Properties of functions generalized convex with respect to a WT-system

Continuity and convergence properties of functions, generalized convex with respect to a continuous weak Tchebysheff system, are investigated. It is shown that, under certain non-degeneracy assumptions on the weak Tchebysheff system, every function in its generalized convex cone is continuous, and pointwise convergent sequences of generalized convex functions are uniformly convergent on compact subsets of the domain. Further, it is proved that, with respect to a continuous Tchebysheff system, L_p-convergence to a continuous function, pointwise convergence and uniform convergence of a sequence of generalized convex functions are equivalent on compact subsets of the domain.     

Mean convergence of an extended Lagrange interpolation process on [0,+∞)

We study some extended Lagrange interpolation processes based on the zeros of the generalized Laguerre polynomials. We give necessary and sufficient conditions such that the convergence of these processes, in suitable L _p weighted spaces on the real semiaxis, is assured for 1<p<+∞.     

On the uniqueness of generalized monosplines of least Lp-norm

The uniqueness of monosplines and perfect splines of leastL _p-norm is treated in the framework of generalized monosplines and total positivity. The analysis is based on the invariance properties of the degree of a certain mapping and on a new composition result for totally positive kernels. For theL _p-case 1<p<∞, uniqueness is shown under the same extra conditions as were previously shown to be needed in theL _p-case. The uniqueness in theL _∞-case is obtained without any restrictions.     

A characterization of multidimensional multi-knot piecewise linear spectral sequence and its applications

We characterize a class of piecewise linear spectral sequences. Associated with the spectral sequence, we construct an orthonormal exponential bases for L2（[0,1）d）, which are called generalized Fourier bases. Moreover, we investigate the convergence of Bochner-Riesz means of the generalized Fourier series.     

Recognition by Spectrum of Some Linear Groups over the Binary Field

We focus our attention on the linear groups L _n (2) and obtain some general properties of these groups. We will show then that the linear groups L _p (2), where 2 is a primitive root mod p (p odd prime), are recognizable by spectrum. For example, the linear groups L ₃(2), L ₅(2), L ₁₁(2), L ₁₃(2), L ₁₉(2), L ₂₉(2), L ₃₇(2), L ₅₃(2), etc. are recognizable by spectrum.     

Random Steiner symmetrizations of sets and functions

In this article we prove almost sure convergence, in the L ₁ distance, of sequences of random Steiner symmetrizations of measurable sets having finite measure to the ball having the same measure. From this result we deduce analogous statements concerning the almost sure convergence to the spherical symmetrization of random Steiner symmetrizations of non negative L _p functions in the natural norm and uniform convergence of non negative continuous functions with bounded support. The latter result is finally used to prove that sequences of random symmetrizations of a compact set converge almost surely in the Hausdorff distance to the ball having the same measure, providing another proof of Mani-Levitska’s conjecture besides the one given in 2006 by Van Schaftingen (Topol Methods Nonlinear Anal 28(1): 61–85, 2006).     

On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length 2p m

In this article, new classes of generalized cyclotomic binary sequences with period 2p ^m are proposed. We determine the linear complexity and autocorrelation of these sequences. The results show that the proposed generalized cyclotomic binary sequences have high linear complexity, but do not have desirable autocorrelation properties.     

Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation

Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a “monotone norm” (e.g., L_p) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k ? 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.     

Propriétés asymptotiques des processus de branchement en environnement aléatoire

Let (Z_n) be a branching process in a random environment. If the process in sub-critical or critical, we study the convergence rate of the survival probability P(Z_n>0) as n→∞; if the process is supercritical, we give a necessary and sufficient condition for the convergence in L^p of the natural martingale, where p>1 is given.     

Sharp estimates for the convergence rate of Fourier series in terms of orthogonal polynomials in L 2((a, b), p(x))

Sharp estimates are given for the convergence rate of Fourier series in terms of classical orthogonal polynomials in some classes of functions characterized by a generalized modulus of continuity in the space L ₂((a, b), p(x)). Expansions in terms of Laguerre, Hermite, and Jacobi polynomials are considered.     

Asymptotics for statistical functionals of long-memory sequences

We present two general results that can be used to obtain asymptotic properties for statistical functionals based on linear long-memory sequences. As examples for the first one we consider L- and V-statistics, in particular tail-dependent L-statistics as well as V-statistics with unbounded kernels. As an example for the second result we consider degenerate V-statistics. To prove these results we also establish a weak convergence result for empirical processes of linear long-memory sequences, which improves earlier ones.     

Fixräume regulärer operatoren und ein Korovkin-Satz

S. J. Bernau has introduced the notion of an exchange subspace of an L_p-space and has shown that the range of a contractive linear projection on an L_p-space (1 ? p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.     

The pointwise convergence of p-adic Mbius maps

The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.     

Calculation of linear bestL p -approximations

In this paper we are concerned with finding theL _p -solution (i.e. minimizing theL _p -norm of the residual vector) to a linear approximation problem or, equivalenty, to an overdetermined system of linear equations. An embedding method is described in which the damped Newton iteration is applied to a series of perturbed problems in order to guarantee convergence and also increase the convergence rate.     

Mean convergence of Hermite-Fejér interpolation

Weighted L^p convergence of Hermite-Fejér interpolation based on the zeros of generalized Jacobi polynomials is investigated. The main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions.     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

Measure-compact operators, almost compact operators, and linear functional equations in L p

Under study are the measure-compact operators and almost compact operators in L _p . We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in L _p , we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in L _p with almost compact or measure-compact operators to equivalent linear integral equations in L _p with quasidegenerate Carleman kernels.     

Sharp interpolation theorems in couples of l p spaces for generalized Lions-Peetre spaces of means

Sharp interpolation theorems for linear operators acting on arbitrary couples of L _p spaces are found in the family of generalized Lions-Peetre spaces of means. This family includes the Lorentz spaces with functional quasi-concave parameters, the Orlicz spaces, and spaces similar to them.     

About computation of the linear complexity of generalized cyclotomic sequences with period p n+1

We propose a computation method for linear complexity of series of generalized cyclotomic sequences with period p ⁿ⁺¹. This method is based on using the polynomial of the classic cyclotomic sequences of period p. We found the linear complexity of generalized cyclotomic sequences corresponding to the classes of biquadratic residues and Hall sequences.