Maciej Paluszynski, Jacek Zienkiewicz
We describe the structure of the resolvent of the discrete rough truncated Hilbert transform under the critical exponent. This extends the results obtained in [8].
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We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in [9] and [10]
We obtain weak type (1,1) estimates for the inverses of truncated discrete rough Hilbert transform. We include an ex- ample showing that our result is sharp. One of the ingredients of the proof are regularity estimates for convolution of singular measure associated with the sequence $[m^α]$.
The fractional Hilbert transforms plays an important role in optics and signal processing. In particular the analytic signal proposed by Gabor has as a key component the Hilbert transform. The higher dimensional Hilbert transform is the Riesz-Hilbert transform which was used by Felsberg and Sommer to construct the monogenic signal. We will construct fractional and quaternionic fractional Riesz-Hilbert transforms based on a eigenvalue decomposition. We will prove properties of these transformations such as shift and scale invariance, orthogonality and the semigroup property. Based on the fracti…
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.
We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators
We identify symmetric quasi-Banach range of the discrete Calderón operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.
Boundedness of the discrete Hilbert transform on discrete weighted Morrey spaces
本文研究了离散Hilbert变换在离散加权Morrey空间上的有界性问题，探讨了其在数字信号处理中的应用价值。
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。
We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for Cesàro bounded operators of fractional order. The results obtained fairly generalize the corresponding ones for power-bounded operators. In passing, we give an extension of the mean ergodic theorem. Examples are given to illustrate the theory.
Optimal distributional estimates of the multiple Hilbert transform
本文通过Calderón算子的d重自复合，给出了多重希尔伯特变换的点态上下分布估计，将A. P. Calderón、D. Boyd和Ch. Fefferman的基本结果推广到任意d∈N。
It is proved that the finite Hilbert transform $T\colon X\to X$, which acts continuously on every rearrangement invariant space $X$ on $(-1,1)$ having non-trivial Boyd indices, is already optimally defined. That is, $T\colon X\to X$ cannot be further extended, still taking its values in $X$, to any larger domain space.
We investigate the spectrum and fine spectra of the finite Hilbert transform acting on rearrangement invariant spaces over $(-1,1)$ with non-trivial Boyd indices, thereby extending Widom's results for $L^p$ spaces. In the case when these indices coincide, a full description of the spectrum and fine spectra is given.
Discrete Hilbert Transform And Discrete Mikhlin Multiplier On Discrete Variable Lebesgue Space
本文在变量离散Lebesgue空间上证明了离散Hilbert变换的有界性，并给出了离散Mikhlin乘子在该空间上有界的充分条件。
We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier transform $F: L_2(Q_p,X)\to L_2(Q_p,X)$ in space of functions, which are square-integrable in Bochner sense and take value in $X$, is a bounded operator.
The T1 theorem for the Hilbert transform fails when p is not 2
本文研究 Hilbert 变换的 T1 定理在 L^{p} 空间中的适用范围，证明除 p=2 外，该定理对 1<p<∞ 均不成立。
The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz-Boas interpolation formulas which are relevant to the Discrete Hilbert and Kak-Hilbert transforms in $l^{2}$.
There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. We study the action of the Bargmann transform on several classical integral operators on $L^2(\R)$, including the fractional Fourier transform, the fractional Hilbert transform, and the wavelet transform.
We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.
We show that if the dyadic Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the Hilbert transform, with a linear relation of the bounds. This result is the counterpart of [arXiv:2212.00090] where the opposite bound was proven.
The paper has been withdrawn by the author.
We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
This article investigates classes of bounded sequences of complex numbers that are universally good for the ergodic Hilbert transform in L_p-spaces, 2\leq p\leq \infty : The class of bounded Besicovitch sequences satisfying a rate condition is among such sequence classes.
This paper has been withdrawn by the author due to some error in the result
Rellich identities for the Hilbert transform
本文利用Rellich恒等式和图Lipschitz区域上的Neumann问题证明涉及共形映射的Hilbert变换恒等式，并由此得到新的加权L^2估计及与Helson-Szegö定理相关的精确范数界。
A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted one-sided ergodic Hilbert transforms on vector-valued $L_p$-spaces. Furthermore, the obtained results are applied to the a.e. convergence of random modulated weighted ergodic series, which is also new in the classical setting.
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 $\mathbb{R}^3_+$.
Weighted discrete Hilbert transforms $(a_n)_n \mapsto \big(\sum_n a_n v_n/(λ_j-γ_n)\big)_j$ from $\ell^2_v$ to $\ell^2_w$ are considered, where $Γ=(γ_n)$ and $Λ=(λ_j)$ are disjoint sequences of points in the complex plane and $v=(v_n)$ and $w=(w_j)$ are positive weight sequences. It is shown that if such a Hilbert transform is unitary, then $Γ\cupΛ$ is a subset of a circle or a straight line, and a description of all unitary discrete Hilbert transforms is then given. A characterization of the orthogonal bases of reproducing kernels introduced by L. de Branges and D. Clark is implicit in these …
A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is cont…
Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E Ψ(N_t) \leq C_{Φ,Ψ,X} \mathbb E Φ(M_t),\;\;\; t\geq 0, $$ where $Φ, Ψ:X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{Φ,Ψ,X}$ coincides with the $Φ,Ψ$-norm of the periodic Hilbert transform. As a corollary, it is shown that the $Φ,Ψ$-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transf…
Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.
We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…
By way of concrete presentations, we construct two infinite-dimensional transforms at the crossroads of Gaussian fields and reproducing kernel Hilbert spaces (RKHS), thus leading to a new infinite-dimensional Fourier transform in a general setting of Gaussian processes. Our results serve to unify existing tools from infinite-dimensional analysis.
We derive a stronger uniqueness result if a function with compact support and its truncated Hilbert transform are known on the same interval by using the Sokhotski-Plemelj formulas. To find a function from its truncated Hilbert transform, we express them in the Chebyshev polynomial series and then suggest two methods to numerically estimate the coefficients. We present computer simulation results to show that the extrapolative procedure numerically works well.
Hilbert transforms on Coxeter groups and groups acting on buildings
本文通过几何对象建立新模型，证明Coxeter群及作用于建筑的群上Hilbert变换满足Cotlar恒等式，从而得到$L_p$-有界性，并首次在具有性质(F$\mathbb{R}$)的群上实现该结果。
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