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.     

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.     

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.     

Modular forms and Eisenstein's continued fractions

Found in the collected works of Eisenstein are twenty continued fraction expansions. The expansions have since emerged in the literature in various forms, although a complete historical account and self-contained treatment has not been given. We provide one here, motivated by the fact that these expansions give continued fraction expansions for modular forms. Eisenstein himself did not record proofs for his expansions, and we employ only standard methods in the proofs provided here. Our methods illustrate the exact recurrence relations from which the expansions arise, and also methods likely similar to those originally used by Eisenstein to derive them.     

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.     

Necessary and sufficient conditions for expansions of Wilson type

We consider expansions of the type arising from Wilson bases. We characterize such expansions for L^2（R）. As an application, we see that such an expansion must be orthonormal, in contrast to the case of wavelet expansions generated by translations and dilation.     

Greedy Algorithms with Prescribed Coefficients

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. We concentrate on studying algorithms that provide expansions into a series. We call such expansions greedy expansions. It was pointed out in our previous article that there is a great flexibility in choosing coefficients of greedy expansions. In that article this flexibility was used for constructing a greedy expansion that converges in any uniformly smooth Banach space. In this article we push the flexibility in choosing the coefficients of greedy expansions to the extreme. We make these coefficients independent of an element f ∈ X. Surprisingly, for a properly chosen sequence of coefficients we obtain results similar to the previous results on greedy expansions when the coefficients were determined by an element f.     

On the spectral problem arising in the solution of a mixed problem for the heat equation with a mixed derivative in the boundary condition

We study issues related to the uniform convergence of the Fourier series expansions of Hölder class functions in the system of eigenfunctions corresponding to a spectral problem obtained from a mixed problem for the heat equation. We prove a theorem on the equiconvergence of these expansions with expansions in a well-known orthonormal basis.     

Expansions about the Gamma for the Distribution and Quantiles of a Standard Estimate

We give expansions for the distribution, density, and quantiles of an estimate, building on results of Cornish, Fisher, Hill, Davis and the authors. The estimate is assumed to be non-lattice with the standard expansions for its cumulants. By expanding about a skew variable with matched skewness, one can drastically reduce the number of terms needed for a given level of accuracy. The building blocks generalize the Hermite polynomials. We demonstrate with expansions about the gamma.     

Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients

We establish Wiener type theorems and Paley type theorems for Laguerre polynomial expansions and disk polynomial expansions with nonnegative coefficients.     

Loaded differential operators: Convergence of spectral expansions

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

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.     

Tensors and likelihood expansions in the presence of nuisance parameters

Stochastic expansions of likelihood quantities are usually derived through oridinary Taylor expansions, rearranging terms according to their asymptotic order. The most convenient form for such expansions involves the score function, the expected information, higher order log-likelihood derivatives and their expectations. Expansions of this form are called expected/observed. If the quantity expanded is invariant or, more generally, a tensor under reparameterisations, the entire contribution of a given asymptotic order to the expected/observed expansion will follow the same transformation law. When there are no nuisance parameters, explicit representations through appropriate tensors are available. In this paper, we analyse the geometric structure of expected/observed likelihood expansions when nuisance parameters are present. We outline the derivation of likelihood quantities which behave as tensors under interest-respectign reparameterisations. This allows us to write the usual stochastic expansions of profile likelihood quantities in an explicitly tensorial form.     

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).     

Dependence of estimates of the local convergence rate of spectral expansions on the distance from an interior compact set to the boundary

We consider the problem on the convergence rate of biorthogonal expansions of functions in systems of root functions of a wide class of ordinary second-order differential operators defined on a finite interval. These expansions are compared with expansions of the same functions in Fourier trigonometric series in an integral or uniform metric on any interior compact subset of the main interval. We find the dependence of the equiconvergence rate of resulting expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

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 expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues

We consider asymptotic expansions for defective and excessive renewal equations that are close to being proper. These expansions are applied to the analysis of processor sharing queues and perturbed risk models, and yield approximations that can be useful in applications where moments are computable, but the distribution is not.     

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.     

Estimates of the local convergence rate of spectral expansions for even-order differential operators

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a wide class of even-order ordinary differential operators defined on a finite interval. These expansions are compared with the trigonometric Fourier series expansions of the same functions in the integral or uniform metric on an arbitrary interior compact set of the main interval as well as on the entire interval. We show the dependence of the equiconvergence rate of these expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the existence of infinitely many associated functions in the system of root functions.