Piatetski-Shapiro现象及其相关问题
Piatetski-Shapiro's phenomenon and related problems
摘要：
这个博士本文是在Alexander Olevskii教授的指导下完成的，主要研究傅里叶分析中的两个领域：三角展开式的唯一性理论和函数空间中的平移不变子空间理论。我们在第一个领域的主要结果扩展到$\ell_q$空间（$q > 2$），这是Piatetski-Shapiro在1954年发现的空间$c_0$的深层现象。我们开发的方法也使我们能够在第二个提到的领域得到一个结果，先验看起来与第一个没有联系。结果（也许有点令人惊讶）是：人们不能用傅里叶变换的零点集来表征$\ell_p(\Z)$或$L^p(\R)$，$1 < p < 2$中的函数，它们的平移跨越整个空间。这应该与$p=1,2$情形相关的经典维纳定理形成对比。
英文摘要：
This Ph.D. thesis, prepared under the supervision of Prof. Alexander Olevskii, is concerned with some problems in two areas of Fourier Analysis: uniqueness theory of trigonometric expansions, and the theory of translation invariant subspaces in function spaces. Our main result in the first area extends to $\ell_q$ spaces ($q > 2$) a deep phenomenon found by Piatetski-Shapiro in 1954 for the space $c_0$. The approach we developed also enabled us to get a result in the second mentioned area, which a priori does not look connected with the first one. The result (maybe, a bit surprising) is: one cannot characterize the functions in $\ell_p(\Z)$ or $L^p(\R)$, $1 < p < 2$, whose translates span the whole space, by the zero set of their Fourier transform. This should be contrasted against the classical Wiener theorems related to the cases $p=1,2$.
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