Infinite Sequence Of Real Numbers Research Articles

Extremal interpolation with the least value of the norm of the second derivative in

In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the -th derivative in , , needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis for the class of interpolated sequences for which the sequence of -th order divided differences belongs to . In the present paper this problem is solved in the case when . The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.

On the Discrepancy of Two Families of Permuted Van der Corput Sequences

AbstractA permuted van der Corput sequenceSbσ$S_b^\sigma$in basebis a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,...,b −1}. These sequences are known to have low discrepancy DN, i.e.t(Sbσ):=lim supN→∞DN(Sbσ)/log N$t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$is finite. Restricting to prime basespwe present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽pin an explicit way. We use this characterization to obtain bounds fort(Spσ)$t\left({S_p^\sigma } \right)$for permutationsσin these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense thatt(Spσ)<t(Spid)$t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$.

Interpolation of infinite sequences by entire functions

The study was originally motivated by a theoretical question raised by Warming and Hyett in a famous publication on the Modified Equation Approach [R.F. Warming, B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput Phys. 14 (1974) 159–179]. Their classical accuracy analysis of a finite-difference method applied to a time-dependent problem implicitly relies on the assumption that a function interpolating the numerical values can be expanded, over an indefinite domain, in Taylor's series of the independent variables x and t. Here, we establish constructively that the problem of interpolation of an arbitrary infinite sequence of real numbers by an entire function of x (and possibly t) admits uncountably many solutions. In the case of a single variable, if the values are bounded, the interpolant can be made bounded, and all its derivatives bounded. Besides, the construction is generalized to the interpolation of the values of the function and its derivatives up to an arbitrarily prescribed order ( Hermitian interpolation). The proposed interpolant depends on a free parameter λ, and its behavior as λ varies is illustrated by a numerical example.

A coinductive calculus of streams

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Nevanlinna Matrices for the Strong Hamburger Moment Problem

Let {c n }∞n=-∞ be a doubly infinite sequence of real numbers. The strong Hamburger moment problem consists of finding positive measures σ on R such that c n = ∫∞ - ∞ t n dσ(t) for n = 0, ±1, ±2,.... The problem is indeterminate if there is more than one solution. For an indeterminate problem there is a one-to-one correspondence between all Pick functions φ and all solutions σ of the moment problem, expressed by ∫∞ - ∞t) -1 dσ(t) = [α(z)φ(z) - γ(z)][β(z)φ(z) - δ(z)] -1 . The functions α,β,γ,δ are holomorphic in the complex plane outside the origin. The purpose of this paper is to study growth properties of these functions a, β,γ,δ, analogous to properties of corresponding entire functions connected with the classical Hamburger moment problem.

Elements of Stream Calculus: (An Extensive Exercise in Coinduction)

Based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers), a coinductive calculus of streams is developed. The main ingredient is the notion of stream derivative, with which both coinductive proofs and definitions can be formulated. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Reducibility and nonreducibility between ℓ^{𝑝} equivalence relations

We show that, for 1 ≤ p > q > ∞ 1 \le p > q > \infty , the relation of ℓ p \ell ^{p} -equivalence between infinite sequences of real numbers is Borel reducible to the relation of ℓ q \ell ^{q} -equivalence (i.e., the Borel cardinality of the quotient R N / ℓ p {\mathbb {R}}^{\mathbb {N}}/\ell ^{p} is no larger than that of R N / ℓ q {\mathbb {R}}^{\mathbb {N}}/\ell ^{q} ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but ‘modular,’ and using this nicer map to derive a contradiction.

Nevanlinna matrices for the strong Stieltjes moment problem

Let ∗c n∗ n=−∞ ∞ be a doubly infinite sequence of real numbers. A solution of the strong Hamburger moment problem is a positive measure σ on (−∞, ∞) such that c n= ∫ −∞ ∞ u n dδ(u) for n = 0, ±1, ±2,…. A solution of the strong Stieltjes moment problem is a positive measure σ on [0, ∞) such that c n= ∫ 0 ∞ u n dδ(u) for n = 0, + 1 ̄ , + 2 ̄ ,… . A moment problem is indeterminate if there exists more than one solution. With an indeterminate strong Hamburger moment problem there is associated a Nevanlinna matrix of functions α, β, γ, δ holomorphic in ofC − ∗0∗. These functions have growth properties partly similar to properties of analogous entire functions associated with an indeterminate classical Hamburger moment problem. In this paper we obtain a stronger growth result in the case where the strong Stieltjes moment problem is solvable.

