Orthogonal Polynomial Series Research Articles

Series of Orthogonal Polynomials as Hyperfunctions

The convergence of series of orthogonal polynomials and of associated functions of the second kind in a general setting is studied. Series whose coefficients satisfy $a_n = O(1 + \varepsilon _n )^n $ are shown to converge in the sense of hyperfunctions in the first case and to holomorphic functions in the second. The latter are shown to be the analytic representations of the former.

On the Weighted Galerkin Method of Numerical Solution of Cauchy Type Singular Integral Equations

The well-known weighted Galerkin method for the direct numerical solution of Cauchy type singular integral equations of the first and the second kind, based on the approximation of the unknown function by a finite series of appropriate orthogonal polynomials, is shown to be identical to the corresponding method based on the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind and the application to the latter of the weighted Galerkin method. This result permits the direct transfer of the whole set of convergence and rate-of-convergence results for the weighted Galerkin method of numerical solution of Fredholm integral equations with regular kernels to the case of Cauchy type singular integral equations. An application to the case where Chebyshev polynomials are used is also made, and the available convergence results for the direct weighted Galerkin method are rederived with essentially no analysis.

Series of Orthogonal Polynomials as Boundary Values

The relations between expansions in orthogonal polynomials of generalized functions on the real axis and certain holomorphic functions in the upper and lower half planes are studied. The holomorphic functions are given by series of functions of the second kind which satisfy the same recurrence formula as the polynomials. A space of generalized functions associated with the polynomials is first introduced. Each element in this space has an analytic representation given by such series whose jump across the real axis is given by the element. Under certain conditions the singularities of the analytic representation may be related to singularities of the associated power series.

Momentum eigenfunctions in the complex momentum plane. IV. The construction of local potential functions. Perturbation series

Expansion of exchange amplitudes, in inverse powers of a momentum, as a means of generating local exchange potentials has been studied further. The first term in such expansions varies inversely as the square of the momentum. Huo [J. Chem. Phys. 67, 5133 (1977)] obtained a local exchange potential by replacing a momentum by its average value obtained from a semiclassical model. The Xα potential with α=1 was obtained. The present research shows that mometum squared should be averaged and this leads to the Xα potential with α=20/27. The potentials were tested numerically for beryllium and although the potential with α=20/27 was superior in some instances to that with α=1, this was not always the case. In fact, for general all around use, α=1 provided best results. This apparent contradiction has been resolved by extending the expansion of exchange amplitude by one more term and using the semiclassical model to obtain the averages. The Xα potential is again obtained but with a α=80/81, a value differing negligibly from unity. Using the known behavior of exchange amplitudes in the complex momentum plane, a method is developed for testing the convergence of expansions in inverse powers of the momentum. The power series expansions are shown to be divergent. This provides a plausible explanation for the infinities at r=0 observed by Huo and shows that inclusion of more and more terms cannot lead to an exact amplitude. Although divergent, the series are asymptotic, i.e., they hold more accurately as the momentum increases as long as only a few terms are used. Therefore, the formulas obtained by Huo are approximately correct since she showed that the average momenta are large. Some different method must be found, however, for improving the accuracy. Two expansions in series of orthogonal polynomials are developed for which the first term gives a local exchange potential. The convergence region of each series is determined.

Edge effect in laminate composite materials

In other studies dealing with the edge effect in composite materials [1-5] each layer was regarded as an anisotropic homogeneous one, all layers rigidly joined together and the entire specimen in a state of generalized plane strain under tension or flexure. The equations of anisotropic elasticity were solved by the finite-difference method [i, 2], by the finite-element method [3, 4], or by expansion into a double series of Legendre polynomials [5]. In the way these methods have been used, however, they hardly yield a true description of the edge effect with respect to stresses, because they are efficacious only when a bounded function will be the solution and not when the stresses can have singularities, as in the case of composite materials. In the application of these methods, the edge stresses remain bounded, but their maximum increases with increasing accuracy of the method (e.g., with an increasing number of terms retained in the series of orthogonal polynomials [5]) and this indirectly confirms the existence of a singularity. Here it will be demonstrated that in composite materials with characteristics close to real, there may appear a stress singularity, then there will be given examples of calculating the degree of such a singularity as a function of the reinforcement factor and of the fiber lay angle in adjacent layers of the composite

Model reduction of dynamic systems over a frequency interval†

Model reduction of linear, time-invariant, single-input, single-output systems over desired frequency intervals (low-pass, band-pass and high-pass) is considered in this paper. Order reduction is effected by manipulating two orthogonal polynomial series, one representing the high-order system and the other representing the approximating low-order model. The method is a generalization of the classical Padé approximations, however, by a special transformation it becomes the simple (classical) Padé problem, thus retaining the computational attractiveness of the latter.

