Invariance of a Shift-Invariant Space

哈代空间上的不变性保持共轭

Invariance Preserving Conjugations on the Hardy Space

研究\(H^2\)上共轭算子\(C\)，刻画其将平移不变子空间映射到平移不变子空间的情况，还探讨了映射到余不变子空间时共轭的存在与否。

Time-Frequency Shift Invariance of Gabor Spaces with an $S_0$-Generator

We consider Gabor Riesz sequences generated by a lattice $Λ\subset \mathbb{R}^2$ and a window function $g \in L^2(\mathbb{R})$ which is well localized in both time and frequency. When $g$ belongs to the Feichtinger algebra, we prove that only those time-frequency shifts with parameters from the lattice $Λ$ leave the corresponding Gabor space invariant. This improves on earlier results where only lattices of rational density were considered. A slightly weaker result is proved - again for lattices of general density - under the regularity assumptions of the classical Balian-Low theorem, where bo…

Extra-invariance of group actions

Given discrete groups $Γ\subset Δ$ we characterize $(Γ,σ)$-invariant spaces that are also invariant under $Δ$. This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal $(Γ,σ)$-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the analogous in the context of shift-invariant spaces in terms of the Fourier transform, which is very well-known. As a consequence of our results, we give a solution for the problem …

Group invariants in algebras of generalized functions

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given.

在连续嵌入的Hilbert空间中随机偏微分方程的不变流形

Invariant manifolds for stochastic partial differential equations in continuously embedded Hilbert spaces

本文研究了在连续嵌入的Hilbert空间中随机偏微分方程子流形的随机不变性条件，并建立了其与有限维SDEs不变性之间的联系，提出了一种新的分析方法。

On a version of Trudinger-Moser inequality with Möbius shift invariance

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk $B\subset\R^2$, recently proved by G. Mancini and K. Sandeep. Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to Möbius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity $\int |u|^{2^*}$ in the higher dimens…

Translation invariance of Fock spaces

We show that there is only one Hilbert space of entire functions that is invariant under the action of naturally defined weighted translations.