A. Bučkovska, S. Pilipović, M. Vuković
An approximation result for the bilinear Hilbert transform is proved and used for the inversion of the bilinear Hilbert transform. Also, p-Lebesgue points $(p\geq 1)$ are analyzed.
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Current work defines Schmidt representation of a bilinear operator $T: H_1 \times H_2 \rightarrow K$, where $H_1, H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.
We prove $L^p$-bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $L^p(\mathbb{R};X)$ into outer Lebesgue spaces on the time-frequency-scale space $\mathbb{R}^3_+$.
Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas in a general Hilbert space.
Given $n$ disjoint intervals $I_j$, on $\mathbb R$ together with $n$ functions $ψ_j\in L^2(I_j)$, $j=1,\dots n$, and an $n\times n$ matrix $Θ$, the problem is to find an $L^2$ solution $\vec φ= {\rm Col} (φ_1,\dots, φ_n)$, $φ_j \in L^2(I_j)$ to the linear system $χΘ\mathcal H \vec φ= \vecψ$, where $\mathcal H = {\rm diag} (\mathcal H_1 ,\dots, \mathcal H_n)$ is a matrix of finite Hilbert transforms and $χ=\text{diag}(χ_1,\dots,χ_n)$ is a matrix of the corresponding characteristic functions on $I_j$, and $\vec ψ={\rm Col} (ψ_1,\dots,ψ_n)$. Since we can interpret $χΘ\mathcal H \vec φ$ as a gener…
In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of ``off-diagonal'' decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.
We develop a theory of Hilbert $\widetilde{\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\widetilde{\C}$-linear functionals and $\widetilde{\C}$-sesquilinear forms. By making use of a generalized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form.
We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrödinger equations.
The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible such domain (i.e, its \textit{optimal domain}). Some classical operators are already optimally defined (e.g., the Hilbert transform in $L^p(\mathbb{R})$, $1<p<\infty$) and others are not (e.g., the Hausdorff-Young inequality in $L^p(\mathbb{T})$, $1<p<2$, or Sobolev's inequality in various spaces). In this paper a detailed investigation is undertaken of the fi…
The finite Hilbert transform on $(-1,1)$
本文综述了有限希尔伯特变换在 $(-1,1)$ 上的重排不变空间中的反演问题的最新进展。
Stability estimates for the regularized inversion of the truncated Hilbert transform
本文研究有限数据计算机断层扫描中截断希尔伯特变换反演问题的稳定性，通过限制重叠区域并利用函数的L2范数或变分先验知识，获得Hölder连续性稳定性结果。
The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincaré-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation $T(f)=g$ whenever the data function $g$ lies in the range of $T$ within $L^1$ (shown to contain $LlogL$). Until now this was only known for $g$ belonging to the union of all $L^p$ spaces with $p>1$. It is established (due to a result of Stein) that $T$ cannot be extended to any domain spa…
Extension and Integral Representation of the finite Hilbert Transform In Rearrangement Invariant Spaces
本文研究重排不变空间中有限Hilbert变换的延拓与反演，通过引入由该算子诱导的向量测度及其积分算子，给出了该算子的积分表示。
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