Effective interactions for dominant shell-model configurations:ds-shell example

Starting from the ds-shell interaction of Kuo, orthogonal polynomial and Brillouin-Wigner expansions for effective interactions in the dominant configuration space for low-lying states of /sup 22/Na (J=2, T=0) have been calculated. Although the states represented have no more than 27% of their strength in the model space, the orthogonal polynomial series converges in low order and good predictions can be obtained with only a third order theory, especially if self consistency is required. For the Brillouin-Wigner expansion, however, we find that higher-order approximations (approx.7th order) are required to achieve reasonable accuracy for states lying below the lowest eigenvalue of the interaction in the excluded space. For higher-lying states the Brillouin-Wigner self-consistent results diverge.

Calculation of Univariate and Bivariate Normal Probability Functions

Mill's ratio is expressed as a convergent series in orthogonal polynomials. Truncation of the series provides an approximation for the complemented normal distribution function $Q(x)$, with its maximum error at a finite value of $x$. The analogous approximation for $xQ(x)$ is used to obtain a new method of calculating the bivariate normal probability function.

Fracture of solids containing arrays of cracks

This survey is largely a collection of the author's recent results on the fracture characteristics of elastic solids containing inhomogeneities in the form of slitlike cracks. The crack tip opening displacement is used as a measure of the fracture behaviour of the solid under three different deformation situations: (a) Opening Mode I; (b) Sliding Mode II; and (c) Tearing Mode III. Various crack geometries have been considered for which closed-form analytical solutions do not seem feasible. To this category belong (a) A stack of cracks (cracks with constant distance of vertical separation) and (b) A doubly-periodic array of cracks forming a rectangular or diamond-shaped pattern. Besides summarizing various results which have been published elsewhere in Scientific Literature, some new results, and necessary amplification of the previous results, are included herein to make the survey as self-contained as possible. It is hoped that it will be found useful by many research workers engaged in crack interaction and propagation problems. The survey is divided into five sections. Section 1 gives a general introduction to the problem at hand and brings out the importance of studying the subject. Sufficient reference is made to the available literature on the subject, without being unduly overemphatic. It is likely that many otherwise good papers have been omitted either through oversight or because they were thought to be peripheral to the subject matter of the survey. Interested readers should, however, be able to find sufficient cross references in the literature cited herein. This section also touches upon the necessity of using the dislocation formalism in solving the problems at hand. For obvious reasons, no attempt has been made to dwell upon this equivalence of slitlike cracks and straight dislocations, the interested reader being again referred to relevant literature. Section 2 formulates the problem in mathematical terms as one consisting in the solution of a singular integral equation. The latter results from the traction-free conditions on the crack faces. The equation is suitably non-dimensionalized and its kernel decomposed into a singular and a non-singular part as dictated by the method of solution. Section 3 presents a perturbation solution for widely spaced cracks. The solution is restricted to a stack of cracks under plane strain conditions. Section 4 deals with an approximate method used to solve the singular integral equation. It is based on an expansion of the non-singular part of the kernel in a series of orthogonal polynomials. The solution of the singular integral equation allows us to calculate the crack tip opening displacement as a function of the externally applied stress and the crack geometry. The results for various loading modes and crack configurations are presented graphically in Section 5 and discussed from the point of fracture initiation from multiple cracks. Where possible, the results are compared with that for an isolated relaxed crack for a better understanding of the change brought about by an array of interacting cracks. In order not to interrupt the text all complicated mathematical expressions have been grouped together and listed at the end of the relevant section.

ON THE CONVERGENCE OF RATIONAL APPROXIMATIONS TO POLYNOMIAL EXPANSIONS IN DOMAINS OF MEROMORPHY OF A GIVEN FUNCTION

We establish analogs of the theorem of Montessus de Ballore for rational approximations (of the type of the Pade approximants) to series of orthogonal polynomials and of Faber polynomials. Bibliography: 11 titles.

Uniqueness of Pade Approximants From Series of Orthogonal Polynomials

Uniqueness of Pade Approximants From Series of Orthogonal Polynomials

Auroral ion velocity distributions for a polarization collision model

We have calculated the effect that convection electric fields have on the velocity distribution of auroral ions at the altitudes where the plasma is weakly-ionized and where the various ion-neutral collision frequencies are much smaller than the ion cyclotron frequencies, i.e. between about 130 and 300 km. The appropriate Boltzmann equation has been solved by expanding the ion velocity distribution function in a generalized orthogonal polynomial series about a bi-Maxwellian weight factor. We have retained enough terms in the series expansion to enable us to obtain reliable quantitative results for electric field strengths as large as 90 mV m −1. Although we have considered a range of ion-neutral scattering mechanisms, our main emphasis has been devoted to the long-range polarization interaction. In general, we have found that to lowest order the ion velocity distribution is better represented by a two-temperature or bi-Maxwellian distribution than by a one-temperature Maxwellian, with there being different ion temperatures parallel and perpendicular to the geomagnetic field. However, the departures from this zeroth-order bi-Maxwellian distribution become significant when the ion drift velocity approaches (or exceeds) the neutral thermal speed.

