Rahul Garg, Sundaram Thangavelu
Considering functions $ f $ on $ \R^n $ for which both $ f $ and $ \hat{f} $ are bounded by the Gaussian $ e^{-{1/2}a|x|^2}, 0 < a < 1 $ we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for $ O(n)-$finite functions thus extending the one dimensional result of Vemuri.
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Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\al\in[-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validate that the known version of Hardy's inequality for the Hermite functions is sharp.
Hardy's inequality for Laguerre expansions of Hermite type with the index $\al\in(\{-1/2\}\cup[1/2,\infty))^d$ is proved in the multi-dimensional setting with the exponent $3d/4$. We also obtain the sharp analogue of Hardy's inequality with $L^1$ norm replacing $H^1$ norm at the expense of increasing the exponent by an arbitrarily small value.
In this article, we give a short proof of Hardy's inequality for Hermite expansions of functions in the classical Hardy spaces $H^p({\mathbb R^n})$, by using an atomic decomposition of the Hardy spaces associated with the Hermite operators. When the space dimension is $1$, we obtain a new estimate of Hardy's inequality for Hermite expansions in $H^p({\mathbb R})$ for the range $0<p<1.$
We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the Dunkl--Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by R. L. Frank, E. H. Lieb and R. Seiringer in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, wh…
On pair correlation of Hermite coefficients of functions from the Hardy class
本文研究 Hardy 类函数的 Hermite 系数成对相关性，在函数及其 Fourier 变换受 Gauss 函数控制的条件下，证明特定 Hermite 系数组合可获得优于已知精确估计的衰减率。
$γ$-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces
本文研究了 Hermite 算子下取值于 UMD Banach 空间的 Littlewood-Paley-Stein 函数，证明了 Hermite 平方函数在 BMO 和 Hardy 空间上的有界性，并给出了这些空间的等价范数及 UMD Banach 空间的新刻画。
Hermite expansions of functions from the weighted Hardy class
本文研究了具有加权Hardy类函数的Hermite展开，通过分析Log型权重和指数型权重，证明了Hermite系数衰减估计及谐振子解的衰减率，并改进了亚临界Hardy不确定性原理。
We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of $p$ and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This invalidates the conventional conjecture which was expected to be true.
Using methods of weight functions, techniques of real analysis as well as the Hermite-Hadamard inequality, a half-discrete Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor related to the Hurwitz zeta function and the Riemann zeta function is obtained. Equivalent forms, normed operator expressions, their reverses and some particular cases are also considered.
By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.
We study weighted Besov and Triebel--Lizorkin spaces associated with Hermite expansions and obtain (i) frame decompositions, and (ii) characterizations of continuous Sobolev-type embeddings. The weights we consider generalize the Muckhenhoupt weights.
In the present paper, we establish that Riesz transforms for Dunkl Hermite expansion as introduced in [4] are singular integral operators with Hörmander's type conditions and we show that are bounded on $L^p(\mathbb{R}^d; dμ_k) 1 < p < 1.
We express Wronskian Hermite polynomials in the Hermite basis and obtain an explicit formula for the coefficients. From this we deduce an upper bound for the modulus of the roots in the case of partitions of length 2. We also derive a general upper bound for the modulus of the real and purely imaginary roots. These bounds are very useful in the study of irreducibility of Wronskian Hermite polynomials. Additionally, we generalize some of our results to a larger class of polynomials.
We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.
Calderón-Zygmund operators and endpoint spaces for Hermite expansions
本文研究了与Hermite算子相关的Calderón-Zygmund型算子，建立了其在Hardy空间和Lipschitz空间等端点空间上有界的充要条件，并应用于分析Riesz变换和伪乘子的有界性。
Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $α\in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, are defined and investigated. To achieve expected goals Schwartz type spaces on $R^d_+$ are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calderón reproducing f…
Using well-known techniques, we establish Hardy-Littlewood-type and Hausdorff-Young-type inequalities for generalized Gegenbauer expansions and their unification.
In this article mixed norm estimates are obtained for some integral operators, from which those for the Hermite semigroup and the Bochner Riesz means associated with the Hermite expansions are deduced. Also, mixed norm estimates for the Littlewood Paley g functions and g* functions for the Hermite expansions are obtained, which lead to those for Hermite multipliers
We prove a Hardy inequality for ultraspherical expansions by using a proper ground state representation. From this result we deduce some uncertainty principles for this kind of expansions. Our result also implies a Hardy inequality on spheres with a potential having a double singularity.
Complete asymptotic expansion for a Durrmeyer variant of operators based on Hermite polynomials
本文研究G. Krech 2016年提出的基于二元埃尔米特多项式的正线性算子的Durrmeyer变体，建立其针对多项式增长局部可积函数的完全渐近展开，系数用函数导数明确表示，还得到Voronovskaja型公式。
In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel--Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the "branch cut" technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for …
Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type parameters $α,β\in(-1,\infty)^d$, whereas in the function systems for $α,β\in[-1/2,\infty)^d$. The ranges of these parameters are the widest in which the corresponding orthonormal bases are composed of bounded functions. Moreover, the sharp $L^1$-analogues of Hardy's inequality are obtained with the same restrictions on the parameters $α$ and $β$.
We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of $ L^2(\R^n) $ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.
Hardy's inequality on $H^p$ spaces, $p\in(0,1]$, in the context of orthogonal expansions is investigated for general basis on a subset of $\mathbb{R}^d$ with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat kernels are proved. Moreover, sharpness of studied Hardy's inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.
Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on $\R^d$ induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical spaces.
