A Method for Incorporating Transcendentally Small Terms into the Method of Matched Asymptotic Expansions

The essential ideas behind a method for incorporating exponentially small terms into the method of matched asymptotic expansions are demonstrated using an Ackerberg–O'Malley resonance problem and a spurious solutions problem of Carrier and Pearson. One begins with the application of the standard method of matched asymptotic expansions to obtain at least the leading terms in outer and inner (Poincaré-type) expansions; some, although not all, matching can be carried out at this stage. This is followed by the introduction of supplementary expansions whose gauge functions are transcendentally small compared to those in the standard expansions. Analysis of terms in these expansions allows the matching to be completed. Furthermore, the method allows for the inclusion of globally valid transcendentally small contributions to the asymptotic solution; it is well known that such terms may be numerically significant.     

Asymptotic expansions and higher order properties of semi-parametric estimators in a system of simultaneous equations

Asymptotic expansions are made for the distributions of the Maximum Empirical Likelihood (MEL) estimator and the Estimating Equation (EE) estimator (or the Generalized Method of Moments (GMM) in econometrics) for the coefficients of a single structural equation in a system of linear simultaneous equations, which corresponds to a reduced rank regression model. The expansions in terms of the sample size, when the non-centrality parameters increase proportionally, are carried out to O(n⁻¹). Comparisons of the distributions of the MEL and GMM estimators are made. Also, we relate the asymptotic expansions of the distributions of the MEL and GMM estimators to the corresponding expansions for the Limited Information Maximum Likelihood (LIML) and the Two-Stage Least Squares (TSLS) estimators. We give useful information on the higher order properties of alternative estimators including the semi-parametric inefficiency factor under the homoscedasticity assumption.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

Arithmetic and metric properties of Oppenheim continued fraction expansions

We introduce a class of continued fraction expansions called Oppenheim continued fraction (OCF) expansions. Basic properties of these expansions are discussed and metric properties of the digits occurring in the OCF expansions are studied.     

Multiplier theorems for special Hermite expansions on C n

The weak type (1,1) estimate for special Hermite expansions on Cⁿ is proved by using the Calder/'on-Zygmund decomposition. Then the multiplier theorem in L^p(lpα) is obtained. The special Mermite expansions in twisted Hardy space are also considered. As an application, the multipliers for a certain kind of Laguerre expansions are given in L^p space.     

ON THE DEGREE OF APPROXIMATION BY WAVELET EXPANSIONS

In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x) ∈ L2 continuous in a finite interval (a,b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.     

Asymptotic expansions of the distributions of estimators in canonical correlation analysis under nonnormality

Asymptotic expansions of the distributions of typical estimators in canonical correlation analysis under nonnormality are obtained. The expansions include the Edgeworth expansions up to order O(1/n) for the parameter estimators standardized by the population standard errors, and the corresponding expansion by Hall's method with variable transformation. The expansions for the Studentized estimators are also given using the Cornish-Fisher expansion and Hall's method. The parameter estimators are dealt with in the context of estimation for the covariance structure in canonical correlation analysis. The distributions of the associated statistics (the structure of the canonical variables, the scaled log likelihood ratio and Rozeboom's between-set correlation) are also expanded. The robustness of the normal-theory asymptotic variances of the sample canonical correlations and associated statistics are shown when a latent variable model holds. Simulations are performed to see the accuracy of the asymptotic results in finite samples.     

Multiplier theorems for special Hermite expansions on Cn

The weak type (1,1) estimate for special Hermite expansions on Cⁿ is proved by using the Calder/'on-Zygmund decomposition. Then the multiplier theorem in L^p(lpα) is obtained. The special Mermite expansions in twisted Hardy space are also considered. As an application, the multipliers for a certain kind of Laguerre expansions are given in L^p space.     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

Localization and convergence of eigenfunction expansions

We state a localization principle for expansions in eigenfunctions of a self-adjoint second order elliptic operator and we prove an equiconvergence result between eigenfunction expansions and trigonometric expansions. We then study the Gibbs phenomenon for eigenfunction expansions of piecewise smooth functions on two-dimensional manifolds.     

Uniform Approximations of Bernoulli and Euler Polynomials in Terms of Hyperbolic Functions

Bernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for B _n ( nz +1/2) and E _n ( nz +1/2) in powers of n ⁻¹, and coefficients are rational functions of z and hyperbolic functions of argument 1/(2 z ). These expansions are uniformly valid for | z ± i /2π|>1/2π and | z ± i /π|1/π, respectively. For a real argument, the accuracy of these approximations is restricted to the monotonic region. The range of validity of the uniformity parameter z is enlarged, respectively, to regions of the form | z ± i /2( m +1)π|>1/2( m +1)π and | z ± i /(2 m +1)π|>1/(2 m +1)π, m =1,2,3,…, by adding certain combinations of incomplete gamma functions to these uniform expansions. In addition, the convergence of these improved expansions is stronger, and for a real argument, the accuracy of these improved approximations is also better in the oscillatory region.     

Minimal expansions in redundant number systems: Fibonacci bases and Greedy algorithms

We study digit expansions with arbitrary integer digits in base q (q integer) and the Fibonacci base such that the sum of the absolute values of the digits is minimal. For the Fibonacci case, we describe a unique minimal expansion and give a greedy algorithm to compute it. Additionally, transducers to calculate minimal expansions from other expansions are given. For the case of even integer bases q, similar results are given which complement those given in [6].     

The Borel Transform and Its Use in the Summation of Asymptotic Expansions

The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to "identically zero" expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x →0, allows us to compute the exponentially small term that these "identically zero" expansions represent.     

The Borel Transform and Its Use in the Summation of Asymptotic Expansions

The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to 'identically zero' expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x →0, allows us to compute the exponentially small term that these 'identically zero' expansions represent.     

Asymptotic Comparisons of Several Variance Estimators and their Effects for Studentizations

In this paper we obtain asymptotic representations of several variance estimators of U-statistics and study their effects for studentizations via Edgeworth expansions. Jackknife, unbiased and Sen's variance estimators are investigated up to the order o^p(n^-1). Substituting these estimators to studentized U-statistics, the Edgeworth expansions with remainder term o(n^-1) are established and inverting the expansions, the effects on confidence intervals are discussed theoretically. We also show that Hinkley's corrected jackknife variance estimator is asymptotically equivalent to the unbiased variance estimator up to the order o^p(n^-1).     

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.     

Asymptotic analysis of a singular Sturm-Liouville boundary value problem

Asymptotic expansions are given for the eigenvalues λ_n and eigenfunctions u_n of the following singular Sturm-Liouville problem with indefinite weight: $$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$ This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration. Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.     

On the distribution of the coefficients of normal forms for Frobenius expansions

Frobenius expansions are representations of integers to an algebraic base which are sometimes useful for efficient (hyper)elliptic curve cryptography. The normal form of a Frobenius expansion is the polynomial with integer coefficients obtained by reducing a Frobenius expansion modulo the characteristic polynomial of Frobenius. We consider the distribution of the coefficients of reductions of Frobenius expansions and non-adjacent forms of Frobenius expansions (NAFs) to normal form. We give asymptotic bounds on the coefficients which improve on naive bounds, for both genus one and genus two. We also discuss the non-uniformity of the distribution of the coefficients (assuming a uniform distribution for Frobenius expansions).     

The geometric structure of the expected/observed likelihood expansions

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.