The complement of certain recursively defined sets

An RD-set (recursively defined) is a minimal set of positive integers containing a given seed and closed under a given set of unary linear operations (x → ax + b). We examine conditions under which the complement of an RD-set may be expressed as the disjoint union of infinite arithmetic progressions.     

The convergence rate of Euler means sequence

Let denote the Euler means of the Fourier series of the 2π-perodic function f(x). For an integer q>0, setting ℰ _nq (ω)=sup‖f−ℰ _n (q,f)‖, the precise value of is obtained.     

Graphs and recursively defined towers of function fields

In this paper we state and explore a connection between graph theory and the theory of recursively defined towers. This leads, among other things, to a generalization of Lenstra's identity (Finite Fields Appl. 8 (2001) 166) and the solution of an open problem concerning the Deuring polynomial posed in (J. Reine Angew. Math. 557 (2003) 53). Further we investigate the effect extension of the constant field has on the limit of certain towers.     

On the rate of convergence to a Poisson process

Let ξ = {ξ₁, ξ₂, ...} be a sequence of independent, spherically symmetric, identically distributed multivariate random vectors such that the condition

On the convergence rate of Fourier series

The rate of convergence of a Fourier‐series representation of a given function depends on the nature of the function and of its derivatives. This is shown by using the graphical outputs of a desk computer for different cases. For full‐range series, the effects of continuity and discontinuity of the function and its first derivatives are shown first. The advantage of half‐range formulae due to the free choice of function in the second half‐range are demonstrated next, along with the importance of choice of sine and cosine series according to the function being represented. Finally, a method is given of modifying the given function in a simple manner whereby a dramatic increase in the rate of convergence of the Fourier series is obtained. Examples of the Gibbs overshoot and its elimination are included.     

On the convergence of a linear recursive sequence with nonnegative coefficients

A simple proof to some known results on the convergence of linear recursive sequences with nonnegative coefficients is given, using the technique of monotone convergence.     

On the convergence rate of the unscented transformation

Nonlinear state-space models driven by differential equations have been widely used in science. Their statistical inference generally requires computing the mean and covariance matrix of some nonlinear function of the state variables, which can be done in several ways. For example, such computations may be approximately done by Monte Carlo, which is rather computationally expensive. Linear approximation by the first-order Taylor expansion is a fast alternative. However, the approximation error becomes non-negligible with strongly nonlinear functions. Unscented transformation was proposed to overcome these difficulties, but it lacks theoretical justification. In this paper, we derive some theoretical properties of the unscented transformation and contrast it with the method of linear approximation. Particularly, we derive the convergence rate of the unscented transformation.     

On the implication of a certain rate of convergence to normality

Probability Theory and Related Fields -     

How does one determine the rate of convergence of a sequence of iterations?

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 96–100, April, 1992.     

On the rate of convergence of iterated exponentials

We study the asymptotic rate of convergence of the sequence of iterated exponentials $\{ z_{1}=a, z_{n+1}=a^{z_{n}}, n \geq 1 \}$ . We show that z _n converges at a linear rate if a is in the interior of the Baker-Rippon convergence region and at a sublinear rate if a is on its boundary. A precise characterization of the rate is explored when the sequence converges sublinearly.     

On some new generalized difference sequence spaces defined by a sequence of moduli

In this paper, we define the new generalized difference sequence spaces [V, λ, F, p, q]₀(Δ _v ^m ), [V, λ, F, p, q]₁(Δ _v ^m ) and [V, λ, F, p, q]_∞(Δ _v ^m ). We also study some inclusion relations between these spaces.     

On the rate of strong convergence for convolutions

An estimate of the rate of strong convergence of convolutions of probability measures is given under the assumption that the components of each convolution converge weakly and in total variation, respectively. Reported at the XVI Seminar on Stability Problems for Stochastic Models, Eger, Hungary, 29 August–3 September 1994. Received by the Editorial Board 15 January, 1996. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

On the convergence rate of a parallel nonoverlapping domain decomposition method

In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and analyzed.It is known that the convergence rate of this method is 1-O(h),where h is mesh size.In this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain size.Counter examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.