We compute Hermite expansions of some tempered distributions by using the Bargmann transform. In other words, we calculate the Taylor expansions of the corresponding entire functions. Our method of computations seems to be superior to the direct computations in the shifts of singularities and the higher dimensional cases.
The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few initial terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation.
Wavelet series expansion in Hardy spaces with approximate duals
该研究给出Hardy空间$H^p(\mathbb{R})$（$0<p\leq1$）中函数构成近似对偶的充分条件，据此得到带近似对偶的小波级数展开，方法无需精确对偶假设且可通过算子范数界检验函数适用性。
On the $L^2$ boundedness of pseudo-multipliers for Hermite expansions
本文研究Hermite伪乘子在$L^2$上的有界性条件，并推广到$L^p$及Gaussian测度设定，核心工具是新的分部积分公式。
Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q…
We develop a wavelet like representation of functions in $L^p(\mathbb{R})$ based on their Fourier--Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the local smoothness of the target function. In the case of continuous functions, a similar expansion is given based on the values of the functions at arbitrary points on the real line. In the process, we give new proofs for the localization of certain kernels, as well as some very classical estimates such as the Markov--Bernstein inequality.
Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$. These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_κ:=-Δ_κ+|x|^2$, where $Δ_κ$ stands for the Dunkl--Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesàro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters …
Appell Polynomials in Shifted Asymptotic Expansions: the Mills ratio, Hermite polynomials, and Stieltjes bounds
本文扩展了移位渐近展开的Appell多项式构造，建立了Borel-Laplace表示，推导了Mills-Hermite展开的余项估计与有限差分公式，将其应用于非中心高斯尾展开，并得到Stieltjes矩序列对应的双侧界层级。
Approximation of almost time and band limited functions I: Hermite expansions
本文研究了近似时间与带限函数在Hermite和缩放Hermite基展开中的逼近质量，推导了函数在具有固定紧支撑的$ L^2 $-Sobolev空间中的收敛速度。
A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for type index $\al\in[-1/2,\infty)^d$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^1$ norms in place of $H^1$ norms are obtained in both settings.
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.
We consider expansions of functions in $L^{p}(\mathbb{R},|x|^{2k}dx)$, $1\leq p<+\infty$ with respect to Dunkl-Hermite functions in the rank-one setting. We actually define the heat-diffusion and Poisson integrals in the one-dimensional Dunkl setting and study their properties. Next, we define and deal with Hilbert transforms and conjugate Poisson integrals in the same setting. The formers occur to be Calderón-Zygmund operators and hence their mapping properties follow from general results.
Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than $100$ the asymptotic methods are enough for a double precision accuracy computation ($15$-$16$ digits) of the nodes and weights of the Gauss--Hermite and Gauss-…
Riesz transforms, Hardy spaces and Campanato spaces associated with Laguerre expansions
本文研究与拉盖尔展开相关的Riesz变换，证明其为Calderón-Zygmund算子以解决公开问题，建立对应Hardy空间与Campanato空间理论，并证明该变换在这些空间上有界，完善了其有界性刻画。
Let $f_k$ be the $k$-th Fourier coefficient of a function $f$ in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on $f$ for the inequality $\sum_{k}|f_k|^2θ^k<\infty$ to hold with $θ>1$. As a by-product new orthogonality relations for the Hermite and Laguerre polynomials are found. The basic machinery for the proofs is provided by the theory of reproducing kernel Hilbert spaces.
Hardy spaces and Campanato spaces associated with Laguerre expansions and higher order Riesz transforms
本文针对拉盖尔微分算子\mathcal{L}_ν，建立了全范围\\\\(p \in (0,1]\\\\)的相关哈代空间理论，证明了对应BMO型空间是其对偶空间，并验证了高阶里斯变换在勒贝格空间等上的有界性。
We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are $L^p-L^q$ bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.
Convergence of Hermite expansions in modulation spaces
本文研究了调和展开在调制空间中的收敛性，证明了在 $1< p<+\infty$ 时收敛，而在 $p = 1,\infty$ 时可能发散，并给出了调和函数扎克变换的上界。
We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(\mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $ \mathbb{C}^n $ and Hermite operator on $ \mathbb{R}^n.$
We study negative powers of Laguerre differential operators in $\R$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution ty…
We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equatio…
In this paper we study linear and non-linear Fourier-Padé approximation for Angelesco systems of functions. This construction is similar to that of Hermite-Padé approximation. Instead of considering power series expansions of the functions in the system, we take their expansion in a series of orthogonal polynomials.
We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. As these series generally have support of order type larger than that of the set of natural numbers, the notion of asymptotic expansion itself needs to be generalized.
Let $μ$ be a probability measure on $\mathbb{T}$ that is singular with respect to the Haar measure. In this paper we study Fourier expansions in $L^2(\mathbb{T},μ)$ using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials $\{z^n\}_{n\in \mathbb{N}}$ is effective in $L^2(\mathbb{T},μ)$, it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame $P_φ(z^n)$ for the model subspace $\mathcal{H}(φ)= H^2 \ominus φH^2$, where $P_φ$ is the orthogonal projectio…
In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces $L^p_α(γ_d)$ are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in t…
In this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the Hausdorff conditions, which play a relevant role in the moment problem theory. We prove that these series converge to functions analytic in cut domains. We are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the Pollaczek polynomials. We can thus furnish a solution for a class of Cauchy integral equations.
The Painleve-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed n by m rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as m and n tend to infinity with r=m/n fixed. Generalizing a method of Bertola and Bothner used to study rational Painleve-II func…
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