On the Classification of the Spectrum of Second Order Difference Operators

We study nonsymmetric second order difference operators acting in the Hilbert space 𝓁² and describe the resolvent set and the essential spectrum of such operators in terms of related formal orthogonal polynomials. As an application, we obtain new results on the growth of orthonormal polynomials outside and inside the support of the underlying measure of orthogonality.     

Matrix Continued Fraction for the Resolvent Function of the Band Operator

The aim of this paper is the expansion of a matrix function in terms of a matrix-continued fraction as defined by Sorokin and Van Iseghem. The function under study is the Weyl function or resolvent function of an operator, given in the standard basis by a bi-infinite band matrix, with p subdiagonals and q superdiagonals, where the p + q – 1 intermediate diagonals are zero. The main goal of this paper is to find, for the moments, an explicit form in terms of the coefficients of the continued fraction, called genetic sums, which lead to a generalization of the notion of a Stieltjes continued fraction. These results are extension of some results already found for the vector case (p = 1) and are a step in the direction towards the solution of the direct and inverse spectral problem. The actual computation of the approximants of the given function as the convergents of the continued fraction is shown to be effective. Some possible extensions are considered.     

Jackson算子关于二阶连续模的最佳逼近常数

求了用Jackson算子Jn（f.,x）逼近函数f（x）（∈C2n）时关于二阶连续模的最佳逼近常数：及用阶数不超过n的三角多项式H对连续函数f（x）的最佳逼近En（f）,的上界估计：     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

On Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by _n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class E_p(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator A_f constructed from fL_p(Γ), 1p<∞, in terms of the best approximation Δ_n,p of f in the space L_p(Γ) by functions belonging to the class _n,p(G).     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

On statistical approximation of a general class of positive linear operators extended in q-calculus

In this paper we present a general class of positive linear operators of discrete type based on q-calculus and we investigate their weighted statistical approximation properties by using a Bohman–Korovkin type theorem. We also mark out two particular cases of this general class of operators.     

Approximation of a function given by its Laurent series

Approximants are defined for a function which is holomorphic in an annulus. They are shown to have good qualitative properties whenf is meromorphic with a fixed number of poles in the annulus. Their denominators are linked to the reverse orthogonal polynomials of dimension 2, or orthogonal polynomials of dimension –2, following the choice of the parameters. Their numerators then follow the same recurrence relation as the denominators.This work was supported by the Human Capital and Mobility Programme of the European Community, Project ROLLS, Contract CHRX-CT93-0416(DG 12 Coma).     

On the Defect Index of Quadratic Self-Adjoint Operator Pencils

Mathematical Notes -     

Criteria for the strong regularity of J-inner functions and γ-generating matrices

The class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A₂) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of γ-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. A determinental criterion is also presented.     

Some properties of the Bézier–Kantorovich type operators

The aim of this paper is to present estimates for the rate of pointwise convergence of the Bézier–Kantorovich modification of the discrete Feller operators in some classes of measurable functions bounded on an interval I, in particular, for functions of bounded pth power variation on I. Our theorems generalize and extend the recent results of Zeng and Piriou (J. Approx. Theory 95(1998) 369; 104(2000) 330) for the kantorovichians of the Bernstein–Bézier operators in the class of functions of bounded variation in the Jordan sense on [0,1].     

On the ω-problem for some types of non-metrizable spaces

The ω-problem on a topological space X consists in finding out whether there exists a function whose oscillation is equal to a given upper semi-continuous (USC) function f:X→[0,∞] vanishing at isolated points of X. If such F exists, we call it an ω-primitive for f. Unlike the case of metrizable spaces, an ω-primitive need not exist if X is not metrizable. We study the ω-problem for f taking the value ∞ in the case of ordinal space, products of regular “constancy” spaces and the wedge sums of such spaces. Some open problems are formulated.     

On the complexity of the approximate table representation of discrete analogs of functions of finite smoothness in the metric ofL P

For discrete analogs of classes of functions of finite smoothness, we study the quantity log Approx characterizing the minimal necessary length of tables that allow us to reconstruct functions from these classes with error not exceeding 1 in the metric of the spaceL ^p . Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 643–647, November, 1998. The author wishes to express his gratitude to his scientific adviser O. B. Lupanov for setting the problem and attention to the author's work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01068.     

On the R-Order of Convergence of a Family of Methods for Simultaneous Extraction of Part of All Roots of Algebraic Polynomials

This note deals with the R-order of convergence of Weierstrass-Durand-Kerner-Dochev type single-step methods for the simultaneous determination of only a part of all roots of algebraic polynomials.     

On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction

In this paper we complement recent work of Maischak and Stephan on adaptive hp-versions of the BEM for unilateral Signorini problems, respectively on FEM-BEM coupling in its h-version for a nonlinear transmission problem modelling Coulomb friction contact. Here we focus on the boundary element method in its p-version to treat a scalar variational inequality of the second kind that models unilateral contact and Coulomb friction in elasticity together. This leads to a nonconforming discretization scheme. In contrast to the work cited above and to a related paper of Guediri on a boundary variational inequality of the second kind modelling friction we take the quadrature error of the friction functional into account of the error analysis. At first without any regularity assumptions, we prove convergence of the BEM Galerkin approximation in the energy norm. Then under mild regularity assumptions, we establish an a priori error estimate that is based on a novel Céa–Falk lemma for abstract variational inequalities of the second kind.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY^′ and ZZ^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=(X_I)_κ=∏_iIX_i in the κ-box topology (that is, with basic open sets of the form ∏_iIU_i with U_i open in X_i and with U_i≠X_i for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and [β<α and λ<κ]β^λ<α) with α regular, be a set of non-empty spaces with each d(X_i)<α, π[Y]=X_J for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is J[I]^<α such that f(x)=f(y) whenever x_J=y_J, and (c) every such f extends to .     

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

For a general (real) parameter, let M _nbe the M-estimator and M _n ⁽¹⁾ be its one-step version (based on a suitable initial estimator M _n ⁽⁰⁾). It is known that, under certain regularity conditions, n(M _n ⁽¹⁾-M _n)=O _p(1). The asymptotic distribution of n(M _n ⁽¹⁾-M _n) is studied; it is typically non-normal and it reveals the role of the initial estimator M _n ⁽⁰⁾.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0387