Lazhar Dhaouadi
In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated $q$-Fourier-Neumen expansion which involves the $q$-little Jacobi polynomials.
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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.
We study the problem of an appropriate choice of derivatives associated with discrete Fourier-Bessel expansions. We introduce a new so-called essential measure Fourier-Bessel setting, where the relevant derivative is simply the ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in this context. Our main results are $L^p$-boundedness of the Riesz transforms (even in a multi-dimensional situation) and an isomorphism between the Sobolev and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect various comments concerning two other closely related Fou…
In this paper we uses an I.I. Hirschman-W. Beckner entropy argument to give an uncertainty inequality for the $q$-Bessel Fourier transform: $$ \mathcal{F}_{q,v}f(x)=c_{q,v}\int_{0}^{\infty}f(t)j_{v}(xt,q^{2})t^{2v +1}d_{q}t, $$ where $j_v(x,q)$ is the normalized Hahn-Exton $q$-Bessel function.
In this note, we shall prove a formula for the Fourier transform of spherical Bessel functions over complex numbers, viewed as the complex analogue of the classical formulae of Hardy and Weber. The formula has strong representation theoretic motivations in the Waldspurger correspondence over the complex field.
We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are of strong type $(p,q)$, of weak type $(p,q)$ and of restricted weak type $(p,q)$. These results may be thought of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of t…
When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{kν}$ and the "shifted" zeros, $qj_{kν}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the "shifted" ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the "shifted" zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.
The aim of this paper is to establish an analogue of Logvinenko-Sereda's theorem for the Fourier-Bessel transform (or Hankel transform) $\ff_α$ of order $α>-1/2$. Roughly speaking, if we denote by $PW_α(b)$ the Paley-Wiener space of $L^2$-functions with Fourier-Bessel transform supported in $[0,b]$, then we show that the restriction map $f\to f|_Ω$ is essentially invertible on $PW_α(b)$ if and only if $Ω$ is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map. As a side result we prove a Bernstein type inequality for the Fourier-Bessel transform.
The aim of this paper is to prove two new uncertainty principles for the Fourier-Bessel transform (or Hankel transform). The first of these results is an extension of a result of Amrein-Berthier-Benedicks, it states that a non zero function $f$ and its Fourier-Bessel transform $\mathcal{F}_α(f)$ cannot both have support of finite measure. The second result states that the supports of $f$ and $\mathcal{F}_α(f)$ cannot both be $(\eps,α)$-thin, this extending a result of Shubin-Vakilian-Wolff. As a side result we prove that the dilation of a $\cc_0$-function are linearly independent. We also exte…
In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura-Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.
We prove qualitatively sharp heat kernel bounds in the setting of Fourier-Bessel expansions when the associated type parameter $ν$ is half-integer. Moreover, still for half-integer $ν$, we also obtain sharp estimates of all kernels subordinated to the heat kernel. Analogous estimates for general $ν> -1$ are conjectured. Some consequences concerning the related heat semigroup maximal operator are discussed.
The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.
Hardy's theorem for the q-Bessel Fourier transform
本文给出了q-Bessel傅里叶变换的Hardy定理q-类比。该定理指出，若函数f及其傅里叶变换满足指数衰减条件，则f为常数乘以高斯函数。
Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform
针对傅里叶-贝塞尔变换，研究人员建立了其定量唯一性与分形不确定性原理，通过适配正则集合的阻尼函数及多尺度迭代等方法，克服了经典傅里叶论证无法直接应用的问题，得到了相关函数估计式。
We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.
In this paper we prove $L^p$ estimates for Stein's square functions associated to Fourier-Bessel expansions. Furthermore we prove transference results for square functions from Fourier-Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of $p$ for the $L^p$-boundedness of Fourier-Bessel Stein's square functions from the corresponding property for Hankel-Stein square functions. Finally, we deduce $L^p$ estimates for Fourier-Bessel multi…
We study Hardy spaces for Fourier--Bessel expansions associated with Bessel operators on $((0,1), x^{2ν+1}\, dx)$ and $((0,1), dx)$. We define Hardy spaces $H^1$ as the sets of $L^1$-functions for which their maximal functions for the corresponding Poisson semigroups belong to $L^1$. Atomic characterizations are obtained.
Variation operators for semigroups associated with Fourier-Bessel expansions
本文研究了 Fourier-Bessel 展开相关半群定义的变差算子，并建立了其 L^p 有界性性质。
A partial converse to the Riemann--Lebesgue lemma for Bessel--Fourier series of order zero
本文证明零阶贝塞尔–傅里叶系数衰减的黎曼–勒贝格引理在 $0\leq α<1/2$ 时存在部分逆命题，并猜想该结论可推广到 $α=1/2$。
On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields
本文证明了复数上贝塞尔函数的Weber-Schafheitlin型积分公式，应用于扩展A.Venkatesh在GL₂上Sym²的超越端oscopy工作至任意数域。
A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform
本文基于控制理论中的衰减率问题，研究了傅里叶-贝塞尔变换的高频不确定性原理，证明了在特定支撑条件下的函数范数估计，并应用于求解阻尼波方程的衰减率。
We establish good numerical estimates for a certain class of integrals involving sixfold products of Bessel functions. We use relatively elementary methods. The estimates will be used in the study of a sharp Fourier restriction inequality on the circle.
Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
Multivariate Bessel functions and multivariate Hankel transforms
本文将贝塞尔函数推广到多维，并相应地将汉克尔变换在微分下的性质推广到多维情形。
New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich-Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.
Partial Fourier transforms are used to find explicit formulas for two remarkable fundamental solutions for a generalized Tricomi operator. These fundamental solutions reflect clearly the mixed type of the operator. In order to prove these results, we establish explicit formulas for Fourier transforms of some type of Bessel functions.
In this paper, an analogous of Heisenberg inequality is established for Laguerre-Bessel transform. Also, a local uncertainty principle for this transform is investigate
Discrete analogs of the index transforms, involving Bessel and Lommel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
Discrete analogs of the index transforms with squares of Bessel functions of the first and second kind $J_ν(z),\ Y_ν(z)$ are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
In this article we prove dimension free $L^p$-boundedness of Riesz transforms associated with a Bessel diferential operator. We obtain explicit estimates of the $L^p$-norms for the Bessel-Riesz transforms in terms of p, establishing a linear behaviour with respect to p. We use the Bellman function technique to prove a bilinear dimension free inequality involving Poisson semigroups defined through this Bessel operator.
In this paper we consider the Bessel-Struve operator $l_α$ and the Bessel-Struve intertwining operator $χ_α$ and its dual, we define and study the Bessel-Struve Sonine transform $S_{α,β}$ on $\mathcal{E}(\mathbb{R})$. We prove that $S_{α,β}$ is a transmutation operator from $l_α$ into $l_β$ on $\mathcal{E}(\mathbb{R})$ and we deduce similar result for its dual $S_{α,β}^*$ on $\mathcal{E}'(\mathbb{R})$. Furthermore, invoking Weyl integral transform and the Dual Sonine transform $^tS_{α,β}$ on $\mathcal{D}(\mathbb{R})$, we get a relation between the Bessel-Struve transforms $\mathcal{F}^α_{BS} $…
A finite transformation method is introduced. This method is equivalent to the $Z$ transform method to a certain extent but generalizes it. By applying the presented method to the Bessel functions, it is possible to solve related ordinary differential equations of order higher than two with given initial conditions.
In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel-Clifford fun…
In this paper, we introduce the Bessel-Struve transform, we establish an inversion theorem of the Weyl integral transform associated with this transform, in the case of half integers, we give a characterization of the range of $\mathcal{D}(\mathbb{R})$ by Bessel-Struve transform and we prove a Schwartz-Paley-Wiener theorem on $\mathcal{E}'(\mathbb{R})$.
We obtain new inequalities for the modified Bessel function of the second kind $K_ν$ in terms of the gamma function. These bounds follow as special cases of inequalities that we derive for the kernel of the Krätzel integral transformation.
New index transforms, involving the square of Bessel functions of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third order partial differential equation, involving the Laplacian, is obtained.
Based on known definite integrals of Bessel functions of the first kind, we obtain exact solutions to unknown definite integrals using the method of integral transforms from Hankel's transform.
Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting
该研究在Dunkl框架下探究Dunkl型Hankel变换的解析性质，引入Bessel函数与K-Bessel函数的Dunkl类似物，研究zeta积分的分布推广及正则性，证明关联zeta分布与Dunkl变换的函数方程。
New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the mod…
Let $λ>0$ and $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. We first introduce and obtain an equivalent characterization of ${\rm CMO}(\mathbb R_+,\, x^{2λ}dx)$. By this equivalent characterization and establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function $b\in {\rm BMO}(\mathbb R_+,\, x^{2λ}dx)$ is in ${\rm CMO}(\mathbb R_+,\, x^{2λ}dx)$ if and only if the Riesz transform commutator $[b, R_{Δ_λ}]$ is compact on $L^p(\mathbb R_+, x^{2λ}dx)$ for any $p\in(1, \infty)$.
In this article we shall study the analytic theory and the representation theoretic interpretations of Hankel transforms and fundamental Bessel kernels of an arbitrary rank over an archimedean field.
On a Bessel function series related to the Riemann xi function
本文研究了与黎曼xi函数相关的贝塞尔函数级数，并证明其解决圆柱热方程的边界值问题。
We determine the asymptotic behaviour of the $n$th derivatives of the Bessel functions $J_ν(a)$ and $K_ν(a)$, where $a$ is a fixed positive quantity, as $n\to\infty$. These results are applied to the asymptotic evaluation of two incomplete Laplace transforms of these Bessel functions on the interval $[0,a]$ as the transform variable $x\to+\infty$. Similar evaluation of the integrals involving the Bessel functions $Y_ν(t)$ and $I_ν(t)$ is briefly mentioned.
Fourier inversion theorems for integral transforms involving Bessel functions
本文研究了广义Weber-Orr变换的逆定理、谱分解及Plancherel-Parseval恒等式，通过偏微分方程推导并完善了经典和广义变换的反演公式。
Motivated by Dunkl operators theory, we consider a generating series involving a modified Bessel function and a Gegenbauer polynomial, that generalizes a known series already considered by L. Gegenbauer. We actually use inversion formulas for Fourier and Radon transforms to derive a closed formula for this series when the parameter of the Gegenbauer polynomial is a strictly positive integer. As a by-product, we get a relatively simple integral representation for the generalized Bessel function associated with even dihedral groups when both multiplicities sum to an integer. In particular, we re…
In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.
Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc
本文评述圆盘上 Bessel–Fourier 级数的收敛问题，概述 Balodis–Córdoba 的混合范数结果，并将其修改推广到 $2\leq p<4$ 时的 $L^p(\mathbb{D},r\mathrm{d}r\mathrm{d}t)$ 范数收敛。
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
We derive integral representations in terms of the Macdonald functions for the square modulus $s\mapsto | Γ( a + i s ) |^2$ of the Gamma function and its Fourier transform when $a<0$ and $a\not= -1,-2,\ldots $, generalizing known results in the case $a>0$. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of the Fokker-Planck equation.
Hardy spaces, Campanato spaces and higher order Riesz transforms associated with Bessel operators
本文针对多元Bessel算子$Δ_ν$，建立了相关Hardy空间与BMO型空间理论，证明其高阶Riesz变换为Calderón-Zygmund算子且在上述空间上有界，推进了Bessel算子框架下的调和分析研究。
Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients of such expansions are integrals over products of Bessel functions in problems involving cylindrical geometry. In this paper, certain identities involving products of two and three general solutions of Bessel's equation have been derived. Some of these identities have been useful in the study of Turner relaxation of annular magnetized plasma [S.K.H. Auluck,…
This paper was published in the special issue of the Journal of Inequalities and Special Functions dedicated to Professor Ivan Dimovski's contributions to different fields of mathematics: transmutation theory, special functions, integral transforms, function theory etc. In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them ar…
In this paper, the mean value formula depends on the Bessel generalized shift operator corresponding to the solutions of the boundary value problem related to the Bessel operator are studied. In addition to, Riesz Bessel transforms related to the Bessel operators are studied. Since Bessel generalized shift operator is translation operator corresponding to the Bessel operator, we construct a family of RBxj by using Bessel generalized shift operator. Finally, we analysis weighted inequalities involving Riesz Bessel transforms .
Discrete analogs of the Lebedev transforms with the product of the modified Bessel functions are introduced and investigated. Several expansions of suitable functions and sequences in terms of the series and integrals, involving the modified and incomplete Bessel functions are established.
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.
In the paper we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singula…
In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.
Recently renewed interest in the Lobachevsky-type integrals and interesting identities involving the cardinal sine motivate an extension of the classical Parseval formula involving both periodic and non-periodic functions. We develop a version of the Parseval formula that is often more practical in applications and illustrate its use by extending recent results on Lobachevsky-type integrals. Some previously known, interesting identities are re-proved in a more transparent manner and new formulas for integrals involving cardinal sine and Bessel functions are given.
Let $\operatorname{St}^{n}_{k}\subset\mathbb{R}^{n\times k}$ be the set of all $n\times k$ matrices whose columns are mutually orthogonal and of unit Euclidean length, and let $μ_{n,k}$ be the surface measure corresponding to this embedding. We calculate the first term of the asymptotic expansion of the Fourier transform of $μ_{n,k}$ for most directions, using the method of stationary phase. The asymptotic behavior near the remaining directions is unknown. We note some interesting connections to trace moments of orthogonal matrices, discrete random walks, Bessel functions, and pose some questi…
Calderón-Zygmund operators in the Bessel setting
本文研究了 Bessel 算子相关的调和分析基本算子，证明其为齐次型空间意义下的 Calderón-Zygmund 算子，从而由一般理论导出其映射性质。
Fix $λ>0$. Consider the Bessel operator $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel--Lizorkin spaces associated with $\triangle_λ$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.
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