Hankel Type Research Articles

Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 > s > 1 0>s>1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 > s > 1 0>s>1 .

Direct Electromagnetic Wave Scattering Calculation Using Methods of Moments through Layered Rough Surface

This thesis focuses on the direct calculation of electromagnetic wave scattering through layered rough surfaces using the Method of Moments. The study aims to contribute to existing knowledge by addressing the problems associated with accurately predicting scattered electromagnetic fields from rough surfaces with layered structures. By using the Method of Moments, this research seeks to provide insights into the behavior of electromagnetic waves when interacting with multi-layered rough surfaces, offering valuable implications for practical engineering applications and the development of advanced computational tools for electromagnetic wave analysis. In practical terms, the surface is typically divided into small patches, and the electromagnetic field at each patch is determined using MoM. This involves solving integral equations describing the interaction between the induced currents on the surface and the incident field. The layered structure may require additional equations to model the reflection of waves and the transmission at the boundaries between the layers. The direct problem is also examined in a two-dimensional picture, where the rough surface profile and the medium are known parameters, and the electromagnetic waves scattered from the surface to the entire space are the unknowns. The rough surface is considered to have a finite length and can be deterministic or arise from a stationary stochastic random process, with a focus on Gaussian-type rough surfaces. The roughness is considered as two basic types: as a perfect electric conductor (PEC) in free space and as a boundary between two dielectric media. The equations of integration obtained are based on the scalar solution of Green's theory, Helmholtz equations, and using Hankel type green functions as kernels due to the one-dimensional nature. The MoM is extensively used to solve these integral equations and is compared to asymptotic approaches and analytical. Various simulations are conducted to test the feasibility of the algorithms, and the results are thoroughly discussed. The study of electromagnetic wave scattering in one-dimensional rough surfaces involves the direct problem, where the field distribution in the whole space is determined using the equation of integration. These equations take into account the surface roughness and medium parameters they are solved using the Method of Moments (MoM) by expressing the field distribution on the surface using basis functions. The direct calculation of electromagnetic wave scattering through a layered rough surface using the Method of Moments (MoM) is a computationally intensive process and complex and. It contains simulating the interaction of EM waves with a complex surface, which presents challenges due to the multiple layers with different properties, and a roughly surface causing and diffraction of incident waves and scattering. When working with a layered rough surface increases the electromagnetic wave interaction complexity. Additional multiple layers introduce considerations, such as layer interfaces and accounting for different material abilities. This requires careful modeling and analysis to exactly capture the behavior of electromagnetic waves as they traverse through and interact with layers. MoM, a commonly used numerical method for solving electromagnetic scattering problems, discreteness the surface into small segments and solves integral equations to determine the fields that is scattered. This approach allows for a maximum analysis of electromagnetic interactions at a grained level.

Characterizations of Hankel type measures

In this paper, motivated by the definition of the Hankel measure and the well-known theory of the Carleson type measure, we introduce and characterize certain Hankel type measures. The characterizations of these measures involve Hardy spaces, Lipschitz spaces, the Zygmund space, Hankel operators, and the balayage of measures. Among positive Borel measures supported on [0,1), we also consider further the relation between Carleson type measures and these Hankel type measures.

ON ARITHMETIC FUNCTIONS, THEIR EXTENDED COEFFICIENTS: VARIOUS RESULTS AND RELATIONS

In this article, we represent a recurrence relation of the arithmetic function connected with an ascending factorial function, Lah and Stirling numbers. We then obtain a relation of harmonic numbers and again extend the coecients of these arithmetic functions involving Bell polynomials through introducing the sequence of Hankel type integrals. On the other hand, making some of the extensions of these arithmetic functions, we derive some more results and the summation formulae in terms of Riemann Zeta function.

Fractionalization of Hankel Type Integral Transforms and Their Relevance

In this paper, the fractionalization of certain types of Hankel transforms is suggested. Barut-Girardello type transforms are then introduced along with the relevant fractional order forms. Finally some further generalizations are suggested.

Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems

This paper is devoted to studying non-commensurate fractional order planar systems. Our contributions are to derive sufficient conditions for the global attractivity of non-trivial solutions to fractional-order inhomogeneous linear planar systems and for the Mittag-Leffler stability of an equilibrium point to fractional order nonlinear planar systems. To achieve these goals, our approach is as follows. Firstly, based on Cauchy’s argument principle in complex analysis, we obtain various explicit sufficient conditions for the asymptotic stability of linear systems whose coefficient matrices are constant. Secondly, by using Hankel type contours, we derive some important estimates of special functions arising from a variation of constants formula of solutions to inhomogeneous linear systems. Then, by proposing carefully chosen weighted norms combined with the Banach fixed point theorem for appropriate Banach spaces, we get the desired conclusions. Finally, numerical examples are provided to illustrate the effect of the main theoretical results.

Existence, uniqueness and asymptotic behavior of solutions to two-term fractional differential equations

In this paper, we give a systematic analysis for solutions to two-term fractional differential equations. Our main contribution is in Section 5. In this section we have shown the global attractiveness, the Mittag-Leffler stability and the local attractiveness of solutions to some classes of two-term fractional differential equations. To do these, our approach is combining Cauchy’s argument principle in complex analysis (to obtain an explicit necessary and sufficient condition for the asymptotic stability of linear equations), the Hankel type contour (to estimate the decay rate of special functions arising in the elementary solutions) and a linearization method developed recently for fractional-order differential equations (to prove the Mittag-Leffler stability of the trivial solution to nonlinear equations and the local attractivity of the origin for the inhomogeneous nonlinear case). Numerical simulations are finally presented to confirm the theoretical findings.

Conversion of a Helical Surface Plasmon Polariton into a Spiral Surface Plasmon Polariton at the Outlet of a Metallic Nanohole.

The conversion of a helical surface plasmon polariton (SPP) creeping out of a circular nanohole in a thick metal (Ag or Au) film into a spiral (Hankel type) SPP outward propagating at the film’s interface is studied theoretically. The dispersion relations of SPPs of various modes in a nanohole, calculated from a transcendental equation, show that the propagation length of an SPP of mode 1 is much larger than the other modes in a specific frequency band, which is dependent on the nanohole size. In this band, the streamlines of the Poynting vector (energy flux) of mode-1 SPP in nanohole exhibit helixes; the surface component of the energy flux is perpendicular to the phase front of the SPP. Numerical results show that, after a helical SPP tunnels through a nanohole, most of the energy flux fans out at the outlet as a dipole radiation. The spatial phase distribution of Ez above the interface indicates that the transmission light carries orbital angular momentum with a topological charge of 1. Additionally, a part of the helical SPP creeping along the edge of an outlet naturally converts into a spiral (Hankel type of order 1) SPP outward propagating at the film’s interface; both SPPs have the same handedness. Moreover, the interferences of multi SPPs generating from two nanoholes and even from a two-dimensional nanohole array are also related to the spiral SPP.

ON HANKEL TYPE CONVOLUTION OPERATORS

Let H′α,β be the Zemanian type space of Hankel transformable generalised functions and let O′α,β,* be the space of Hankel convolution operators for H′α,β . This H′α,β is the dual of a subspace H′α,β of O′α,β,* for which O′α,β,* is also the space of Hankel convolution. In this paper the elements of O′α,β,* are characterised as those in L(Hα,β) and in L(H′α,β) that commute with Hankel translations. Moreover, necessary and sufficient condition on the generalised Hankel type transform h′α,βS of S ∈ O′α,β,* are established in order that every T ∈ O′α,β,* such that S ∗ T ∈ Hα,β lie in H′α,β.

The linear canonical Hankel type transformations associated with translation and convolution

This paper is the study of Hankel type translation and Hankel type convolution for linear canonical Hankel type transformations. In this paper, I have studied some inequalities associated with Hankel type translation and Hankel type convolution. Also, I have studied some applications of linear canonical Hankel transformation to a canonical convolution integral equation and a generalized nonlinear parabolic equation.

HANKEL CONVOLUTORS ON CERTAIN HANKEL TYPE TRANSFORMABLE FUNCTION SPACES

In the present paper the spaces of Hankel type convolutors are introduced and we have characterized the dual spaces of certain Hankel type transformable function spaces as Spaces of Hankel type convolutors.

