论文标题
傅立叶和STFT乘数之间的比较：短时傅立叶变换的平滑效果
Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform
论文作者
论文摘要
我们研究sTft乘数$ a^{g_1，g_2} _ {1 \ otimes m} $具有Windows $ g_1，g_2 $，符号$ a（x，x，ω）=（1 \ otimes m）（x，x，ω）= m（x，ω）= m（ω）$（ω）$，$（x，x，Ω）乘数$ t_ {m_2} $带有符号$ m_2 $ on $ \ mathbb {r}^d $。我们在符号上找到足够且必要的条件。对于$ m = m_2 $，以前的平等仅适用于调制空间中窗口功能的特定选择，而它从未发生在lebesgue空间的领域中。通常，STFT乘数$ a^{g_1，g_2} _ {1 \ otimes m} $，也称为本地化操作员，由于所谓的两次短期傅立叶变换而产生了平滑效果，该效应进入了$ a^{g_1，g_1，g_1，g_1，g_2} _______________________} $ M}的定义。作为副产品，我们证明了反踢键运算符的连续性$ a^{g，g} _ {1 \ otimes m}的必要条件：l^p \ to l^q $具有乘数$ m $ M $ in弱$ l^r $空间。最后，我们展示了其离散对应物的相关结果：在此设置中，STFT乘数称为Gabor乘数，而傅立叶乘数则称为线性时间不变（LTI）过滤器。
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.
