {-1/2,1/2}$是由于Schur。
{"title":"Irreducibility of extensions of Laguerre polynomials","authors":"S. Laishram, Saranya G. Nair, T. Shorey","doi":"10.7169/facm/1748","DOIUrl":"https://doi.org/10.7169/facm/1748","url":null,"abstract":"For integers $a_0,a_1,ldots,a_n$ with $|a_0a_n|=1$ and either $alpha =u$ with $1leq u leq 50$ or $alpha=u+ frac{1}{2}$ with $1 leq u leq 45$, we prove that $psi_n^{(alpha)}(x;a_0,a_1,cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore all the exceptions other than $n=2^{12},alpha=89/2$ are necessary. The above result with $0leqalpha leq 10$ is due to Filaseta, Finch and Leidy and with $alpha in {-1/2,1/2}$ due to Schur.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2019-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48545905","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":"Conical measures and closed vector measures","authors":"S. Okada, W. Ricker","doi":"10.7169/FACM/1711","DOIUrl":"https://doi.org/10.7169/FACM/1711","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41764774","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}
A. Defant, Domingo García, M. Maestre, P. Sevilla-Peris
{"title":"Dirichlet series from the infinite dimensional point of view","authors":"A. Defant, Domingo García, M. Maestre, P. Sevilla-Peris","doi":"10.7169/FACM/1741","DOIUrl":"https://doi.org/10.7169/FACM/1741","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1741","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46551474","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 consider Toepliz operators with (locally) integrable symbols acting on Bergman spaces Ap (1
本文考虑具有（局部）可积符号的Toepliz算子作用于复平面开单位圆盘的Bergman空间Ap（1
{"title":"On compactness of Toeplitz operators in Bergman spaces","authors":"J. Taskinen, J. Virtanen","doi":"10.7169/FACM/1727","DOIUrl":"https://doi.org/10.7169/FACM/1727","url":null,"abstract":"In this paper we consider Toepliz operators with (locally) integrable symbols acting on Bergman spaces Ap (1<p<∞) of the open unit disc of the complex plane. We give a characterization of compact Toeplitz operators with symbols in L1 under a mild additional condition. Our result is new even in the Hilbert space setting of A2, where it extends the well-known characterization of compact Toeplitz operators with bounded symbols by Stroethoff and Zheng.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1727","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45441393","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}
Weakly compactly generated Banach spaces and their subspaces are characterized by the presence of projectional skeletons with some additional properties. We work with real spaces. However the presented statements can be extended, without much extra effort, to complex spaces.
{"title":"WCG spaces and their subspaces grasped by projectional skeletons","authors":"M. Fabian, V. Montesinos","doi":"10.7169/FACM/1721","DOIUrl":"https://doi.org/10.7169/FACM/1721","url":null,"abstract":"Weakly compactly generated Banach spaces and their subspaces are characterized by the presence of projectional skeletons with some additional properties. We work with real spaces. However the presented statements can be extended, without much extra effort, to complex spaces.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1721","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44446676","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":"Operators on Fock-type and weighted spaces of entire functions","authors":"Ó. Blasco","doi":"10.7169/FACM/1708","DOIUrl":"https://doi.org/10.7169/FACM/1708","url":null,"abstract":"","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.7169/FACM/1708","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46143248","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 study the derivatives of an integer-valued polynomial of a given degree. Denoting by $E_n$ the set of the integer-valued polynomials with degree $leq n$, we show that the smallest positive integer $c_n$ satisfying the property: $forall P in E_n, c_n P' in E_n$ is $c_n = mathrm{lcm}(1 , 2 , dots , n)$. As an application, we deduce an easy proof of the well-known inequality $mathrm{lcm}(1 , 2 , dots , n) geq 2^{n - 1}$ ($forall n geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order $k$ and then we give two divisibility properties for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest positive integer $lambda_n$ satisfying the property: $forall P in E_n$, $forall k in mathbb{N}$: $lambda_n P^{(k)} in E_n$. In particular, we show that: $lambda_n = prod_{p text{ prime}} p^{lfloorfrac{n}{p}rfloor}$ ($forall n in mathbb{N}$).
