Hilbert Operators Research Articles

Bounds of operators on the Hilbert sequence space

Abstract The author has computed the bounds of the Hilbert operator on some sequence spaces [18, 19]. Through this study the author has investigated the bounds of operators on the Hilbert sequence space and the present study is a complement of those previous research.

Hankel matrices acting on the Hardy space $$H^1$$ H 1 and on Dirichlet spaces

If $$\,\mu \,$$ is a finite positive Borel measure on the interval $$\,[0,1)$$ , we let $$\,\mathcal {H}_\mu \,$$ be the Hankel matrix $$\,(\mu _{n, k})_{n,k\ge 0}\,$$ with entries $$\,\mu _{n, k}=\mu _{n+k}$$ , where, for $$\,n\,=\,0, 1, 2, \ldots $$ , $$\mu _n\,$$ denotes the moment of order $$\,n\,$$ of $$\,\mu $$ . This matrix induces formally the operator $$\,\mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n\,$$ on the space of all analytic functions $$\,f(z)=\sum _{k=0}^\infty a_kz^k\,$$ , in the unit disc $$\,\mathbb {D} $$ . When $$\,\mu \,$$ is the Lebesgue measure on $$\,[0,1)\,$$ the operator $$\,\mathcal {H}_\mu \,$$ is the classical Hilbert operator $$\,\mathcal {H}\,$$ which is bounded on $$\,H^p\,$$ if $$\,1<p<\infty $$ , but not on $$\,H^1$$ . J. Cima has recently proved that $$\,\mathcal {H}\,$$ is an injective bounded operator from $$\,H^1\,$$ into the space $$\,\mathscr {C}\,$$ of Cauchy transforms of measures on the unit circle. The operator $$\,\mathcal {H}_\mu \,$$ is known to be well defined on $$\,H^1\,$$ if and only if $$\,\mu \,$$ is a Carleson measure and in such a case we have that $$\mathcal {H}_\mu (H^1)\subset \,{\mathscr {C}}$$ . Furthermore, it is bounded from $$\,H^1\,$$ into itself if and only if $$\,\mu \,$$ is a 1-logarithmic 1-Carleson measure. In this paper we prove that when $$\,\mu \,$$ is a 1-logarithmic 1-Carleson measure then $$\,{\mathcal {H}}_\mu \,$$ actually maps $$\,H^1\,$$ into the space of Dirichlet type $$\,{\mathcal {D}}^1_0\,$$ . We discuss also the range of $$\,{\mathcal {H}}_\mu \,$$ on $$\,H^1\,$$ when $$\,\mu \,$$ is an $$\alpha $$ -logarithmic 1-Carleson measure ( $$0<\alpha <1$$ ). We study also the action of the operators $$\,{\mathcal {H}}_\mu \,$$ on Bergman spaces and on Dirichlet spaces.

The Cauchy dual and 2-isometric liftings of concave operators

We present some 2-isometric lifting and extension results for Hilbert space concave operators. For a special class of concave operators we study their Cauchy dual operators and discuss conditions under which these operators are subnormal. In particular, the quasinormality of compressions of such operators is studied.

Some Dynamic Inequalities Involving Hilbert and Hardy–Hilbert Operators with Kernels

In this paper, we prove some new dynamic inequalities involving Hilbert and Hardy–Hilbert operators with kernels and use them to establish general forms of multiple Hilbert and Hardy–Hilbert type inequalities on time scales.

Grand Lebesgue sequence spaces

Abstract We introduce grand Lebesgue sequence spaces and study various operators of harmonic analysis in these spaces, e.g., maximal, convolution, Hardy, Hilbert, and fractional operators, among others. Special attention is paid to fractional calculus, including the density of the discrete version of a Lizorkin sequence test space in vanishing grand spaces.

