On a conservative integral equation with two kernels

We study the solvability of the integral equation

On the Integral Operators of the Third Kind

We establish some necessary and sufficient conditions for an operator in the space of square summable functions to be representable as a sum of multiplication by a bounded function and an integral operator.     

A degenerate kernel method for eigenvalue problems of compact integral operators

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by a degenerate kernel method. By interpolating the kernel of the integral operator in both the variables, we prove that the error bounds for eigenvalues and for the distance between the spectral subspaces are of the orders h ^2r and h ^r respectively. By iterating the eigenfunctions we show that the error bounds for eigenfunctions are of the orders h ^2r . We give the numerical results.      

Norm of a multilinear integral operator from product of weighted-type spaces to weighted-type space

We investigate the following multilinear integral operator $$T_K^m(\overrightarrow{f})(x)={\int }_0^{\infty }\dots {\int }_0^{\infty }K(x,t_1,\dots ,t_m){\displaystyle \prod _{j=1}^m}{f}_j(t_j)d{t}_1\dots d{t}_m,$$ where $m\in ?$ and $K:{?}_{+}^{m+1}\to {?}_{+}$ is a continuous kernel function satisfying the condition $$K(x,g_1(x)s_1,\dots ,g_m(x)s_m)=h(x)K(1,s_1,\dots ,s_m),$$ for some functions $g_j,j=\stackrel{￣}{1,m}$, which are continuous, increasing, $g_j({?}_{+})={?}_{+},j=\stackrel{￣}{1,m}$, and a function $h:{?}_{+}\to {?}_{+}$, from a product of weighted-type spaces to weighted-type spaces of real functions. We calculate the norm of the operator, extending and complementing some results in the literature. We also give an explanation for a relation between integrals of an L^p integrable function and its radialization on ${?}^n$.     

Consistency of general bootstrap methods for degenerate -type and -type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.     

Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

Solution of certain weakly singular integral equations of the first kind with indefinite parameters

In this paper we study a two-dimensional weakly singular integral equation of the first kind with logarithmic kernel. We construct a pair of spaces of the desired elements and the right-hand sides, where we prove the correctness of the problem under consideration and obtain inversion formulas for the integral operator.     

Bounds for Non-Gaussian Approximations of U-Statistics

Let X, ,X ₁,...,X _n be i.i.d. random variables taking values in a measurable space ( ). Consider U-statistics of degree two

Optimality of function spaces for kernel integral operators

We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space X we characterize its optimal range partner, that is, the smallest r.i. space Y such that the operator is bounded from X to Y. We apply the general results to Lorentz spaces to illustrate their strength.     

Generalized Hodges-Lehmann estimators for the analysis of variance

For X_i, …, X_n a random sample and K(·, ·) a symmetric kernel this paper considers large sample properties of location estimator satisfying , . Asymptotic normality of is obtained and two forms of interval estimators for parameter θ satisfying EK(X₁ − θ, X₂ − θ) = 0, are discussed. Consistent estimation of the variance parameters is obtained which permits the construction of asymptotically distribution free procedures. The p-variate and multigroup extension is accomplished to provide generalized one-way MANOVA. Monte Carlo results are included.     

On some properties of systems of Volterra integral equations of the fourth kind with kernel of convolution type

We consider the system of integral equations of the form Ax +V x = Ψ, where V is the Volterra operator with kernel of convolution type and A is a constant matrix, det A = 0. We prove an existence theorem and establish necessary and sufficient conditions for the kernel of the operator of the system to be trivial.     

Sampling Properties of U-statistics for a Class of Stationary Nonlinear Processes

We consider the sampling properties of U-statistics based on a sample of realization from a class of stationary nonlinear processes which include, in particular, linear, bilinear and finite order volterra processes. It is shown that if the size n of the realization tends to infinity then certain normalized versions of the U-statistics tend to be distributed normally with zero means and finite variances.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

The LIL for <Emphasis Type="Italic">U</Emphasis>-Statistics in Hilbert Spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest. R. Adamczak’s research partially supported by MEiN Grant 2 PO3A 019 30. R. Latała’s research partially supported by MEiN Grant 1 PO3A 012 29.     

Convergence in the integral metric of the general projective method for solving singular integral equations of the first kind with the Cauchy kernel

We study a projective method for solving singular integral equations of the first kind with the Cauchy kernel. Depending on the index of the equation, we introduce pairs of weight spaces which represent a restriction of the space of summable functions. We prove the correctness of the stated problem. We obtain sufficient conditions for the convergence of the projective method in the integral metric.     

On integrability in the LIL for degenerateU-statistics

It is shown by example that square integrability of the kennelh of a completely degenerateU-statistic is not a necessary condition for the law of the iterated logarithm to hold. Necessary integrability conditions are given onh which are in a sense the strongest possible. Sufficient conditions for a degenerateU-statistic of order 2 to satisfy the LIL, weaker than square integrability of the kernel, are also given.Research partially supported by NSF Grant No. DMS-93-00725. Part of this author's work was done at the University of Bielefeld, supported by the Deutsche Forschungsgemeinshaft, Sonderforschungbereich 343, Diskrete Structuren in der Mathematik.Research partially supported by the Army Research Office and the National Security Agency.     

Convergence of self-normalized generalizedU-statistics

Conditions are given for almost certain and distribution convergence of self-normalized generalizedU-statistics composed of random variables without particular probabilistic structure. The set of almost certain limit points of some classicalU-statistics is obtained. A variant of theU-statistic involving squares of some of the random variables is also treated. Applications include Martingale differences, stationary sequences, and the classical i.i.d. case where a Marcinkiewicz-Zygmund-type strong law is obtained.     

Limits of canonicalU-processes andB-valuedU-statistics

It is shown that the limite law of canonicalU-process is the law of a chaos process which has a versio with bounded and ·₂ continuous paths. This is also true forB-valued canonicalU-statistics with values in a separable Banach space. Some properties of Banach spaces of type 2 related withU-statistics are presented.Research partially supported by NSF Grants No. DMS-9000132 and No. DMS-8505550 and carried out in the University of Connecticut and the MSRI.     

Estimate for a class of multilinear fractional integral operator with rough kernel

In this paper, we prove the boundedness from H1(Rn) to L n/n-α ,∞(Rn) for the multilinear fractional integral operator with rough kernel related to the mi-th remainder of Taylor series of Ai at x about y for i = 1,2, ,l.