本文研究了给定阶的整数值多项式的导数。用$E_n$表示次为$leq n$的整值多项式的集合，我们证明了满足性质$forall P in E_n, c_n P' in E_n$的最小正整数$c_n$是$c_n = mathrm{lcm}(1 , 2 , dots , n)$。作为一个应用，我们推导出了一个众所周知的不等式$mathrm{lcm}(1 , 2 , dots , n) geq 2^{n - 1}$ ($forall n geq 1$)的简单证明。在论文的第二部分，我们推广了给定阶导数$k$的结果，然后给出了所得数$c_{n , k}$的两个可整除性质(推广了$c_n$的性质)。根据这一研究，我们通过确定给定自然数$n$满足性质:$forall P in E_n$, $forall k in mathbb{N}$: $lambda_n P^{(k)} in E_n$的最小正整数$lambda_n$来总结本文。特别地，我们显示:$lambda_n = prod_{p text{ prime}} p^{lfloorfrac{n}{p}rfloor}$ ($forall n in mathbb{N}$)。
{"title":"On the derivatives of the integer-valued polynomials","authors":"Bakir Farhi","doi":"10.7169/facm/1786","DOIUrl":"https://doi.org/10.7169/facm/1786","url":null,"abstract":"In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by $E_n$ the set of the integer-valued polynomials with degree $leq n$, we show that the smallest positive integer $c_n$ satisfying the property: $forall P in E_n, c_n P' in E_n$ is $c_n = mathrm{lcm}(1 , 2 , dots , n)$. As an application, we deduce an easy proof of the well-known inequality $mathrm{lcm}(1 , 2 , dots , n) geq 2^{n - 1}$ ($forall n geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order $k$ and then we give two divisibility properties for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest positive integer $lambda_n$ satisfying the property: $forall P in E_n$, $forall k in mathbb{N}$: $lambda_n P^{(k)} in E_n$. In particular, we show that: $lambda_n = prod_{p text{ prime}} p^{lfloorfrac{n}{p}rfloor}$ ($forall n in mathbb{N}$).","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47332835","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 $F in mathbb{R}[x_1, ldots, x_n]$ be a homogeneous form of degree $d > 1$ satisfying $(n - dim V_{F}^*) > 4$, where $V_F^*$ is the singular locus of $V(F) = { mathbf{z} in {mathbb{C}}^n: F(mathbf{z}) = 0 }$. Suppose there exists $mathbf{x}_0 in (0,1)^n cap (V(F) backslash V_F^*)$. Let $mathbf{t} = (t_1, ldots, t_n) in mathbb{R}^n$. Then for a smooth function $varpi:mathbb{R}^n rightarrow mathbb{R}$ with its support contained in a small neighbourhood of $mathbf{x}_0$, we prove $$ Big{|} int_{0}^{infty} cdots int_{0}^{infty} varpi(mathbf{x}) x_1^{i t_1} cdots x_n^{i t_n} e^{2 pi i tau F(mathbf{x})} d mathbf{x} Big{|} ll min { 1, |tau|^{-1} }, $$ where the implicit constant is independent of $tau$ and $mathbf{t}$.
设$F in mathbb{R}[x_1, ldots, x_n]$为满足$(n - dim V_{F}^*) > 4$的次$d > 1$的齐次形式，其中$V_F^*$为$V(F) = { mathbf{z} in {mathbb{C}}^n: F(mathbf{z}) = 0 }$的奇异轨迹。假设存在$mathbf{x}_0 in (0,1)^n cap (V(F) backslash V_F^*)$。让$mathbf{t} = (t_1, ldots, t_n) in mathbb{R}^n$。然后，对于支持包含在$mathbf{x}_0$的小邻域内的光滑函数$varpi:mathbb{R}^n rightarrow mathbb{R}$，证明了其隐式常数与$tau$和$mathbf{t}$无关的$$ Big{|} int_{0}^{infty} cdots int_{0}^{infty} varpi(mathbf{x}) x_1^{i t_1} cdots x_n^{i t_n} e^{2 pi i tau F(mathbf{x})} d mathbf{x} Big{|} ll min { 1, |tau|^{-1} }, $$。
{"title":"On an oscillatory integral involving a homogeneous form","authors":"S. Yamagishi","doi":"10.7169/facm/1775","DOIUrl":"https://doi.org/10.7169/facm/1775","url":null,"abstract":"Let $F in mathbb{R}[x_1, ldots, x_n]$ be a homogeneous form of degree $d > 1$ satisfying $(n - dim V_{F}^*) > 4$, where $V_F^*$ is the singular locus of $V(F) = { mathbf{z} in {mathbb{C}}^n: F(mathbf{z}) = 0 }$. Suppose there exists $mathbf{x}_0 in (0,1)^n cap (V(F) backslash V_F^*)$. Let $mathbf{t} = (t_1, ldots, t_n) in mathbb{R}^n$. Then for a smooth function $varpi:mathbb{R}^n rightarrow mathbb{R}$ with its support contained in a small neighbourhood of $mathbf{x}_0$, we prove $$ Big{|} int_{0}^{infty} cdots int_{0}^{infty} varpi(mathbf{x}) x_1^{i t_1} cdots x_n^{i t_n} e^{2 pi i tau F(mathbf{x})} d mathbf{x} Big{|} ll min { 1, |tau|^{-1} }, $$ where the implicit constant is independent of $tau$ and $mathbf{t}$.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2018-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41257188","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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