The aim of this paper is to continue our investigation of the Lebesgue function of weighted Lagrange interpolation by considering Erdős weights on ℝ and weights on [−1, 1]. The main results give lower bounds for the Lebesgue function on large subsets of the relevant domains.

Some new results concerning tubular sets are presented, with applications to the convergence of the Polya algorithm in the contexts of simultaneous approximation and approximation of multivariate functions by univariate functions. (The Polya algorithm constructs a best uniform approximation from the limit, as p → ∞, of best Lp approximations.)

We generalise the classical Bernstein's inequality: . Moreover we obtain a new representation formula for entire functions of exponential type.

If a weighted Euler transformation is applied to the asymptotic series for ezE1(z) the remainder can be expressed as an integral. Examination of this integral shows that for a transformation of given order the smallest term of the resulting series remains at approximately a constant distance from the start of the series. If, however, there is no restriction on the order of transformation the remainder may be decreased to zero by increasing the number of terms used, but if z is close to the negative real axis the rate of decrease is small. A more general theorem for alternating real series and Taylor's series is also given.

We obtain various refinements and generalizations of a classical inequality of S. N. Bernstein on trigonometric polynomials. Some of the results take into account the size of one or more of the coefficients of the trigonometric polynomial in question. The results are obtained using interpolation formulas.

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set for functions in the Sobolev space with norm , where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with . Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of satisfies , where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.

Approximations for the Stieltjes integral with (φ,Φ)-Lipschitzian integrators are given. Applications for the Riemann integral of a product and for the generalized trapezoid and Ostrowski inequalities are also provided.

In this paper we introduce a set of orthonormal functions, , where ϕn[r] is composed of a sine function and a sigmoidal transformation γr of order r>0. Based on the proposed functions ϕn[r] named by sigmoidal sine functions, we consider a series expansion of a function on the interval [−1,1] and the related convergence analysis. Furthermore, we extend the sigmoidal transformation to the whole real line ℝ and then, by reconstructing the existing sigmoidal cosine functions ψn[r] and the presented functions ϕn[r], we develop two kinds of 2-periodic series expansions on ℝ. Superiority of the presented sigmoidal-type series in approximating a function by the partial sum is demonstrated by numerical examples.

In this paper a method is developed for the asymptotic expansion of some classes of integral as a parameter k → 0+. The procedure is analogous to the method of inner and outer sums for treating certain types of infinite series whose terms contain a small parameter, and can involve heavy algebra. However, this aspect of the process can be delegated to a symbolic manipulation package.

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

In this paper, we obtain sharp estimates for the expected payoffs and prices of European call options on an asset with an absolutely continuous price in terms of the price density characteristics. These techniques and results complement other approaches to the derivative pricing problem. Exact analytical solutions to option-pricing problems and to Monte-Carlo techniques make strong assumptions on the underlying asset's distribution. In contrast, our results are semi-parametric. This allows the derivation of results without knowing the entire distribution of the underlying asset's returns. Our results can be used to test different modelling assumptions. Finally, we derive bounds in the multiperiod binomial option-pricing model with time-varying moments. Our bounds reduce the multiperiod setup to a two-period setting, which is advantageous from a computational perspective.

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

We consider the problem of predicting integrals of second order processes whose covariances satisfy some Hölder regularity condition of order α > 0. When α is an odd integer, linear estimators based on regular sampling designs were constructed and asymptotic results for the approximation error were derived. We extend this result to any α > 0. When 2K < α ≤ 2K + 2, K a non-negative integer, we use an appropriate predictor based on the Euler-MacLaurin formula of order K with regular sampling designs. We give the corresponding result for the mean square error.

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.

In this paper the method of inner and outer sums [5], together with the computational power of computer symbolic manipulation, are used to extend to high order the asymptotic expansions in an appropriate limit of some infinite series arising in low Reynolds-number fluid mechanics. The enhanced applicability of the expansions is demonstrated, and the method is extended to treat alternating series.

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure µ on x = [0,1]q , q 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on , w 2, · ·· }, subject to an ‘ϵ-contractivity' condition, is employed. Thus, only an optimization over the associated probabilities pi is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the pi which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.