An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.     

On sharp estimates of the convergence of double Fourier–Bessel series

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.     

Bessel框架及其摄动

本文进一步研究了Bessel框架的性质,给出其稳定性和摄动的几个有意义的结果。     

Estimate of the Fourier–Bessel Multipliers for the Poly-axially Operator

We prove the analogue of H?rmander–Mikhlin multiplier theorem for the multidimensional Fourier–Bessel transform associated with the Poly-axially operator.     

On Inversion Of Bessel Potentials Associated With The Laplace–Bessel Differential Operator

Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated.     

Wiener’s lemma for singular integral operators of Bessel potential type

In this paper, we introduce an algebra of singular integral operators containing Bessel potentials of positive order, and show that the corresponding unital Banach algebra is an inverse-closed Banach subalgebra of ${\mathcal {B}} (L^q_w)$ , the Banach algebra of all bounded operators on the weighted space $L_w^q$ , for all $1\le q<\infty $ and Muckenhoupt $A_q$ -weights $w$ .     

Bessel场位与奥尔里奇空间

本文引进并研究了带奥尔里奇范数的Bessel场位(?)_n~φ(R~n),并证明了,在一定条件下,当k为正整数时,(?)_k~φ(R~n)与Orlicz-Sobolev空间L_k~φ(R~n)等价.从而推广了Calderon在[4]中关于(?)_a~p(R~n)的定理.     

仿射Bessel序列的判别

通过引进Bessel控制函数,并对它的充分性和必要性作研究,建立了对仿射Bessel序列的新的判别法.最后提出几个未解决的问题.     

Uniform convergence of basic Fourier–Bessel series on a q-linear grid

We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.

     

关于Bessel函数的素性

本文中我们证明了:Bessel函数J_ν(z)当ν为奇数时是素的,而当ν为偶数时,z~2是J_ν(z)仅有的右因子。     

Bessel变换的一个下界

Bessel逆问题在物理、化学和工程学等诸多领域有重要应用.解决线性逆问题的传统方法不适合处理具有奇异性曲线边缘的二元函数.鉴于切波对这一类函数的最优表示能力,相关文献采用切波方法研究Bessel逆问题,构造了目标函数的切波域值估计器,得到了它在函数空间V中积分均方差收敛阶的上界.在此基础上利用统计理论给出其最小最大风险的一个下界,证明了在估计Bessel逆问题时此估计器是最优的.     

A variant of Hardy's and Miyachi's theorems for the Bessel–Struve transform

In this paper, we establish new versions of Hardy's and Miyachi's theorems for the Bessel–Struve transform.     

Approximation of the Bessel function Jν(x) by numerical integration

An approximation is given for the Bessel function J_ν(x), with real ν, based on the application of a quasi-gaussian (i.e. gaussian with preassigned nodes in ?1 and +1) quadrature formula to the integral representation

$\text{J}{}_{\mathrm{\nu }}\text{(x)=}\text{(}\text{x}\text{2}\text{)}{}^{\mathrm{\nu }}\text{π}\text{Φ(ν+}\text{1}\text{2}\text{)}\text{∫}\text{?1}\text{1}\text{(1?t}{}^2\text{)ν?}\text{1}\text{2}\text{cos xt dt}$

In the case of integer values of ν, we compare our results with the approximation obtained by applying the trapezoidal rule to the representation

$\text{J}{}_{\mathrm{<mtext>n</mtext>}}\text{(x)=}\text{1}\text{π}\text{∫}\text{0}\text{π}\text{cos}\text{(}\text{x sin}\text{θ ?}\text{n}\text{θ)}\text{d}\text{θ}$

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Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe？cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Existence and stability of vector solitary waves for nonlinear Schrödinger systems of Hartree-type with Bessel potential

In this paper, we study a class of two-dimensional nonlinear Schrödinger systems of Hartree-type with Bessel potential kernel, which models the propagation and interaction of two-color light beams in nematic liquid crystals. The global well-posedness is proved by using fixed point argument, Gagliardo-Nirenberg inequality and conservation laws. In addition, we also obtain the existence and orbital stability of ground state vector solitary waves applying variational methods and Concentration-compactness Lemma.     

Bessel过程的位势及有关问题

<正> R~+=[0,+∞)上的连续保守扩散称为具有指数 v 的 Bessel 过程(简称 BE~v 或BE),如其无穷小生成在 C~2(0,+∞)(二次连续可微函数全体)上重合于微分算子     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

Hardy–Rellich identities with Bessel pairs

Archiv der Mathematik - We prove an identity that implies the classical Rellich inequality as well as several improved versions of Rellich type inequalities. Moreover, our equality gives a simple...