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 generalized vector multi-interval finite Hilbert transform, we call the formula for the solution as "the inversion formula" and the necessary and sufficient conditions for the existence of a solution as the "range conditions". In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix $Θ$ is symmetric and positive definite, and; b) all the entries of $Θ$ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. When the matrix $Θ$ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann-Hilbert Problem. We also discuss other cases of the matrix $Θ$.
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 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…
In this paper we study the small-$λ$ spectral asymptotics of an integral operator $\mathscr{K}$ defined on two multi-intervals $J$ and $E$, when the multi-intervals touch each other (but their interiors are disjoint). The operator $\mathscr{K}$ is closely related to the multi-interval Finite Hilbert Transform (FHT). This case can be viewed as a singular limit of self-adjoint Hilbert-Schmidt integral operators with so-called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when $\text{dist}(J,E)>0$, and $\mathscr{K}$ is …
Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1π\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an approp…
Stability estimates for the regularized inversion of the truncated Hilbert transform
Seminar for Applied Mathematics, ETH Zürich（苏黎世联邦理工学院应用数学研讨会）；Department of Nuclear Medicine, Vrije Universiteit Brussel（布鲁塞尔自由大学核医学系）；Department of Mathematics, University of Central Florida（中佛罗里达大学数学系）
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…
The minimal representation $π$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on $\mathbb R^m$ with respect to the measure $|x|^{-1} dx_1... dx_m$. This article gives an explicit integral formula for the holomorphic extension of $π$ to a holomorphic semigroup of $O(m+3, \mathbb C)$ by means of the Bessel function. Taking its `boundary value', we also find the integral kernel of the `inversion operator' corresponding to the inversion element on the Minkowski space $\mathbb R^{m,1}$.
The Hilbert matrix $\mathcal{H}_{n,m} = (n+m+ 1)^{-1}$ has been extensively studied in previous literature. In this paper we look at generalized Hilbert operators arising from measures on the interval $[0, 1]$, such that the Hilbert matrix is obtained by the Lebesgue measure. We provide a necessary and sufficient condition for these operators to be bounded in $\ell^p$ and calculate their norm.
A Hilbert space frame on $R^n$ is {\it scalable} if we can scale the vectors to make them a tight frame. There are known classifications of scalable frames. There are two basic questions here which have never been answered in any $R^n$: Given a frame in $R^n$, how do we scale the vectors to minimize the condition number of the frame? I.e. How do we scale the frame to make it as tight as possible? If we are only allowed to use scaling numbers from the interval $[1-ε,1+ε]$, how do we scale the frame to minimize the condition number? We will answer these two questions in $R^2$ to begin the proces…
Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this problem has a unique answer (the nullspace is trivial) but is severely ill-posed. We isolate the difficulty and show that by restricting $f$ to functions with controlled total variation, reconstruction becomes stable. In particular, for functions $f \in H^1(I)$, we show that $$ \|Hf\|_{L^2(J)} \geq c_1 \exp{\left(-c_2 \frac{\|f_x\|_{L^2(I)}}{\|f\|_{L^2(I)}}\ri…
Let $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_μ=(μ_{n,k})_{n,k\geq 0}$ with entries $μ_{n,k}=μ_{n+k}$, where $μ_{n}=\int_{[0,1)}t^ndμ(t)$, induces formally the operator as $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^\infty\left(\sum_{k=0}^\infty μ_{n,k}a_k\right)(n+1)z^n , z\in \mathbb{D},$$ where $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$. In this paper, we characterize those positive Borel measures on $[0, 1)$ for which $\mathcal{DH}_μ$ is bounded (resp. compact) from Dirichlet spaces $\mathcal{D}_α( 0<α\leq2 )$ into $\mathcal…
We study a restriction of the Hilbert transform as an operator $H_T$ from $L^2(a_2,a_4)$ to $L^2(a_1,a_3)$ for real numbers $a_1 < a_2 < a_3 < a_4$. The operator $H_T$ arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). There, the reconstruction requires recovering a family of one-dimensional functions $f$ supported on compact intervals $[a_2,a_4]$ from its Hilbert transform measured on intervals $[a_1,a_3]$ that might only overlap, but not cover $[a_2,a_4]$. We show that the inversion of $H_T$ is ill-posed, which is why…
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.
The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $σ$ and $w$ on $\mathbb R$. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where $σ$ and $w$ are integrated over complementary intervals only. A key novely of the proof is a two-weight inequality for the Poisson integral with `holes…
In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form $\left\Vert S f\right\Vert_{L^2(K)}\geq C(I,K)\left\Vert f\right\Vert_{L^2(I)}$ where $I$ and $K$ are two dyadic intervals and $f$ supported in $I$. If $I\subset K$ such bound exist while in the other cases $K\subsetneq I$ and $K\cap I=\emptyset$ such bounds are only available under additional constraints on the derivative of $f$. In the later case, we establish a bound of the form $\left\Vert S f\right\Vert_{L^2(K)}\geq C(I,K)|\left\langle f\right\rangle_I|$ where $\left\langle f\right\rangle_I$ is th…
If $μ$ is a positive Borel measure on the interval $[0, 1)$, the Hankel matrix $\mathcal H_μ=(μ_{n,k})_{n,k\ge 0}$ with entries $μ_{n,k}=\int_{[0,1)}t^{n+k}\,dμ(t)$ induces formally the operator $$\mathcal{H}_μ(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D} $. In this paper we describe those measures $μ$ for which $\mathcal{H}_μ$ is a bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty $. We also characterize the measures $μ$ for which $\mathcal H_μ$ lies…
We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. The wavelet transform of a signal is an $L^2$-function on an appropriately chosen group, while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps $L^2$-functions on a group unitarily to fields of Hilbert-Schmidt operat…
