Application of Laguerre matrix polynomials to the numerical inversion of Laplace transforms of matrix functions

This paper presents an application of Laguerre matrix polynomial series to the numerical inversion of Laplace transforms of matrix functions.     

Application of high order numerical quadratures to numerical inversion of the Laplace transform

In this paper, new algorithms are proposed for Fredholm integral equations of the first kind corresponding to the inverse Laplace transform. We apply high order numerical quadratures to the truncated integral equation and apply regularization to the discretized linear systems. The resulted regularized least square problems are then solved by the reduced QR factorization method. Several examples taken from the literature are tested. Numerical results show that the approximate inverse Laplace transform obtained by our approach can be very accurate.     

An exact far-field inversion for the Born approximation and plane wave decompositions for Hankel functions

We consider the Born approximation (representative for first-order approximations) of the scattering problem for the scalar Helroholtz equation with a fixed real-valued free-space wavenumber and a complex-valued compactly supported potential. The boundary condition is the Sommerfeld radiation condition. We derive an exact series-integral representation of the potential from the Fourier coefficients of its far-field pattern, suitable for discussion of the connected stability problem. Furthermore we stress the connection between this representation and some plane wave decompositions for Hankel functions. Without loss of generality we restrict ourselves to the case of two space dimensions.     

On certainp-harmonic functions in the plane

A representation theorem is given for certainp-harmonic functions (1<p<) in the plane, near a corner of the domain of definition. The proof uses stream functions and the hodograph transformation. The stream function of ap-harmonic function isp-harmonic, where 1/p+1/p=1. Solutions of the formr ^k f(ø) (polar coord.) are crucial for the representation.     

The numerical approximation of analytic functions in the complex domain

In this paper we are concerned with the numerical approximation of analytic functions in the complex domain. This approach leads to an efficient technique for deriving an interpolation polynomial as well as a useful error analysis.     

A method for the numerical inversion of Laplace transforms

In this paper a numerical inversion method for Laplace transforms, based on a Fourier series expansion developed by Durbin [5], is presented. The disadvantage of the inversion methods of that type, the encountered dependence of discretization and truncation error on the free parameters, is removed by the simultaneous application of a procedure for the reduction of the discretization error, a method for accelerating the convergence of the Fourier series and a procedure that computes approximately the ‘best’ choice of the free parameters. Suitable for a given problem, the inversion method allows the adequate application of these procedures. Therefore, in a big range of applications a high accuracy can be achieved with only a few function evaluations of the Laplace transform. The inversion method is implemented as a FORTRAN subroutine.     

The boundary Harnack inequality for infinity harmonic functions in the plane

We prove the boundary Harnack inequality for positive infinity harmonic functions vanishing on a portion of the boundary of a bounded domain under the assumption that is a quasicircle.

     

Generalization of the Neumann problem to harmonic functions outside cuts on the plane

We consider a boundary value problem for harmonic functions outside cuts on the plane. The jump of the normal derivative and a linear combination of the normal derivative on one side with the jump of the unknown function are given on each cut. The problem is considered with three conditions at infinity, which lead to distinct results on the existence and number of solutions. We obtain an integral representation of the solution in the form of potentials whose density satisfies a uniquely solvable Fredholm integral equation of the second kind.     

Polynomial approximation of functions from Hardy classes in simply-connected domains in the plane

We study the question of the connection between properties of the domain G and the density of polynomials in the Hardy classes HP(G) and also establish Jackson and Bernshtein type approximation theorems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1266–1271, September, 1990.     

Projective equivalence of smooth functions on the plane

In this paper, we find differential invariants for the action of the projective group on the space of smooth functions on the plane and propose a classification of orbits of this action.     

The boundary behavior of functions meromorphic in bounded plane regions

The object of this note is to extend the classical theorems of Fatou-Nevanlinna, Riesz-Nevanlinna, Riesz-Tsuji, Lusin-Privalov and Lusin-Privalov-Tsuji to bounded plane regions.     

The method of step by step inversion for numerical solution of the Biharmonic equation

The research was supported by the Russian Foundation for Fundamental Research (Grant 93-01-00494).     

Spline regularization of numerical inversion of Mellin transform

A method is described for inverting the Mellin transform which uses an expansion in Laguerre polynomials and converts the Mellin transform to the Laplace transform, then the Laplace transform is converted to the first kind convolution integral equation by a suitable substitution. The integral equation so obtained is an ill-posed problem and we use the spline regularization to solve it. The performance of the method is illustrated by the inversion of the test functions available in the literature [J. Inst. Math. & Appl. 20 (1977), p. 73], [J. Math. Comp. 53 (1989), p. 589], [J. Sci. Stat. Comp. 4 (1983), p. 164]. The effectiveness of the method is shown by results obtained demonstrated by means of tables and diagrams.     

Fourier inversion for multidimensional characteristic functions

A density functionf(x),xR ⁿ is said to bepiecewise smooth if for eachxR ⁿ , the mean value function is piecewiseC with compact support. (d is normalized surface measure on the unit sphere). The Fourier transform is with spherical partial sum . Theorem. For suchf, lim _r f _R (x)=M ₀+f(x) if and only ifrM _r f(x) hask=[(n–3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim where 0 is uniquely determined.     

Global inversion of functions: an introduction

This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications and we give a self-contained elementary exposition.The first part is devoted to the celebrated Hadamard-Caccioppoli theorem on proper local homeomorphisms treated in the framework of the Hausdorff spaces. In the proof, the concept of -limit set is used in a crucial way and this is perhaps the novelty of our approach.In the second part we deal with open sets in Banach spaces. The concept of attraction basin here is the main tool of our exposition which also shows a few recent results, here extended from finite dimensional to general Banach spaces, together with the classical theorem of Hadamard-Levy which assumes that the operator norm of the inverse of the derivative does not grow too fast (roughly at most linearly).Dedicated to Roberto Conti on the occasion of his 70th birthdaySupported by the Ministero dell'Università e della Ricerca Scientifica e Technologica, Progetto 40% Equazioni Differenziali Ordinarie e Applicazioni.     

New quadrature formulas for the numerical inversion of the Laplace transform

New quadrature formulas for the evaluation of the Bromwich integral, arising in the inversion of the Laplace transform are discussed. They are obtained by optimal addition of abscissas to Gaussian quadrature formulas. A table of abscissas and weights is given.