Spectrum of the nonself-adjoint Schrödinger operator in unbounded regions

It is proved that the discrete spectrum of the operator – + q(x) in the space L₂(E₂k) (k 1) where q(x) is a measurable complex-valued function satisfying the condition ¦q(x) ¦ Ce^–¦x¦, having no finite limit points, and for k=1 the discrete spectrum consists of a finite number of points.Translated from MatematicheskieZametki, Vol. 9, No. 1, pp. 19–26, January, 1971.In conclusion the author wishes to thank M. A. Naimark for directing this work.     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

Piecewise polynomial approximation

For best piecewise polynomial approximation _n=_n (f; [0, 1]) of a functionf, which is continuous on the interval [0, 1] and admits a bounded analytic continuation onto the disk K=z:¦z–1¦<, the relation _n=o[ _f (e ^–n )] is valid.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 129–134, February, 1972.     

Приближение непреры вных функций средним и Эйлера

Let _n(1,f,x)=1/2 ⁿ _k=1 ⁿ C _k ⁿ S_k(x,f) denote the Euler means of the Fourier series of the 2-periodic functionf(x). For a function the main term of deviationf(x)– _n (1,f, x) is calculated. Asymptotically exact order of decrease of the upper bound of such deviations over the classH ⁽⁾ is also obtained.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x ₁,x ₂,..., _n }I=[–1, 1] and . ForfC ₁(I) definef* byf–p _f =f*, wherep _f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT _n (x) and obtain x ^m –p _x ^m8eE _n–1(x ^m ), whereE _n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE _n (f) that omits theassumptionf ⁽ⁿ⁺¹⁾>0 in a similar estimate of Meinardus.     

The approximation of functions of many variables by their Féjér sums

We construct elliptic Féjér polynomials K_n(x) of m variables. We prove some of their properties: a) the Féjér polynomials are positive on the m-dimensional torus T^m, K_n(x)0, b) (x)=o(n^–1), as n, c) we calculate their norms in the spaces L[T^m] and C[T^m]. We estimate the deviation of the Féjér sum _n(x,f) from the functionf(x). For the space C[T^m]: where c_,m c_1,m are constants.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 817–828, June, 1973.In conclusion, the author wishes to express his gratitude to S. B. Stechkin for help with the paper.     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

Conditioned limit theorems for random walks with negative drift

Summary In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S _n is a random walk with negative mean and finite variance then there is a constant so that (S _[n.]/n ^1/2¦N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES ₁=–a<0, ES ₁ ² <, and there is a slowly varying function L so that P(S ₁>x)x ^–q L(x) as x then (S _[n.]/n¦S _n >0) and (S _[n.]/n¦N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1–(x/a)^–q )⁺) and are otherwise linear with slope –a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second.The research for this paper was started while the author was visiting W. Vervaat at the Katholieke Universiteit in Nijmegen, Holland, and was completed while the author was at UCLA being supported by funds from NSF grant MCS 77-02121     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

Some best possible bounds concerning the traces of finite sets II

LetX be ann-element set and be a family of its subsets. Consider the family _x = {F – {x} : F } for a givenx X. We write(m, n) (m – k, n – 1), when for all with || m, there exists an elementx ofX such that| _x| m – k. We show that (m, n) (m – 10,n – 1) for allm 5n and (m, n) (m – 13,n – 1) for allm 29n/5.     

Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.     

Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

Minimal submanifolds in a Riemannian manifold with pinched positive curvature

In this paper, we prove the following result: LetM be ann-dimensional compact curvature-invariant minimal subinanifoid immersed in a(> 2n–2/5n–2)- pinched Riemannian manifoldN. Denote by the second fundamental form ofM. If ¦¦²<3/9((5n–2)–2(n–1)), thenM is totally geodesic. This result generalizes the Simons pinching theorem.Supported by the JSPS postdoctoral fellowship and NSF of China.     

On the uniqueness of a Walsh series converging on subsequences of partial sum

We show that if a Walsh series whose coefficients tend towards zero is such that the subsequence of its partial sums indexed by n_k, where n_k satisfies the condition 2^k–1k2^k (k=0, 1, 2, ...), tends everywhere, except possibly for a denumerable set, towards a bounded functionf(x), then this series is the Fourier series of the functionf(x).Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 27–32, July, 1974.     

On the set of positive deflections of functions meromorphic in the unit disk

For an arbitrary (0, 1/2) we construct a functionf(z) of lower order [f]=. that is meromorphic in the disk D = {z ¦z¦< 1} and such that the set (f) of positive deflections of the functionf (in the sense of V. P. Petrenko) has positive logarithmic capacity.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 48–53.