论文标题
某些指数系统的封闭跨度$e_λ$在空间中$ l^p（γ，β）$，生物三相家族的属性到$e_λ$ in $ l^2（γ，β）$，MONM问题和Carleson的微分方程
The closed span of some Exponential system $E_Λ$ in the spaces $L^p(γ,β)$, properties of a Biorthogonal family to $E_Λ$ in $L^2(γ,β)$, Moment problems, and a differential equation of Carleson
论文作者
论文摘要
一组复数$λ= \ {λ_n，μ_n\} _ {n = 1}^{\ infty} $，带有多个项\ [\ [\ {λ_n，μ_n\} _ {n = 1} \ {\ Underbrace {λ_1，λ_1，\ dots，λ_1} _ {μ_1-时代}，\ Underbrace {λ_2，λ_2，\ dots，λ_2} \ UnderBrace {λ_k，λ_k，\ dots，λ_k} _ {μ_k-times}，\ dots \} \]被说属于$ \ bf abc $ class，如果满足三个条件：$ \ bf（a）$ \ bf（a）$ \ bf（a）$美元类型零。假设$λ\在\ bf abc $中，我们以müntz-szászTheorem的精神为特征，其关联的指数系统的封闭跨度\ [e_λ：= \ {x^k e^e^e^{λ_nx} k = 0,1,2，\ dots，μ_n-1 \} \] banach空间$ l^p（γ，β）$，其中$ - \ infty <γ<γ<β<β<\ infty $和$ p \ ge 1 $。与$e_λ$相关，我们探索其唯一的生物表达序列\ [r_λ= \ {r_ {r_ {n，k}：\，n \ in \ in \ mathbb {n}，\，k = 0,1，\ dots，\ dots，μ_n-1 \} $ l^2（γ，β）$。 As a result, we find a solution to the Moment problem \[ \int_γ^β f(t)\cdot t^k e^{\overline{λ_n} t}\, dt=d_{n,k},\qquad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k = 0,1，\ dots，μ_n-1，\ quad d_ {n，k} = o（e^{a \reλ_n}）\，\，\，for \，\，\，a <β。 \]最后，我们表征了L. Carleson研究的无限顺序微分方程的解空间。
A set of complex numbers $Λ=\{λ_n,μ_n\}_{n=1}^{\infty}$ with multiple terms \[ \{λ_n,μ_n\}_{n=1}^{\infty}:= \{\underbrace{λ_1,λ_1,\dots,λ_1}_{μ_1 - times}, \underbrace{λ_2,λ_2,\dots,λ_2}_{μ_2 - times},\dots, \underbrace{λ_k,λ_k,\dots,λ_k}_{μ_k - times},\dots\} \] is said to belong to the $\bf ABC$ class if it satisfies three conditions: $\bf (A)$ $\sum_{n=1}^{\infty}μ_n/|λ_n|<\infty$, $\bf (B)$ $\sup_{n\in\mathbb{N}}|\argλ_n|<π/2$, $\bf (C)$ $Λ$ is an interpolating variety for the space of entire functions of exponential type zero. Assuming that $Λ\in\bf ABC$, we characterize in the spirit of the Müntz-Szász theorem, the closed span of its associated exponential system \[ E_Λ:=\{x^k e^{λ_n x}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,μ_n-1\} \] in the Banach spaces $L^p(γ,β)$, where $-\infty<γ<β<\infty$ and $p\ge 1$. Related to $E_Λ$, we explore the properties of its unique biorthogonal sequence \[ r_Λ=\{r_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots,μ_n-1\}\subset\overline{\text{span}}(E_Λ) \] in $L^2(γ,β)$. As a result, we find a solution to the Moment problem \[ \int_γ^β f(t)\cdot t^k e^{\overline{λ_n} t}\, dt=d_{n,k},\qquad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,\dots ,μ_n-1,\quad d_{n,k}=O(e^{a\Reλ_n})\,\, for\,\, a<β. \] Finally, we characterize the solution space of a differential equation of infinite order, studied by L. Carleson.
