Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

Summary LetL(x, T),xR ^d ,TR ₊ ^N , be the local time of theN-parameter Wiener processW taking values inR ^d . Even in the distribution valued casedd2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|0 and/orT and of related occupation integrals asT. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X _T (f), and of the second order, i.e. normalized convergence laws forL(x, T)–E(L(x, T)) resp.X _T (f)–E(X _T (f)).This research was made during a stay at the LMU in München supported by DAAD     

On the Normal Order of ϕk+1(n)/ϕk(n), where ϕk is the k-Fold Iterate of Euler's Function

Let be the j-fold iterated function of . Let and > 0 be fixed, Q be a prime, and let N _k(Q|x) denote the number of those nx for which Q . We give the asymptotics of N _k(Q|x) in the range .     

Mean value estimates in lattice point theory

LetQ(u) be a positive definite quadratic form inr2 variables with a real symmetric coefficient matrix of determinantD. Given a real vectorb with 0b _j <1, forx>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)x, letV(x) be the volume of this ellipsoid andP(x)=A(x)–V(x). Let . By introduction of a parameter we shall show how the treatment of estimates onP(x) and onM(x) can be unified.     

On the existence of value in the Varaiya-Lin sense in differential games of pursuit and evasion

This paper is strictly related to Ref. 1. A pursuit-evasion game described in part by the system and is considered. The state variablesx andy are restricted, in the sense that (x(t),t) N ₁ and (y(t),t) N ₂. The existence of a value in the sense of Varaiya and Lin is proved under the assumption that the sets of all admissible trajectories for the two players are compact and the lower value is not greater than the upper value.     

Matrix summability and a generalized Gibbs phenomenon

Letf be a real-valued function sequence {f _k } that converges to on a deleted neighborhoodD of . If there is a subsequence {f _k(j) } and a number sequencex such that lim _j x _j = and either lim _j f _k(j) (x _j )>lim sup _x (x) or lim _j f _k(j) (x _j ) x (x), thenf is said to display theGibbs phenomenon at . IfA is a (real) summability matrix, thenAf is a function sequence given by(Af) _n (x)= _k=0 a _n,k f _k (x). IfAf displays the Gibbs phenomenon wheneverf does, thenA is said to beGP-preserving. By replacingf _k (x) withf _k (x _j )F _k,j , the Gibbs phenomenon is viewed as a property of the matrixF, andGP-preserving matrices are determined by properties of the matrix productAF. The general results give explicit conditions on the entries {a _n,k } that are necessary and/or sufficient forA to beGP-preserving. For example: if(x)0 thenF displaysGP iff lim _k,j F _k,j 0, and ifA isGP-preserving then lim _n,k A _n,k 0. IfA is a triangular matrix that is stronger than convergence, thenA is notGP-preserving. The general results are used to study the preservation of the Gibbs phenomenon by matrix methods of Nörlund, Hausdorff, and others.     

Problem of the extremal decomposition of a closed plane

One considers the problem of the maximum of the product of powers of conformal radii of nonoverlapping domains in the following formulation. Let A=a₁, ..., a_n and B=(b₁, ..., b_m be systems of distinguished points in ¯C and let ={₁,..., _m} be a system of positive numbers. ByU(D,b ) we denote the reduced modulus of the simply connected domain D relative to the pointb D. Find the maximum of the sum in the familyD of all systems of nonoverlapping simply connected domains D_j, j=1, ..., m, satisfying the following condition: the domain D_j does not contain points b_i B, different from b_j, and some collection A_j, for each domain, of points from A, _j=1 ^m A _j =A. The solution of this problem is obtained by the simultaneous use of the method of variation and of the method of the moduli of families of curves and is given by Theorem 1 of the present paper. As consequences of Theorem 1 one obtains Theorems 2 and 3, strengthening the corresponding results of a previous paper of the author.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 149–154, 1985.     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

Fast Preconditioned Iterative Methods for Convolution-Type Integral Equations

We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + _j=1 ^m µ_j x(t – t _j) + ₀ k(t – s)x(s) ds = g(t), 0 t , where t _j (–, ), for 1 j m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

---

Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

The fractional parts of a sum of polynomials

LetX=(x ₁,...,x _s ) be a vector ofs real components and , whereP _j (x _j ) are polynomials of exact degree k with real coefficients and without constant terms. The authors extend a result of Davenport and obtain an approximation on f(X) where t means the distance fromt to the nearest integer.     

