Pointwise convergence of a class of non-orthogonal wavelet expansions

Non-orthogonal wavelet expansions associated with a class of mother wavelets is considered. This class of wavelets comprises mother wavelets that are not necessarily integrable over the whole real line, such as Shannon's wavelet. The pointwise convergence of these wavelet expansions is investigated. It is shown that, unlike other wavelet expansions, the ones under consideration do not necessarily converge pointwise to the functions at points of continuity, unless a more stringent condition, such as bounded variation, is imposed.

     

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.     

High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers

High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth–Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth–Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers outperform the corresponding two-impulse, low energy transfers in terms of propellant fraction, and (b) compared with the traditional Hohmann-like transfers, both the two-impulse, low energy and low-thrust, low energy transfers perform very efficiently, at the cost of flight time.     

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.     

Large degree asymptotics of generalized Bessel polynomials

Asymptotic expansions are given for large values of n of the generalized Bessel polynomials . The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the z-plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points z=±i/n are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed.     

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

Transplantation and multiplier theorems for Fourier-Bessel expansions

Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.

     

Uniform asymptotic methods for integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson’s lemma, Laplace’s method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.     

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.     

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 the distribution functions of Pickands-type estimators

The asymptotic expansions for the distribution functions of Pickands-type estimators in extreme statistics are obtained. In addition, several useful results on regular variation and intermediate order statistics are presented. Project supported by the National Natural Science Foundation of China (Grant No. 19601007) and Doctoral Program Foundation of Higher Education of China.     

On the equisummability of Hermite and Fourier expansions

We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions.     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

A note on the Landau constants

In this paper we establish new asymptotic expansions for the nth Landau constant G_n. An open problem is indicated as well.     

Bounding differences in Jager Pairs

Symmetrical subdivisions in the space of Jager Pairs for continued fractions-like expansions will provide us with bounds on their differences. Results will also apply to the classical regular and backwards continued fractions expansions, which are realized as special cases.     

Metric Properties and Exceptional Sets of the Oppenheim Expansions over the Field of Laurent Series

We investigate metric properties of the polynomial digits occurring in a large class of Oppenheim expansions of Laurent series, including Lüroth, Engel, and Sylvester expansions of Laurent series and Cantor infinite products of Laurent series. The obtained results cover those for special cases of Lüroth and Engel expansions obtained by Grabner, A. Knopfmacher, and J. Knopfmacher. Our results applied in the cases of Sylvester expansions and Cantor infinite products are original. We also calculate the Hausdorff dimensions of different exceptional sets on which the above-mentioned metric properties fail to hold.     

Two-Point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to two points. 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 two points.     

Methods de laplace et de la phase stationnaire sur l'espace de wiener

We obtain, using large deviations principles, full asymptotic expansions of functionals of Laplace type on Wiener space for general (i.e. degenerate) diffusions extending the results of Schilder [12], Azencott [2] and Doss [7, 20]. The variational hypothesis (non degeneracy of the minima) used here is shown to be optimal. The first term of the expansion is explicitly computed. Using the integration by parts of Malliavin calculus the stationary phase method is also developed. The results of this paper are the basic fact used (in [5]) to obtain the asymptotic expansion for small time of the density of a degenerate diffusion, they are also relevant for semi-classical expansions     

Some asymptotic expansions of moments of order statistics

Series expansions of moments of order statistics are obtained from expansions of the inverse of the distribution function. They are valid for certain types of distributions with regularly varying tails. We show that the expansions converge quickly when the sample size is moderate to large, and we obtain bounds on the rate of convergence. The special case of the Cauchy distribution is treated in more detail.