Uniform Asymptotic Expansions for Meixner Polynomials

Meixner polynomials m _n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m _n (nα;β,c) as . One holds uniformly for , and the other holds uniformly for , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases. April 16, 1996. Date revised: October 30, 1996.     

Averages Using Translation Induced by Laguerre and Jacobi Expansions

Approximation by averages of the generalized translation induced by Laguerre and Jacobi expansions will be shown to satisfy a strong converse inequality of type B with the appropriate K -functional. April 9, 1998. Date revised: February 22, 1999. Date accepted: March 5, 1999.     

On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros

An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M _n (nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α _{n, s} denotes the s th zero of M _n (nα;δ, η) , counted from the right, and if α˜ _n,s denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α _n,s and α˜ _n,s as n→∞ . December 28, 1998. Date revised: June 4, 1999. Date accepted: September 6, 1999.     

Asymptotic Expansions for the Stochastic Approximation Averaging Procedure in Continuous Time

This paper considers the statistical estimation problem of the root of a nonlinear function under observations of the nonlinear regression model in continuous time. Asymptotic expansions for stochastic approximation averaging procedure are constructed, the gain factor, which minimizes the asymptotic bias is found, asymptotic normality for the procedure is proved. This revised version was published online in June 2006 with corrections to the Cover Date.     

Uniform Asymptotic Expansion of the Voltage Potential in the Presence of Thin Inhomogeneities with Arbitrary Conductivity

Asymptotic expansions of the voltage potential in terms of the "radius" of a diametrically small(or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold,σ, are examples of inhomogeneities for which the convergence to the background potential,or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.The purpose of this paper is to find a "simple" replacement for the background potential, with the following properties:(1) This replacement may be(simply) calculated from the limiting domain Ω\σ, the boundary data on the boundary of Ω, and the right-hand side.(2) This replacement depends on the thickness of the inhomogeneity and the conductivity,a, through its boundary conditions on σ.(3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0.     

Asymptotic expansions for derivatives of stable laws

For the derivativesp ^(k)(x; α, γ) of the stable density of index α asymptotic formulae (of Plancherel Rotach type) are computed ask→∞ thereby exhibiting the detailed analytic structure for large orders of derivatives. Generalizing known results for the special case of the one-sided stable laws (O<α<1, γ=-α) the whole range for the index of stability and the asymmetry parameter γ is covered.     

Asymptotic expansions for the coefficients of analytic generating functions

PairsF(x), G(x) of analytic generating functions that satisfy relations such as 1+G(x)=exp(F(x)) are studied. It is shown that, ifF(x) satisfies fairly mild regularity conditions, such as those imposed by Hayman in his study of coefficients of some general classes of functions, thenG(x) satisfies the much stricter conditions imposed by Harris and Schoenfeld. This enables one to obtain complete asymptotic expansions for the coefficients ofG(x). Applications of this result are made to enumerations of trees.Dedicated to Professor Janos Aczél on his 60th birthday     

Wavelet Expansions and Fractal Dimensions

The bounds of the upper and lower box dimensions of the graph of a function in terms of the coefficients in its wavelet decomposition are given. An example, that the formula for upper box dimension, given in [4], does not hold, is presented. March 10, 1997. Date revised: March 3, 1998. Date accepted: March 3, 1998.     

On Cayley-Transform Methods for the Discretization of Lie-Group Equations

In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials. Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits) is attained when exact integrals are replaced by certain quadrature formulas. March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001.     

Uniform Approximation by Entire Functions That Are All Bounded on a Given Set

For two closed sets F and G in the complex plane C, G C , we solve the following problem Under what conditions on F and G can every function f , continuous on F and analytic in its interior, be uniformly approximated by entire functions, each of which is bounded on G ? February 7, 1995. Date revised: October 31, 1995.     

Asymptotic summation of power series

Summary. We give an asymptotic expansion in powers of of the remainder , when the sequence has a similar expansion. Contrary to previous results, explicit formulas for the computation of the coefficients are presented. In the case of numerical series (), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of , which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series. Received April 22, 1997     

Approximation in L 1 -Norm by Elements of a Uniform Algebra

In this paper we deal with the following situation (for the terminology see [G]). Let A be a uniform algebra on the metric compact space Γ and let be a complex homomorphism of A . Suppose the set of representing measures of has finite dimension. Let be a core measure and let us assume that any real annihilating measure for A has the form S d μ , where S $\in$ L ^∞ _R (μ) . March 4, 1996.     

Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

For a compact set K\subset R ^d with nonempty interior, the Markov constants M _n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M _n (K) = n ² /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B ^d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] ^d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.     

Exponentially small expansions in the asymptotics of the Wright function

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, _pΨ_q(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions.     

On Rational Interpolation to |x|

We consider Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes, and we show that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. Date received: August 18, 1995. Date revised: January 10, 1996.     

Asymptotic expansions of the representation formulae for (C 0) m-parameter operator semigroups 1 . 2

In this paper, we expand asymptotically the general representation formulae for (C _o) m-parameter operator semigroups. When we consider special semigroups, our results yield the asymptotic expansions for multivariate Feller operators. In particular, the asymptotic expansions for univariate and multivariate Bernstein operators are reobtained. See the related examples at the end.     

Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces

Summary.   Let X,X ₁,X ₂,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ ^d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ ^d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S _N ] of the normalized sums S _N =N ^−1/2(X ₁+⋯+X _N ) and show that

Uniform Estimates and the Whole Asymptotic Series of the Heat Content on Manifolds

We give a recursive algorithm for the computation of the complete asymptotic series, for small time, of the amount of heat inside a domain with smooth boundary in a Riemannian manifold; we consider arbitrary smooth initial data, and we impose Dirichlet condition on the boundary. When the Ricci curvature of the domain and the mean curvature of its boundary are both nonnegative, we also give sharp upper and lower bounds of the heat content which hold for all values of time. These estimates extend to convex sets of the Euclidean space having arbitrary boundary.     

The Remainder in Ruijsenaars’ Asymptotic Expansion of Barnes Double Gamma Function

We show that the remainder in Ruijsenaars’ asymptotic expansion of the logarithm of Barnes double gamma function gives rise to a completely monotone function. Fourier expansions of the multiple Bernoulli polynomials are also obtained. Research supported by the Carlsberg Foundation.