Expansion of automorphic functions

This paper proposes a method of expansion for automorphic functions on Lie groups, which generalizes the usual Fourier series expansion of automorphic functions on SL₂()/SO(2). Detailed consideration is given to the special case of the expansion of functions on SL₃()/SO(3), which are automorphic with respect to SL₃().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 119–141, 1982.     

Lagrange Inversion and Schur Functions

Macdonald defined an involution on symmetric functions by considering the Lagrange inverse of the generating function of the complete homogeneous symmetric functions. The main result we prove in this note is that the images of skew Schur functions under this involution are either Schur positive or Schur negative symmetric functions. The proof relies on the combinatorics of Lagrange inversion. We also present a q-analogue of this result, which is related to the q-Lagrange inversion formula of Andrews, Garsia, and Gessel, as well as the operator of Bergeron and Garsia.     

More on the weeks method for the numerical inversion of the Laplace transform

Summary Most of the numerical methods for the inversion of the Laplace Transform require the values of several incidental parameters. Generally, these parameters are related to the properties of the algorithm and to the analytical properties of the Laplace Transform functionF(s).One of the most promising inversion methods, the Weeks methods, computes the inverse functionf(t) as a series expansion of Laguerre functions involving two parameters, usually denoted by andb. In this paper we characterize the optimal choiceb _opt ofb, which maximizes the rate of convergence of the series, in terms of the location of the singularities ofF(s).     

Automorphic functions on SL(m,ℝ)

One presents a method for the expansion of functions F:SL(m,)^h that are automorphic with respect ot the congruence subgroup in SL(m,). For m=2 the method gives the usual expansion into Fourier series. One gives a comparison with the expansion presented previously by the author in the paper: The expansion of automorphic functions, J. Sov. Math.,26, No. 3 (1984).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 164–176, 1983.     

Sampling expansions for functions having values in a Banach space

A sampling expansion for vector-valued functions having values in a Banach space, together with an inversion formula, is derived. The proof uses the concept of framing models of Banach spaces that generalizes the notion of frames in Hilbert spaces. Two examples illustrating the results are given, one involving functions having values in , and the second involving functions having values in for

     

On the injectivity of a class of Wiener-Hopf integral operators in a Hilbert space

A class of Wiener-Hopf integral operators, with kernels vanishing along the positive real axis, is obtained from considering weighted transaxial line-integrals of rotationally symmetric functions defined on ². An analysis of these operators is given when acting in, the Hilbert space L²(⁺). A necessary and sufficient condition for injectivity is established and inversion formulas are provided in some cases. A specific operator falling into this class, the so-called incomplete Abel transform., is presented and an inversion formula is given. This inversion formula makes precise a formal result previously established in Dallaset al. [J. Opt. Soc. Am. A4, 2039 (1987)] and it is also shown to be consistent with an inversion formula derived by Hansen [J. Opt. Soc. Am. A9, 2126 (1992)].This research was supported by NIH/NCI Grant R01 CA49261     

A Ray Method for the Asymptotic Solution of the Diffusion Equation

A general method is presented for finding asymptotic solutionsof problems posed for linear partial differential equationscontaining lower order (dispersive) terms. Powers of a largeparameter appear in these equations multiplying the lower orderterms. The expansion procedure is a "ray method", i.e. all ofthe functions that appear in the expansion satisfy ordinarydifferential equations along certain curves called rays. Special attention is paid to the equation of heat conduction.In this case, 1/ is related to the thermal conductivity of themedium. For this equation several problems are considered inwhich the parameter enters the data or the inhomogeneous (source)term in various ways.     

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.     

Scalar Products of Symmetric Functions and Matrix Integrals

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions, and integrals defining matrix-model partition functions. Using the fermionic Fock space representation, we prove an expansion of an associated class of KP and 2-Toda tau functions _r,n in a series of Schur functions generalizing the hypergeometric series and relate it to the scalar product formulas. We show how special cases of such tau functions can be identified as formal series for partition functions. A closed form expansion of log _r,n in terms of Schur functions is derived.     

