The repetition property for sequences on tori generated by polynomials or skew-shifts

The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schrödinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schrödinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.     

The limits of sequences of iterated overshoot distribution functions

In this paper, we deal with sequences of iterated overshoot distribution functions. Under certain norming conditions and under the assumption that these sequences are convergent, the limits are completely characterized. The paper can be considered as a continuation of the work by Harkness and Shantaram [4].As has been indicated by the referee, Shantaram and Harkness recently published a continuation of their work (see [5]). The present paper, however, is more general than [5]. The attention of the authors was directed to this problem while studying the behaviour of market demand transmitted through a chain of stock points.     

Uniform approximation by generalised polynomials

We here describe methods both numerical and analytic to solve the problem of finding the best uniform approximations to a continuous function by a finite dimensional linear space of functions which does not necessarily satisfy the Haar condition. We show how a knowledge ofH-sets is essential and how the theory is simplified by the use of this concept.     

Nonstandard arithmetic of iterated polynomials

LetK be an algebraic number field of finite degree andf(X,T) a polynomial overK. For eachφ(X)∈Z[X], we denote byE(φ) the set of all integersa with φ ^m (a) =φ ⁿ (a) for somem≠n. In this paper, we give a condition for a polynomialφ(X)∈Z[X] to satisfy the following; If forn∈N, there existr∈K anda∈Z−E(φ) such thatf r, φ ^m (a)=0, then there exists a rational functiong(X) overK andk∈N such thatf(g(T)), φ ^k (T))=0 .     

Uniform approximation by combinations of Bernstein polynomials

We prove a pointwise characterization result for combinations of Bernstein polynomials. The main result of this paper includes an equivalence theorem of H. Berens and G. G. Lorentz as a special case.     

On quadratic fields generated by polynomials

Let be a polynomial of degree d ≥ 2 without multiple roots. Under the assumption of the ABC-conjecture, an asymptotic formula for the number of distinct fields among for has recently been given by Cutter, Granville, and Tucker. We use bounds for character sums to obtain an unconditional lower bound on the number of such fields for . Received: 19 November 2007     

Wavelets generated by the Rudin-Shapiro polynomials

In this paper, we consider the well-known Rudin-Shapiro polynomials as a class of constant multiples of low-pass filters to construct a sequence of compactly supported wavelets.     

Differential polynomials generated by linear differential equations

This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.     

On semiiterative methods generated by Faber polynomials

Numerische Mathematik - Consider a nonsingular system of linear algebraic equationsA x=b, or in fixed point formx=Tx+c, where the eigenvalues ofT are contained in some compact subset Ω of the...     

Polynomial normal densities generated by Hermite polynomials

It is known that for a normalN(0, 1) random variable (r.v.) Y₀ the expectation of the Hermite polynomial H_n in Y₀ is equal to zero, i.e.,E[H_n(Y₀)]=0, n≥1. We give examples of other distributions satisfying this condition as well as some characterizations of these distributions. We show that for some subsets of Hermite polynomials the orthogonality measure is not unique. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Analogs of the arcsine distribution for sequences linearly generated by independent random variables

Let {ξ_k}, kz ...?1,0,1, ..., be a sequence of independent identically distributed random variables with . Let {C_k} be a numerical sequence such that \(\Sigma _{ - \infty }^\infty c_k^2< \infty \) Let $$X_n = \sum\limits_{ - \infty }^\infty {c_{k - n} \xi _k } , S_n = \sum\limits_1^n {X_k } $$ . This article investigates the limit behavior of the distributions of functionals of the following type: $$\mathcal{V}_n = \tfrac{1}{n}\sum\limits_1^n {h\left( {S_k } \right)} $$ , where h is a bounded function on R¹.     

Uniform approximation by polynomials on compacta of special form

A lower estimate of the least deviations is obtained and polynomials of the best uniform approximation are found for some functions given on compact sets of the complex plane containing complete preimages Q ^?1(v _j ) of several points v _j ?? ? for some polynomial Q(z) of complex variable.