Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

The maximal accuracy of stable difference schemes for the wave equation

We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation,u _tt =c ² u _xx , given by

On the set visited once by a random walk

Summary In this paper we prove the following statement. Given a random walk ,n=1, 2, ... where ₁, ₂ ... are i.i.d. random variables, let (n) denote the number of points visited exactly once by this random walk up to timen. We show that there exists some constantC, 0 <C < , such that with probability 1. The proof applies some arguments analogous to the techniques of the large deviation theory.Research supported by the Hungarian National Foundation for Scientific Research, Grant N^o # 819/1     

Reducibility of lacunary polynomials,VII

Letk>1 and let be non-zero algebraic numbers contained in the field . It is shown that for almost all, in the sense of density integer vectorsn ₁,...,n _k the polynomial becomes irreducible over on dividing by the product of all factorsx–, where is a root of unity.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

On the tail behaviour of the distribution function of multiple stochastic integrals

Summary Let F _n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define , and consider the integrals where f is a bounded measurable function. We give a good upper bound on the probability . An analogous estimate is given for multiple integrals with respect to a Poisson process.     

Beste Quadraturformeln für Integrale mit einer Gewichtsfunktion

Let be a fixed matrix with elements that are 0 or 1 and letX be a fixed set ofm+1 different knots. The problem is to find necessary and sufficient conditions for (E, X) to guarantee the existence of a quadrature formula with a remainder term of type for any choice of a weight functionw(t) and satisfyingR(f)=0 forf a polynomial of degree at mostn–1. The result generalizes the corresponding result ofI. J. Schoenberg for the special case of quasi-Lagrange-matricesE. —in case of the existence ofR it is possible to calculate the best quadrature formulaR ^* in the sense ofSard by integrating splines of degree 2n–1. But ifE contains onlyn ones it is sufficient to integrate polynomials of degreen–1.     

An algorithm for a linear diophantine equation and a problem of Frobenius

Summary We solve the diophantine equation for nonnegative variablesx _j , wherea _j andL are positive integers. We characterize both the values ofL that lead to solutions and those that do not lead to solutions. We solve the Frobenius problem of finding the largest value ofL for which no solution exists.     

The range of two-parameter random walk in space

Summary Let {X _ij; i>0, j>0} be a double sequence of i.i.d. random variables taking values in the d-dimensional integer lattice E _d . Also let . Then the range of random walk {S _mn: m>0, n>0} up to time (m, n), denoted by R _mn , is the cardinality of the set {S _pq: 0m, n). In this paper a sufficient condition in terms of the characteristic function of X ₁₁ is given so that a.s. as either (m, n) or m(n) tends to infinity.     

Easy transportation-like problems onK-dimensional arrays

TheK-dimensional version of two transportation-like problems is posed and efficiently solved. The caseK=2 goes as follows: Given an (m,n)-matrixA of reals, a realm-vectoru, and a realn-vectorv, find a real (m,n)-matrixX minimizing

Strong law of large numbers for functionals of products of random matrices

For the sequence A₁, A₂, ..., A_n, ... of m×m independent random matrices such that for each k there exists a joint density function P_k(X) of the elements _ij ^k, we prove the following theorem: if and for some positive constants ₁ and ₂, then with probability 1,

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.     

Strong consistency of nearest neighbor kernel regression estimation for stationary dependent samples

Under quite mild conditions onk _n . the strong consistency is proved for the nearest neighbor density, the nearest neighbor kernel regression and the modified nearest neighbor kernel regression of an a-mixing stationary sequence in time series context. The condition imposed on the mixing coefficients is , . which is simple and weak.     

On quasi degree quadrature rules

Summary In this paper we examine quadrature rules for the integral which are exact for all with +d. We specify three distinct families of solutions which have properties not unlike the standard Gauss and Radau quadrature rules. For each integerd the abscissas of the quadrature rules lie within the closed integration interval and are expressed in terms of the zeros of a polynomialq _d(y). These polynomialsq _d(y), (d=0, 1, ...), which are not orthogonal, satisfy a three term recurrence relation of the type Q_d+1(y)=(y+_d+1)q_d(y)–_d+1yq_d–1(y) and have zeros with the standard interlacing property.This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Коэффициенты рядов Ф урье и класс Липшица

We generalize and sharpen certain results concerning Fourier series from the Lipschitz class. In particular, for sinnx we prove the following: Let ¦b_n¦n^–2L(n) where L(x) is a continuous and slowly oscillating function. Then

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Existence of Positive Solutions for a Class of Higher Order Neutral Functional Differential Equations

The higher order neutral functional differential equation

Change of variables in a multiple integral for local fields

A proof of the formula for locally compact fields andC ¹-isomorphisms :UV, whereU andV are open subsets of , was never published. In this paper we give two short proofs, one of them is a more elementary variant of the other.