傅里叶乘子与短时傅里叶乘子的比较：短时傅里叶变换的平滑效应
Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform
AI总结：
本文比较傅里叶乘子与STFT乘子，给出二者相等的充分必要条件，并揭示STFT乘子因双窗短时傅里叶变换而具有平滑效应，同时得到反Wick算子连续性的必要条件及离散Gabor乘子的对应结果。
摘要：
我们研究具有窗函数 $g_1,g_2$、符号 $a(x,ω)=(1\otimes m)(x,ω)=m(ω)$（其中 $(x,ω)\in\mathbb{R}^{2d}$）的 STFT 乘子 $A^{g_1,g_2}_{1\otimes m}$ 与 $\mathbb{R}^d$ 上符号为 $m_2$ 的傅里叶乘子 $T_{m_2}$ 之间的联系。我们找到了符号 $m,m_2$ 和窗函数 $g_1,g_2$ 上使得等式 $T_{m_2}= A^{g_1,g_2}_{1\otimes m}$ 成立的充分必要条件。当 $m=m_2$ 时，前述等式仅在调制空间中特定选择的窗函数下成立，而在勒贝格空间的范畴内则从不出现。一般而言，STFT 乘子 $A^{g_1,g_2}_{1\otimes m}$（也称为局部化算子）呈现出一种平滑效应，这源于进入 $A^{g_1,g_2}_{1\otimes m}$ 定义中的所谓双窗短时傅里叶变换。作为附带结果，我们证明了具有弱 $L^r$ 空间中乘子 $m$ 的反 Wick 算子 $A^{g,g}_{1\otimes m}: L^p\to L^q$ 连续性的必要条件。最后，我们展示了其离散对应情形的相关结果：在此设置下，STFT 乘子被称为 Gabor 乘子，而傅里叶乘子则更常被称为线性时不变（LTI）滤波器。
英文摘要：
We study the connection between STFT multipliers $A^{g_1,g_2}_{1\otimes m}$ having windows $g_1,g_2$, symbols $a(x,ω)=(1\otimes m)(x,ω)=m(ω)$, $(x,ω)\in\mathbb{R}^{2d}$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\mathbb{R}^d$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $g_1,g_2$ for the equality $T_{m_2}= A^{g_1,g_2}_{1\otimes m}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A^{g_1,g_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called two-window short-time Fourier transform which enters in the definition of $A^{g_1,g_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{g,g}_{1\otimes m}: L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.
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