A generalization of the inversion formulas of systems of power series in systems of implicit functions

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx _j =_j(w)=_j(w₁,...,w_m), j=1,...,n, defined by the system of equations f_j(w, x)=F_j (w₁,...,w_m:z₁,...,x _n )=0, j=1,...,n,f _j , (0, 0)=0, F_j(0, 0)/z_k=_jk in a neighborhood of the point (0, 0)C _(w,x) ^m+n , in terms of the coefficients of the power series of the functions F_j(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978.     

Die konvexe Hülle von n rotationssymmetrisch verteilten Punkten

Summary The number of edges A _n , the length l _n and the surface F _n of the convex hull of n independent, identically distributed random points in the plane are considered under the assumption of rotational symmetry. The asymptotic behaviour of the expectations E(A _n ), E(l _n ) and E(F _n ) is studied according to the behaviour of the function Pr( as x 1 (distributions on the unit disc) or x (distributions on the whole plane).

---

---

Herrn Prof. H. Hadwiger zu seinem 60. Geburtstag gewidmet.     

Dual-orthogonal polynomials

Let (x, ) and (x,) be two functions,x[a, b] and { _j } _j=1 and { _j } _j=1 be two sequences where _{i j} and _{i j} whenij. We define the vector spacesU _k =span{(x, _j )} _j=1 ^k andV _k =span{(x, _j )} _j=1 ^k where we assume thatdim(U _k )=dim(V _k )=k,k1. We then look for the generalized polynomialsp _m xU _m+1\U _m so that _a ^b p _m (x)(x, _j )d(x)=0,j=1,2,...,m. If such generalized polynomials exist for allm1 we say that {p _m } _m=1 is a dual-orthogonal polynomial sequence from {(x, _j )} _j=1 to {(x, _j )} _j=1 with respect to the distribution (x),x[a, b]. In this article we present existence theorems for dual-orthogonal polynomials, explicit formulae forp _m(x), theorems about the zeros ofp _m(x), and, in the end, a Gauss-type quadrature formula for dual-orthogonal polynomials.     

Interval-dividing processes

Summary If and then P(n ^–1·[(Y ₁)++(Y _n )] converges to cnts. law on R ¹) = P(n ^–1·[(Y ₁)++(Y _n )] converges to a cnts. law on R ¹). Thus if ,n then n ^–1[(X ₁)+...+(X _n )] converges a.s. The main result here generalizes this: Let X ₍₁₎ ⁿ , X ₍₂₎ ⁿ ,..., X _(n) ⁿ be the order statistics associated with X ₁, X ₂,,X _n. Define random variables Z ₁,Z ₂, by {Z _n =i}={X _n =X _(i) ⁿ }. Then if Z ₁,Z ₂,Z ₃, are independent and P(Z_ni)i/n, and {X _i} is bounded, n ^–1·[(X ₁)++(X _n)] converges a.s.     

Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method

Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.     

Necessary and sufficient conditions for the convergence of integrated and mean-integratedr-th order error of histogram density estimates

Suppose thatX _l ,..., X _n are samples drawn from a population with density functionf andf _n (x)=f _n (x;X _l ,..., X _n is an estimate off(x), Denote bym _nr =|f _n (x)–f(n)| ^r dx andM _nr =E(m _nr) the Integratedr-th Order Error and Mean Integratedr-th Order Error off _n for somer1 (whenr=2,they are the familiar and widely studied ISE and MISE), In this paper the same necessary and sufficient condition for and a.s. is obtained whenf _n (x) is the ordinary histogram estimator.The Project supported by National Natural Science Foundation of China.     

A Tauberian Theorem for Ergodic Averages,Spectral Decomposability,and the Dominated Ergodic Estimate for Positive Invertible Operators

Suppose that (,) is a -finite measure space, and 1 < p < . Let T:L^p( L ^p() be a bounded invertible linear operator such that T and T ^–1 are positive. Denote by _n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {T^j+T^–j}_{j =1} . Martín-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of E_n(T)_n=1 ; (b) a uniform A _p p estimate in terms of discrete weights generated by the pointwise action on of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {T^j+T^–j}_j=1 already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform A _p weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages E_n(T)_{n = 1} .     

Sur les plus grands facteurs premiers d'un entier

Let and, for each integern such that (n)k, denote byP _k (n) itsk ^th largest prime factor. Further, given a set of primesQ of positive density <1 satisfying a certain regularity condition, defineP(n, Q), as the largest prime divisor ofn belonging toQ, assuming thatP(n,Q)=+ if no such prime factor exists. We provide estimates of , fork2, and of . We also study the median value of the functionP(n,Q) and that of the functionP _k (n) for eachk1.