Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2021-03-16 , DOI: 10.1007/s00041-020-09804-z José Luis Romero
We show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values \(\{|f(\lambda )|: \lambda \in \Lambda \}\) on any set \(\Lambda \subseteq {\mathbb {R}}\) with lower Beurling density \(D^{-}(\Lambda )>2\).
We consider a totally positive function of Gaussian type, i.e., a function \(g \in L^2({\mathbb {R}})\) whose Fourier transform factors as
$$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$(1)with \(\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}\), and the shift-invariant space
$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$generated by its integer shifts within \(L^\infty ({\mathbb {R}})\). As a consequence of (1), each \(f \in V^\infty (g)\) is continuous, the defining series converges unconditionally in the weak\(^*\) topology of \(L^\infty \), and the coefficients \(c_k\) are unique [6, Theorem 3.5].
中文翻译：
具有完全正生成器的平移不变空间中的符号检索
我们表明，由高斯型的全正函数生成的平移不变空间中的实值函数f由其绝对值\（\ {| f（\ lambda）|：在具有较低Beurling密度\（D ^ {-}（\ Lambda）> 2 \）的任何集\（\ Lambda \ subseteq {\ mathbb {R} } \）上的lambda \ in \ Lambda \} \）。
我们考虑一个高斯型的完全正函数，即函数\（g \ in L ^ 2（{\ mathbb {R}}）\），其傅立叶变换因子为
$$ \ begin {aligned} \ hat {g}（\ xi）= \ int _ {{\ mathbb {R}}} g（x）e ^ {-2 \ pi ix \ xi} dx = C_0 e ^ { -\ gamma \ xi ^ 2} \ prod _ {\ nu = 1} ^ m（1 + 2 \ pi i \ delta _ \ nu \ xi）^ {-1}，\ quad \ xi \ in {\ mathbb { R}}，\ end {aligned} $$（1）与\（\增量_1，\ ldots，\增量_M \在{\ mathbb {R}}，C_0，\伽马> 0，M \在{\ mathbb {N}} \杯\ {0 \} \），和位移不变空间
$$ \ begin {aligned} V ^ \ infty（g）= \ Big \ {f = \ sum _ {k \ in {\ mathbb {Z}}}} c_k \，g（\ cdot -k）：c \ in \ ell ^ \ infty（{\ mathbb {Z}}）\ Big \}，\ end {aligned} $$由\（L ^ \ infty（{\ mathbb {R}}）\）中的整数移位生成。作为（1）的结果，每个\（f \ in V ^ \ infty（g）\）是连续的，定义级数在\（L ^ \ infty \）的弱\（^ * \）拓扑中无条件收敛。，并且系数\（c_k \）是唯一的[6，定理3.5]。
京公网安备 11010802027423号