A KERNEL-TYPE ESTIMATOR OF A QUANTILE FUNCTION UNDER RANDOMLY TRUNCATED DATA

A kernel-type estimator of the quantile function Q(p) = inf {t : F(t)≥p}, 0≤p≤1, is proposed based on the kernel smoother when the data are subjected to random truncation. The Bahadur-type representations of the kernel smooth estimator are established, and from Bahadur representations the authors can show that this estimator is strongly consistent, asymptotically normal, and weakly convergent.     

Some properties of analytic in a disk functions with applications to the study of the behavior of singular integrals

We obtain representations for an analytic in a disc function such that its real part has a zero of an integer order at a fixed boundary point. We consider certain applications of these representations for studying properties of singular integrals with Hilbert kernel.     

Representations of metabelian groups

A knowledge of the simple representation theory of finite abelian groups is useful for understanding the representations of solvable groups, since these provide the one-dimensional representations. The representation theory of metabelian groups (those G with abelian commutator subgroup G′) would seem to be a natural next level.In this paper we shall show that these representations, too, may be simply described in several ways: they are induced from linear representations of some explicity defined subgroups; their degrees may be calculated from a knowledge of the subgroups of G; these degrees depend only on the kernel of the representation (in fact, only on the intersection of this kernel with G′). As an application of these results, we can calculate for metabelian groups a certain measure of group-commutativity studied in an earlier paper [4].     

A note on linearized polynomials and the dimension of their kernels

Recently explicit representations of the class of linearized permutation polynomials and the number of such polynomials were given in Zhou (2008) [4] and Yuan and Zeng (2011) [3]. In this paper, we generalize this result to linearized polynomials with kernel of any given dimension, solving an open problem in Charpin and Kyureghyan (2009) [1]. Moreover, more explicit representations of such polynomials are given and several classes of explicit linearized polynomials with kernel of any given dimension are presented.     

A Domain Integral Equation for the Bergman Kernel

Certain integral operators involving the Szegö, the Bergman and the Cauchy kernels are known to have the reproducing property. Both the Szegö and the Bergman kernels have series representations in terms of an orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting a reproducing property for a space associated with the Bergman kernel. The construction leads to a domain integral equation for the Bergman kernel.^{1 2}     

A new construction of the Clifford-Fourier kernel

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.     

Harmonic analysis and systems of covariance for phase space representation of the poincaré group

Continuing some earlier work on the Galilei group, the spectral resolution of phase space representations of the Poincaré group is achieved by deriving all possible decompositions into irreducible representations corresponding to reproducing, kernel Hilbert spaces. Systems of covariance related to quantum measurements performed with extended test particles are analyzed, and questions of global unitarity discussed.Supported in part by NSERC Research Grants.     

一种基于Bhattacharyya核的SVM

包括图像识别在内的很多应用领域里,把单个样本表示成向量的集合的形式是很自然的想法,利用一个合适的核函数我们可以把这些向量映射到一个更高维的Hilbert空间,在这个高维空间里用Kernel PCA方法找到样本的高斯分布族,这样就可以把样本上的核函数定义成它们所服从的高斯分布密度函数的Bhattacharrya仿射.这样得到的核函数具有比较好的性质,比如说在各种变换下有稳定性表现,从而也说明了即使还有别的表示样本的方法,用向量集合的形式来表示单个的样本也是具有合理性的.     

Tensor product characterizations of mixed intersections of non quasianalytic classes and kernel theorems

Mixed intersections of non quasi‐analytic classes have been studied in [12]. Here we obtain tensor product representations of these spaces that lead to kernel theorems as well as to tensor product representations of intersections of non quasi‐analytic classes on product of open or of compact sets (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

Dilations of Some VH-Spaces Operator Valued Invariant Kernels

We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifies many dilation results, in particular B. Sz.-Nagy??s and Stinesprings?? dilation type theorems.     

Calkin Representations of Unbounded Operator Algebras Acting on Non-Separable Domains

There are constructed representations of unbounded operator algebras which generalize representations of B(H) constructed by J. W. CALKIN and H. BEHNCKE. For a large class of unitary spaces D, each uniformly closed two-sided ideal of the maximal Op*-algebra L⁺ (D) appears as kernel of such a representation. Irreducibility of the representations is characterized in terms of properties of ultrafilters which define the representations.     

Probability density estimation with surrogate data and validation sample

The probability density estimation problem with surrogate data and validation sample is considered. A regression calibration kernel density estimator is defined to incorporate the information contained in both surrogate variates and validation sample. Also, we define two weighted estimators which have less asymptotic variances but have bigger biases than the regression calibration kernel density estimator. All the proposed estimators are proved to be asymptotically normal. And the asymptotic representations for the mean squared error and mean integrated square error of the proposed estimators are established, respectively. A simulation study is conducted to compare the finite sample behaviors of the proposed estimators.     

Generalized power functions

In this paper some relations for the kernels of the Carleman–Vekua equation, in particular the representations of these kernels in the form of generalized power functions completely analogous to the well-known elementary Cauchy kernel expansion, are studied. The obtained results are applied to some problems of the theory of generalized analytic functions.     

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups \(\mathrm {PO}(p,q)\) by considering their action on the associated pseudo-Riemannian hyperbolic space \(\mathbb {H}^{p,q-1}\) in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.     

Automatic Feature Selection via Weighted Kernels and Regularization

Selecting important features in nonlinear kernel spaces is a difficult challenge in both classification and regression problems. This article proposes to achieve feature selection by optimizing a simple criterion: a feature-regularized loss function. Features within the kernel are weighted, and a lasso penalty is placed on these weights to encourage sparsity. This feature-regularized loss function is minimized by estimating the weights in conjunction with the coefficients of the original classification or regression problem, thereby automatically procuring a subset of important features. The algorithm, KerNel Iterative Feature Extraction (KNIFE), is applicable to a wide variety of kernels and high-dimensional kernel problems. In addition, a modification of KNIFE gives a computationally attractive method for graphically depicting nonlinear relationships between features by estimating their feature weights over a range of regularization parameters. The utility of KNIFE in selecting features through simulations and examples for both kernel regression and support vector machines is demonstrated. Feature path realizations also give graphical representations of important features and the nonlinear relationships among variables. Supplementary materials with computer code and an appendix on convergence analysis are available online.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

Differential operators for a scale of Poisson type kernels in the unit disc

In this paper, we introduce a scale of differential operators which is shown to correspond canonically to a certain scale of solution kernels generalizing the classical Poisson kernel for the unit disc. The scale of kernels studied is very natural and appears in many places in mathematical analysis, such as in the theory of integral representations of biharmonic functions in the unit disc.