Uniqueness of Padé approximants from series of orthogonal polynomials

It is proved that whenever a nonlinear Padé approximant, derived from a series of orthogonal polynomials, exists, it is unique.

ORTHOGONAL POLYNOMIAL SOLUTIONS TO SOME MIXED BOUNDARY-VALUE PROBLEMS IN ELASTICITY THEORY

Crack and frictionless punch problems in two- and three-dimensional elasticity have been extensively studied and complete solutions are available. In general these solutions involve the computation of awkward definite integrals. In this paper it is shown that if the specified pressure over a crack or displacement over a punch are expressed as series in certain orthogonal polynomials, the surface displacements and stresses may also be expressed in simple series form.

L 1-Convolution algebras: Representation and factorization

Let * denote a convolution with respect to which l1 becomes a Banach algebra. Necessary and sufficient conditions are given for (l1, *) to be represented by pointwise products of series of orthogonal polynomials. Properties of the polynomials are related to properties of the convolution; and, in the case of positive convolutions, an analogue of Hincin's factorization theorem is obtained through the use of Delphic semigroups.

Use of generalized orthogonal polynomial solutions of Boltzmann’s equation in certain aeronomy problems: Auroral ion velocity distributions

We present a method of solution of Boltzmann's equation that is valid for conditions where the species velocity distribution function departs significantly from a Maxwellian. This method should be particularly useful for problems in aeronomy and space physics where there are large or rapidly varying forces acting on a neutral or ionized gas or for problems dealing with highly rarefied gases where the mean free path for collisions is comparable to or greater than the characteristic scale length of the problem. The method of solution, which was first introduced by Mintzer, corresponds to expanding the species distribution function in a generalized orthogonal polynomial series with an arbitrary weight factor. The specific form of the weight factor depends on the details of the problem. The method is illustrated by calculating auroral ion velocity distributions for a relaxation collision model, and the results are compared with the velocity distributions obtained from an exact solution to Boltzmann's equation.

Padé approximation of Stieltjes series

Using Nuttall's compact formula for the [ n, n − 1] Pad'e approximant, the authors show that there is a natural connection between the Padé approximants of a series of Stieltjes and orthogonal polynomials. In particular, we obtain the precise error formulas. The [ n, n − 1] Padé approximant in this case is just a Gaussian quadrature of the Stieltjes integral. Hence, analysis of the error is now possible and under very mild conditions it is shown that the [ n, n + j], j ⩾ − 1, Padé approximants converge to the Stieltjes integral.

A technique for analysing integral rate data in non‐ideal flow reactors to estimate unknown parameters

AbstractA technique is described for analysing integral rate data taken from a non‐ideal flow chemical reactor characterised by the axial dispersion model, using a weighted residual approach.This approach avoids the formidable computational difficulties involved with the solution of the boundary value problem associated with the model differential equations. The measured concentration profile is approximated by a series of orthogonal polynomials. Model parameters are estimated by specifying that the fitting polynomial satisfy certain moment relations derived from the mass balance equation and boundary conditions.If the form of the rate equation is known, a priori, the estimation of the rate constant and the dispersion coefficient becomes explicit. In the more general case, in which the reaction order is also unknown, the estimation problem reduces to a uni‐dimensional search for the reaction order parameter (or the adsorption parameter in the case of complex kinetics). The method is applied to both linear and non‐linear systems and conclusions are drawn regarding the sensitivity of the chemical kinetic and transport parameters to experimental error.

Aberration balancing in rotationally symmetric lenses*

The expansion of the aberration function of lens systems in analytic form is considered. The results are used to obtain an expansion of the aberration function of a rotationally symmetric system, with either a circular or annular pupil, in a series of orthogonal polynomials. The significance of this for aberration balancing is discussed, and algorithms for obtaining such an expansion numerically are described.

An Equiconvergence Theorem for a Class of Eigenfunction Expansions

A recent result of Muckenhoupt concerning the convergence of the expansion of an arbitrary function in terms of the Hermite series of orthogonal polynomials is generalised to a class of orthogonal expansions which arise from an eigenfunction problem associated with a second-order linear differential equation.