Asymmetric Pulse Modeling for FRI Sampling

We consider sampling and reconstruction of finite-rate-of-innovation (FRI) signals such as a train of pulses, where the pulses have varying degrees of asymmetry. We address the problem of asymmetry modeling starting from a given symmetric prototype. We show that among the class of unitary operators that are linear and invariant to translation and scale, the fractional Hilbert (FrH) operator is unique for parametrically modeling pulse asymmetry. The FrH operator is obtained by a trigonometric interpolation between the standard Hilbert and identity operators, where the interpolation weights are determined by the degree of asymmetry. The FrH operators are also steerable , which allows for estimation of the asymmetry factors, in addition to the delays and amplitudes, using the high-resolution spectral estimation techniques that are used for solving standard FRI problems. We also develop the discrete counterpart using discrete FrH operators and show that all the desirable properties carry over smoothly to the discrete setting as well. We derive closed-form expressions for the Cramer–Rao bounds and Hammersley–Chapman–Robbins bound, on the variances of the estimators for continuous and discrete parameters, respectively. Experimental results show that the proposed estimators have variances that meet the lower bounds. We demonstrate an application of the proposed discrete FrH methodology on real electrocardiogram (ECG) signals in the presence of noise. Specifically, we show how the asymmetry of QRS complexes in various channels of an ECG signal could be modeled accurately.

Encoding TLA+ into unsorted and many-sorted first-order logic

is a specification language designed for the verification of concurrent and distributed algorithms and systems. We present an encoding of a non-temporal fragment of into (unsorted) first-order logic and many-sorted first-order logic, the input languages of first-order automated theorem provers. The non-temporal subset of is based on untyped set theory and includes functions, arithmetic expressions, and Hilbert's choice operator. The translation, based on encoding techniques such as boolification, injection of unsorted expressions into sorted languages, term rewriting, and abstraction, is the core component of a back-end prover based on first-order theorem provers and smt solvers for the Proof System. • A TLA + Proof System back-end prover based on first-order provers and SMT solvers. • New techniques for encoding set theory, functions and Hilbert's operator into FOL. • Encoding of an untyped language (TLA + ) into many-sorted FOL. • Improved verification of safety properties of distributed systems in TLA + . • Reduce effort required to write interactive proofs in the TLA + Proof System.

A Hankel Matrix Acting on Spaces of Analytic Functions

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $\mu_n$ denotes the moment of order $n$ of $\mu $. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D $. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_\mu $ on Hardy spaces and on M\"obius invariant spaces.

A generalized Hilbert operator acting on conformally invariant spaces

If μ is a positive Borel measure on the interval [0,1), we let Hμ be the Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k=μn+k, where, for n=0,1,2,…, μn denotes the moment of order n of μ. This matrix formally induces the operator Hμ(f)(z)=∑n=0∞(∑k=0∞μn,kak)zn on the space of all analytic functions f(z)=∑k=0∞akzk, in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators Hμ on Hardy spaces has been recently studied. This article is devoted to a study of the operators Hμ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the Qs-spaces.

Hilbert series and operator basis for NRQED and NRQCD/HQET

We use a Hilbert series to construct an operator basis in the 1/m expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET). We present a list of effective operators with mass dimension d≤8. Comparing to the current literature, our results for NRQED agree for d≤8, but there are some discrepancies in NRQCD/HQET at d=7 and 8.

Fractional Riesz–Hilbert-Type Transforms and Associated Monogenic Signals

The fractional Hilbert transforms play a significant role in optics and signal processing because they interpolate between the near field and the far field. Mathematically, they interpolate between the identity operator and the Hilbert transform. The higher dimensional analog of the Hilbert transform is the Riesz–Hilbert transform. In this paper, we construct fractional Riesz–Hilbert operators, which also interpolate between the identity operator and the Riesz–Hilbert transform. The fractional Riesz–Hilbert transforms form a semigroup that is generated by the Riesz–Hilbert transforms. Based on the fractional Riesz–Hilbert transform we construct a fractional monogenic signal and generalize the construction of the monogenic signal. Using quaternions instead of complex numbers as eigenvalues of the transform Riesz–Hilbert transform the interpolation path can be generalized and de.pends on the chosen unit quaternion. That leads to the quaternionic fractional Riesz–Hilbert transform and a quaternionic fractional monogenic signal.

Hilbert matrix on spaces of Bergman-type

It is well known (see [8,14]) that the Libera operator L is bounded on the Besov space Hνp,q,α if and only if 0<κp,α,ν:=ν−α−1p+1. We prove unexpected results: the Hilbert matrix operator H, as well as the modified Hilbert operator H˜, is bounded on Hνp,q,α if and only if 0<κp,α,ν<1. In particular, H, as well as H˜, is bounded on the Bergman space Ap,α if and only if 1<α+2<p and is bounded on the Dirichlet space Dαp=A1p,α if and only if max⁡{−1,p−2}<α<2p−2. Our results are substantial improvement of [11, Theorem 3.1] and of [6, Theorem 5].