Parabolic Boundary Value Problems in Unlimited Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in unbounded by variable z piecewise-homogeneous by radially variable r wedge-shaped by an angularly variable φ hollow cylinder were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet — Neumann, Neumann — Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a Fourier integral transform on a Cartesian axis by an applicative variable and a hybrid integral transform of the Hankel type of the second kind on a segment of the polar axis with n points of conjugation were used to construct classic solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image.

Toeplitz type operator and Gleason's problem on Hp,q,s(B) of Cn

ABSTRACT In this paper, the authors study the boundedness of the Toeplitz type operator and the Hankel type operator on . Further, the authors discuss the solvability of the Gleason's problem on . Finally, the inclusion relations of the space and some typical function spaces are also considered.

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

PARABOLIC BOUNDARY VALUE PROBLEMS IN A PIECEWISE HOMOGENEOUS WEDGE-SHAPED SOLID CYLINDER

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous wedge-shaped solid cylinder were constructed at first time by the method of integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices). The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable $\varphi \in (0; \varphi_0)$, a Fourier integral transform on the Cartesian segment $(-l_1;l_2)$ by an applicative variable $z$ and a hybrid integral transform of the Hankel type of the first kind on a segment $(0;R)$ of the polar axis with $n$ points of conjugation by an radial variable $r$ were used to construct solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image. The structure of the solution of the problem in the case of setting the Neumann boundary conditions on the wedge edges is analyzed. Exact analytical formulas for the components of the main solutions are written and the theorem on the existence of a single bounded classical solution of the problem is formulated. The obtained solutions are algorithmic in nature and can be used (using numerical methods) in solving applied problems.

Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula

Abstract This paper considers two finite integral transforms of Fourier-type, in view to propose a set of eight new convolutions, and to analyze the solvability of a class of the integral equations of Wiener-Hopf plus Hankel type, defined on finite intervals, which is involved in engineering problems. The solvability and solution of the considered equations are investigated by means of Fourier-type series, and a Shannon-type sampling formula is obtained. Some concluding remarks with respect to theoretical issues and engineering applications are emphasized in the last section, along with the analysis of some illustrative cases, which exemplify that the present method solves cases which are not under the conditions of previously known techniques.

Concentration of the empirical spectral distribution of random matrices with dependent entries

We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_{n}$ be a Hermitian random matrix of the size $n\times n$ that can be split into independent blocks of the size at most $d_{n}=o(n^{2})$. We prove that under some mild conditions on the distribution of the entries of $X_{n}$, the empirical spectral measure of $X_{n}$ concentrates around its mean. The main theorem is a strengthening of the recent result by Kemp and Zimmerman, where the size of the blocks grows as $o(\log n)$. As an application, we are able to upgrade the results of Schenker and Schulz on the convergence in expectation to the semicircle law of a class of random matrices with dependent entries to weak convergence in probability. Other applications include patterned random matrices, e.g. matrices of Toeplitz, Hankel or circulant type and matrices with heavy tailed entries in the domain of attraction of the Gaussian distribution.

Pseudo-differential type operators and Gevrey spaces

In this paper, the pseudo-differential type operator [Formula: see text] associated with the Bessel type operator [Formula: see text] defined by (2.3) involving the symbol [Formula: see text] whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces. It is shown that the operator [Formula: see text] is a continuous linear map of one Gevrey space into another Gevrey space. A special pseudo-differential type operator called the Gevrey–Hankel type potential is defined and some of its properties are investigated. A variant of [Formula: see text] is also studied.

Integral equation of Toeplitz plus Hankel's type and parabolic equation related to the Kontorovich‐Lebedev‐Fourier generalized convolutions

In this paper, we consider a class of integral equations with two Toeplitz plus Hankel's type kernels, and a class of parabolic equations. Using the Kontorovich‐Lebedev transform method, we obtain the closed form of solutions related to the Kontorovich‐Lebedev‐Fourier generalized convolutions. Furthermore, the estimate of solution in Lp with weight is established.

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω1,..., ωn), ωj ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U n such that $$||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } $$ , where dm2n(z) is the 2ndimensional Lebesgue measure on U n and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B p (ω) and L p (ω) (the weighted L p -space).