A new application of Lagrange-Bürmann expansions. I. General principle

A new scheme is developed for improving the convergence of slowly convergent series solutions. The method is based on a transformation of variables of similarity form whereby the resulting composite function is constructed by its Lagrange-Bürmann expansion. It is the improved convergence of the new expansion that we take most advantage of in this method. The convergence of the Lagrange-Bürmann expansion as well as its inversion scheme is proved for analytic (object) functions. The inversion is required to recover from the Lagrange-Bürmann expansion the object function which is imbedded in the mapping functions. Several numerical examples demonstrate the improved convergence of the new method. The improvement owes much to the invariance properties of the mapping function under a group and the “built-in” feature of analytic continuation of the method. These features are elucidated in detail.     

Thermodynamics of the two-dimensional Heisenberg ferromagnet in the framework of the renormalization-group approach

A method of describing the thermodynamic functions of two-dimensional isotropic Heisenberg ferromagnets based on a generalization of the 2 + expansion is proposed. Interpolation formulas constructed in the framework of this approach make possible a smooth passage to the results obtained by the methods of high-temperature expansions.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 530–537, September, 1995.     

Expansions over a Simplex of Real Functions by Means of Bernoulli Polynomials

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval [a,b]R that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of C ^m -functions on a 2-dimensional simplex; a method to generalize the expansion on an N-dimensional simplex is also discussed. This new expansion is applied to find new cubature formulas for 2-dimensional simplex.     

On a Uniform Treatment of Darboux’s Method

Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darbouxs method can be used to give an asymptotic expansion for the coefficient of zⁿ in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.     

Modified ‐expansion method for finding exact wave solutions of the coupled Klein–Gordon–Schrödinger equation

The coupled Klein–Gordon–Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using Lie classical symmetries, and various solutions of the nonlinear ODE are obtained by the modified ‐expansion method proposed recently. With the aid of solutions of the nonlinear ODE, more explicit traveling wave solutions of the coupled Klein–Gordon–Schrödinger equation are found out. The traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. Copyright © 2012 John Wiley & Sons, Ltd.     

A Singular Expansion of Solution for a Regularized Compressible Stokes System

A compressible Stokes system is considered in a sector of the plane. The continuity equation is regularized by adding the diffusion term . We give a high-order expansion of corner singularities for the regularized system when the corner singularities for the Laplacian are implemented. A solution formula is constructed in an abstract way, and new associate singular functions are introduced for extracting high-order corner singularities. In the expansion, the smoother parts of the associate singular functions are used. Communicated by Rafael D. Benguriasubmitted 14/11/02, revised 12/08/03, accepted 04/10/03     

Asymptotic Solution of the Problem on an Accelerated Source of High-Frequency Oscillations

The problem of the wave field of a source moving in an inhomogeneous medium is considered. It is assumed that the velocity of the source is less than the velocity of sound for t < 0 and is greater than this velocity for t > 0 (subsonicsupersonic transition). An asymptotic expansion for the wave field in a neighborhood of the source is constructed on the basis of the well-known Hadamard ansatz. The expansion derived is uniform with respect to the velocity of the source and contains new special functions. These functions are generalizations of the Hankel and Bessel functions and possess some remarkable properties. Bibliography: 7 titles.     

Local asymptotic types

The local theory of asymptotic types is elaborated. It appears as coordinate-free version of part of Gohberg-Sigals theory of the inversion of finitely meromorphic, operator-valued functions at a point.Mathematics Subject Classification(2000): Primary 47A56; Secondary 06A11, 47L15     

The least square inversion method for the exterior ray transforms of 3D scalar and vector fields

Inversion of the scalar and vector ray transforms is performed in domain , ie, with the presence of an obstacle or singularity in the origin. Initially, the ray transforms of the basis functions for the scalar and vector fields are evaluated in an analytical form, and next, the inversion procedure is reduced to a linear system of equations by the use of the least squares method.     

Über die Radon-Transformation kreissymmetrischer Funktionen und ihre Beziehung zur Sommerfeldschen Theorie der Hankelfunktionen

For the Radon transform of functions with circular symmetry an inversion formula is proved in a new and elementary way. The inversion formula combined with Fourier theory is applied to Sommer-feld's integral for H, yielding a representation of products which generalizes Nicholson's integral for |H| ².     

Positive Harmonic Singularity at Infinity

When an inversion is taken, an equivalent version of the classical Bôcher theorem gives a representation for positive harmonic functions defined outside a compact set in ⁿ. This is generalized in an axiomatic potential theory (which includes Riemann surfaces), using the Martin integral representation.