Large deviations for moving average processes

Let Z = { hellip;, − 1, 0, 1, …}, ξ, ξ n, n ϵ Z a doubly infinite sequence of i.i.d. random variables in a separable Banach space B, and a n , n ϵ Z, a doubly infinite sequence of real numbers with 0 ≠ ∑ n ϵ z | a n | < ∞. Set X n = ∑ iϵz a i ξ i + n , n ⩾ 1. In this article, we prove that (X 1 + X 2 + … + X n) n, n ⩾ 1 satisfies the upper bound of the large deviation principle if and only if E exp q k ( ξ) < ∞, for some compact subset K of B, where q k (·) is the Minkowski functional of the set K. Interestingly enough, however, the lower bound holds without any conditions at all! We will also present an asymptotic property of the corresponding rate function.

The stability and instability of partial realizations

A finite or infinite sequence of real numbers is said to be stable if it admits minimal realization by a stable linear system. It is shown that the preservation of stability as such a sequence is truncated or extended is not a generic property even among stable sequences. This is of interest for identification of linear systems from partial data and for partial realization of random systems, in which cases it constitutes a negative result. Certain finite sequences have infinitely many minimal realizations each having different stability properties. In this case, a graphical criterion in the spirit of the Nyquist criterion is derived to exploit this lack of uniqueness in order to determine whether one can achieve a stable pole placement by a judicious choice of partial realization.

On irregularities of distribution of real sequences

A natural measure of the amount of unavoidable clustering that must occur in any bounded infinite sequence of real numbers is studied. We determine the extreme value for this measure and exhibit sequences that achieve this value.

Almost Automorphic Integrals of Almost Automorphic Functions

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that(i) exits for every real t but no kind of uniformity of convergence is stipulated;(ii) exits for every t;(iii) for every t.

Maximizing Stationary Utility in a Constant Technology

Abstract : This paper is concerned with a problem in the optimal control of a nonstochastic process over time. It can also be looked on as a problem in convex programming in a space of infinite sequences of real numbers. The literature on optimal economic growth contains several papers in which a utility function of the form (1) U(x1,x2,...) = Summation, t=1 to t=infinity, of alpha (superscript(t-1)) u(x sub t), Oalpha1, is maximized under given conditions of technology and population growth. Here xt is per capita consumption in period t, and u(x) is a strictly concave, increasing, single-period utility function. Alpha is called a discount factor. A generalization of (1) has been proposed under the name stationary utility, and is definable by a recursive relation (2) U(x1, x2, x3,...) = V(x1, U(x2, x3,...)). One obtains (1) by V(x, U) = u(x) + alpha U. The natural generalization of alpha in (1) to stationary utility is the function (2a) alpha(x) = (the partial derivative of V(x,U) with respect to U) subscript U = U(x,x,x,...). In this paper we study the maximization of (2) under production assumptions.

On the inversion of moving averages

Abstract 1. Let an infinite sequence of real numbers be given by (1) {ηt } =[…, η t-2, η t-1, η t , η t+1, η t+2, . . . ] and let (b) = (b 0, b 1,.., bh ) represent real constants. The sequence(2) {ζ t } = […, ζ t-2, ζ t-1, ζt , ζ t+1, ζ t+2,…] defined for every t by the relation(3) ζ t = b 0 · η t + b 1 · η t-1 — . . . + b h-1 · η t-h+1 + b h · η t-h is said to be a moving average of {η t } with weights (b i ) The variable t, which is restricted to integral values 0, ± 1, ± 2 etc., will in the sequel be spoken of as representing time.