A note on property ( $${W_E}$$ W E )

We investigate a new spectrum property (\({W_E}\)), which extends the generalized Weyl theorem. Using the property of consistence in Fredholm and index, we establish for a bounded linear operator T defined on a Hilbert space sufficient and necessary conditions for which the property \({(W_E)}\) holds. We also explore conditions on Hilbert operators T and S so that property \({(W_E)}\) holds for \({T\oplus S}\) . Moreover, we study the permanence of property \({(W_E)}\) under perturbations by power finite rank operators commuting with T and discuss the relation between property (\({W_E}\)) and hypercyclic operators.

Some q-inequalities for Hausdorff operators

We calculate the sharp bounds for some q-analysis variants of Hausdorff type inequalities of the form $$\int_0^{ + \infty } {{{\left( {\int_0^{ + \infty } {\frac{{\phi \left( t \right)}}{t}f\left( {\frac{x}{t}} \right){d_q}t} } \right)}^p}{d_q}x} \leqslant {C_\phi }\int_0^b {{f^p}\left( t \right)} {d_q}t$$ . As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Polya operator.

Infinite-dimensional Hilbert tensors on spaces of analytic functions

In this paper, the $m-$order infinite dimensional Hilbert tensor (hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the spaces of analytic functions, which is called Hilbert tensor operator. The boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$ ($p>2(m-1)$). On the base of the boundedness, two positively homogeneous operators are introduced to the spaces of analytic functions, and hence the upper bounds of norm of such two operators are found on Bergman spaces $A^p$ ($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $\pi$ and $\pi^\frac1{m-1}$, respectively.

Inequalities for Hilbert operator and its extensions: The probabilistic approach

We present a probabilistic study of the Hilbert operator \[Tf(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{f(y)\,\mathrm{d}y}{x+y},\qquad x\geq0,\] defined on integrable functions $f$ on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and $\Phi$-inequalities for $T$. We also show similar estimates for higher dimensional analogues of the Hilbert operator, and by the further careful modification of martingale methods, we obtain related sharp localized inequalities for Hilbert and Riesz transforms.

SOME INEQUALITIES FOR THE NUMERICAL RADIUS FOR HILBERT SPACE OPERATORS

We introduce some new refinements of numerical radius inequalities for Hilbert space invertible operators. More precisely, we prove that if $T\in {\mathcal{B}}({\mathcal{H}})$ is an invertible operator, then $\Vert T\Vert \leq \sqrt{2}\unicode[STIX]{x1D714}(T)$.

A multichannel time–frequency and multi-wavelet toolbox for uterine electromyography processing and visualisation

The uterine electromyogram, also called electrohysterogram (EHG), is an electrical signal generated by the uterine contractile activity. The EHG has been considered a promising biomarker for labour and preterm labour prediction, for which there is a demand for accurate estimation methods. Preterm labour is a significant public health concern and one of the major causes of neonatal mortality and morbidity [1]. Given the non-stationary properties of the EHG signal, time–frequency domain analysis can be used. For real life signals it is not generally possible to determine a priori the suitable quadratic time–frequency kernel or the appropriate wavelet family and relative parameters, regarding, for instance, the adequate detection of the signal frequency variation in time. There has been a lack of a comprehensive software tool for the selection of the appropriate time frequency representation of a multichannel EHG signal and extraction of relevant spectral and temporal information. The presented toolbox (Uterine Explorer) has been specifically designed for the EHG analysis and exploration in view of the characterisation of its components. The starting point is the multichannel scalogram or spectrogram representation from which frequency and time marginals, instantaneous frequency and bandwidth are obtained as EHG features. From this point the detected components undergo parametric and non-parametric spectral estimation and wavelet packet analysis. Intrauterine pressure estimation (IUP) is obtained using the Teager, RMS, wavelet marginal and Hilbert operators over the EHG. This toolbox has been tested to build up a dictionary of 288 EHG components [2], useful for research in preterm labour prediction.

About the Behavior of the Hilbert Operator in Some Spaces of Hӧlder Type

About the Behavior of the Hilbert Operator in Some Spaces of Hӧlder Type

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions