Ashish Pathak, Dileep Kumar
In this paper, we extend the concept of continuous Bessel wavelet transform in $L^p$-space and derived the Parseval's as well as the inversion formulas. By using Bessel wavelet coefficients we characterized the Besov- Hankel space.
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In this paper we consider the generalized little Hankel and Berezin type operators on Besov spaces of holomorphic functions on polydiscs and give results about the boundedness of these operators.
Pseudo-differential operators associated with the fractional Hankel-Bessel transform
本文提出了一种基于分数Hankel-贝塞尔变换的伪微分算子理论，通过引入分数傅里叶型相位，构建了新的符号类并研究了其在加权L^p空间和分数Sobolev空间上的有界性。
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。
Besov spaces and product BMO over polydisc
该研究证明了解析Besov空间$B_{p,p}^{1/p}(\mathbb{C}_+^n)$连续嵌入Chang–Fefferman乘积BMO，进而得出有限Schatten类小Hankel算子具有乘积BMO符号的结论，还给出Fejér均值的相关条件。
Besov spaces and Schatten class Hankel operators for Hardy and Paley--Wiener spaces in higher dimensions
该研究针对高维凸区域上的Paley–Wiener空间，建立Paley–Wiener型Besov空间理论，明确Hankel算子属于Schatten类Sᵖ的充要条件，并拓展了相关结果的适用范围。
The Pseudo-Differential Operator (p.d.o.) $h_{μ,a}$ associated with the Bessel Operator involving the symbol $a(x,y)$ whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. $h_{μ,a}$ on Hankel translation $τ$ and Hankel convolution of Gevrey functions is continuous linear map into another Gevrey spaces.
Besov spaces in operator theory
本综述探讨了Besov类在算子理论中的多种应用，包括描述Schatten-von Neumann类的Hankel算子、Hilbert空间上幂有界算子多项式的范数估计，以及在算子扰动理论中的重要作用。
In the paper we investigate the properties of spaces with generalized smoothness, such as Calderón spaces that include the classical Nikolskii-Besov spaces and many of their generalizations, and describe differential properties of generalized Bessel potentials that include classical Bessel potentials and Sobolev spaces. Kernels of potentials may have non-power singularity at the origin. With the help of order-sharp estimates for moduli of continuity of potentials, we establish the criteria of embeddings of potentials into Calderón spaces, and describe the optimal spaces for such embeddings.
Within the new concept of a local iterated function system (local IFS), we consider a class of attractors of such IFSs, namely those that are graphs of functions. These new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature. For a class of local fractal functions, we derive explicit conditions for them to be elements of Besov and Triebel--Lizorkin spaces. These two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical …
If $E$ is an operator space, the non-commutative vector valued $L^p$ spaces $S^p[E]$ have been defined by Pisier for any $1 \leq p \leq \infty$. In this paper a necessary and sufficient condition for a Hankel matrix of the form $(a_{i+j})_{0 \le i,j}$ with $a_k \in E$ to be bounded in $S^p[E]$ is established. This extends previous results of Peller where $E=\C$ or $E=S^p$. The main theorem states that if $1 \leq p < \infty$, $(a_{i+j})_{0 \le i,j}$ is bounded in $S^p[E]$ if and only if there is an analytic function $ϕ$ in the vector valued Besov Space $B_p^{1/p}(E)$ such that $a_n = \hat ϕ(n)$…
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and pote…
Local spectral density and Slepian concentration for spherical Fourier-Bessel truncation spaces
研究\(\mathbb{R}^d\)中球傅里叶 - 贝塞尔截断空间的对角核渐近性和浓度谱，在耦合渐近区域\(N/K\to\kappa\)确定归一化对角再生核极限分布，证明渐近双峰特征值分布和香农数公式，揭示密度转变及与\(\kappa\)的关系。
Recently, with the progress of science and the characteristic properties that distinguish the Slepian system called Prolate spheroidal wave functions from the others orthonormal systems, it became clear its important contributions in several areas such as mathematical statistics, signal processing, numerical analysis etc...The main issue of this work is to establish the convergence quality of the truncated error of a function $f$ from the Hankel Sobolev space $\mathcal{H}_α^r$, $r > 0$ and $α\geq -1/2$, by a generalized prolate spheroidal wave functions basis in $L^2$ norm. In the meantime, we…
Hankel Multipliers of Laplace Transform Type
本文证明了 $(0,1)^n$ 上 Laplace 变换类型的 Hankel 乘子具有弱型 (1,1) 性质，并分析了 Bessel 算子虚幂的 Lp 有界性。
We will prove an analogue of Landau's necessary conditions [Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967).] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier-Bessel frames are obtained.
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.
We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. More Precisely, we investigate the inequalities \begin{equation*} \big\|f\big\|_{\dot{k}_{v,σ}^{α_{1},r}}\leq c\big\|f\big\|_{\dot{K}_{u}^{α_{2},δ}}^{1-θ}\big\|f\big\|_{\dot{K}_{p}^{α_{3},δ_{1}}A_{β}^{s}}^{θ}, \end{equation*} and \begin{equation*} \big\|f\big\|_{\mathcal{E}_{p,2,u}^{σ}}\leq c\big\|f\big\|_{\mathcal{M}_{μ}^{δ}}^{1-θ}\big\|f\big\|_{\mathcal{N}_{q,β,v}^{s}}^{θ}, \end{equation*} with some appropriate assumptions on the para…
In this paper we determine the asymptotics of the determinant of Bessel operators for sufficiently smooth generating functions. These operators are similar to Wiener-Hopf operators with the Fourier transform replaced by the Hankel transform and thus the asymptotics of the determinanst are similar to the well-known Szegö-Akhiezer-Kac formula for truncated Wiener-Hopf determinants. In order to compute the above, we also show that the Bessel operators differ from the Wiener-Hopf by a Hilbert-Schmidt operator.
Discrete Besov spaces via semigroups associated to the discrete Laplacian and regularity of non-local operators
本文通过离散拉普拉斯算子的热与泊松半群刻画离散Besov空间，并据此证明离散拉普拉斯分数幂和贝塞尔势的正则性，同时给出离散热核导数的新估计。
In a recent paper we unified Bessel functions of different orders .Here we extend the unification to other linairely independant solutions to Bessel equation, Neumann's and Hankel's functions
We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $\R^d$, equipped with power weights $w(x) = |x|^γ$, $γ>-d$. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.
We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the C*-algebra O_{ν+1} arising from this multiresolution analysis.
In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted $L_{q}$-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt $A_{\infty}$-class. In Besov space case we have the restriction that the microscopic parameter equals to $q$. Going beyond the $A_{p}$-range, where $p$ is the integrability paramet…
The paper is devoted to Mellin convolution operators with meromorphic kernels in Bessel potential spaces. We encounter such operators while investigating boundary value problems for elliptic equations in planar 2D domains with angular points on the boundary. Our study is based upon two results. The first concerns commutants of Mellin convolution and Bessel potential operators: Bessel potentials alter essentially after commutation with Mellin convolutions depending on the poles of the kernel (in contrast to commutants with Fourier convolution operatiors.) The second basic ingredient is the resu…
Schatten--Lorentz characterization of Riesz transform commutator associated with Bessel operators
本文用鞅和二进分析刻画了与Bessel算子相关的Riesz变换换位子的Schatten--Lorentz范数，建立了其与符号振荡空间及Besov范数的等价关系，并确定了临界指标为Bessel测度的下维数。
For a fixed reals $c>0$, $a>0$ and $α>-\frac{1}{2}$, the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions as some authors call it, are the eigenfunctions of the finite Hankel transform operator, denoted by $\mathcal{H}_c^α$, which is the integral operator defined on $L^2(0,1)$ with kernel $H_c^α(x,y)=\sqrt{cxy}J_α(cxy)$. Also, they are the eigenfunctions of the positive, self-adjoint compact integral operator $\mathcal{Q}_c^α=c\mathcal{H}_c^α\mathcal{H}_c^α.$ The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a re…
$A_\infty$-invariance of oscillatory norms, and Schatten characterisations of commutators
本文扩展已有交换子Schatten类刻画的抽象框架，引入$A_\infty$等价测度对放宽正则性假设，用实变调和分析方法复现并简化了Bessel-Riesz变换交换子的相关刻画结果。
A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weighted function spaces. Besides of the derivation of Sobolev-type embedding results, sharp trace theorems, point-wise multiplier properties, and interpolation characterizations particular emphasize is put on spaces distinguished by boundary conditions. This work is the fundament for…
We show that smoothness implies norm-controlled inversion: the smoothness of an element $a$ in a Banach algebra with a one-parameter automorphism group is preserved under inversion, and the norm of the inverse $a^{-1}$ is controlled by the smoothness of $a$ and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra-smoothness, we resort to Dales-Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm c…
The purpose of this paper is to establish L^p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L^p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L^p norm of the function itself. An important step in its proof involves measuring the L^p stability of functions in the approximating space in terms of the l^p norm of the coefficients involved. As an application of the B…
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