Abstract: Since they are rather important and quite accessible, we repeat the general theoretical facts concerning weights, moments and polynomials pertaining to fourth order differential equations. We then briefly discuss the squares of the differential equations of the second order, giving a number of easily derived examples of fourth order problems. This is followed by three new orthogonal polynomial sets satisfying fourth order differential equations, but which do not satisfy second order differential equations.

Abstract: This is not a book for the beginner. The text is intended for a course at advanced postgraduate level. The pre-requisites are a thorough grounding in modern pure mathematics and a knowledge of the more elementary aspects of electromagnetic theory. The author has selected a wide variety of topics from electromagnetic theory and general relativity and packed them into a little over two hundred pages. This makes the development in the text rather condensed and many points are relegated to the exercises. The mathematical approach throughout is modern based on 'index-free' differential geometry with special emphasis on differential forms. There is much in this book to stimulate the specialist. The detailed treatments of the diffraction of electromagnetic waves at a wedge and a cylinder are unusual in a book of this length. My only major criticism is of the chapter on general relativity. In attempting to cover a wide range of topics (Robertson-Walker metrics, plane waves, spherical blackholes, spherically symmetrical interior solutions and the existence of singularities) the reader's appetite is often whetted but rarely satisfied. One is left wishing the author had treated these topics more extensively. The translation by Dr Evans Harrell is excellent although at one point the algebra of forms is described as 'graduated' rather than the more usual 'graded'. The book is well produced by Springer with remarkably few misprints for a book of this typographical complexity.

Abstract: The purpose of this paper is to introduce a class of orthogonal functions similar in many respects to the classical orthogonal polynomials [6]. These functions are linear combinations of integral (positive, negative and zero) powers of a single complex variable z. Hence they are called Laurent polynomials.

Abstract: The properties of matrix orthogonal polynomials on the unit circle are investigated beginning with their recurrence formulas. The techniques of scattering theory and Banach algebras are used in the investigation. A matrix generalization of a theorem of Baxter is proved.

Abstract: A numerical procedure using orthogonal polynomials is used for extracting the background in measured spectra. An X-ray spectrum obtained with the energy-dispersive method is considered as an example.

Abstract: The construction of a class of invariant polynomials in several matrices extending the zonal polynomials is discussed. The method adopted generalized the orginal group-theoretic approach of James [9]. A table of three-matrix polynomials up to degree 5 is presented.

Abstract: A filter-design oriented theory is presented for polynomials with prescribed phase properties. The phase function is specified by its values and/or higher derivative values at a set of given frequencies. The polynomials are generated by recurrence formulae whose coefficients are calculated by a recursive algorithm. Formulae are presented to calculate the higher derivatives of some composite functions of the phase which are utilized in the process of flat phase approximations.

Abstract: An algorithm is presented for the computation of normalrzed Legendre polynomials. In order to permit wide ranges of argument, degree, and order of these functions, an \"extended-range\" arithmetic is introduced whereby a separate storage location is allocated to the exponent of a floatmg-pomt number. Since this device may have other applications, separate subroutines are developed for addition of extended-range numbers and also for conversion to and from ordinary floating-point form.

Abstract: An inequality is established which provides a unifying principle for the distribution of zeros of real polynomials and certain entire functions. This inequality extends the applicability of multiplier sequences to the class of all real polynomials. The various consequences obtained generalize and supplement several results due to Hermite-Poulain, Laguerre, Marden, Obreschkoff, Polya and Schur. 1* Introduction* In the vast literature dealing with the distribution of zeros of real polynomials and real entire functions, an important role is played by linear transformations T which possess the following property: (1)

Abstract: 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.

Abstract: Theq-Krawtchouk polynomials are the spherical functions for three different Chevalley groups over a finite field. Using techniques of Dunkl to decompose the irreducible representations with respect to a maximal parabolic subgroup, we derive three addition theorems. The associated polynomials are related to affine matrix groups.

Abstract: It is shown that Xx¡$ increases as X increases for 0 < X < 1, k = I, 2,..., [|J where xfy is the kth positive zero of ultraspherical polynomial P?\x). The aim of this work is to prove the following Theorem. Let xjfy, k = 1, 2, . . ., [f ], be the zeros of the ultraspherical polynomial F„(X)(x) in decreasing order on (0, 1), where 0 0, XxfX < (X + e)x£r\ *=l,2,...,[f]. Remark 1. For our purposes the following form of Sturm comparison theorem will prove useful. This formulation differs from the usual formulation [2, p. 19] in that fix) < F(x) is hypothesized for the interval a < x < Xm, rather than for the larger interval a < x < xm. (See the work [1] for the proof of this formulation of Sturm theorem.) Lemma. Let the functions y(x) and Y(x) be nontrivial solutions of the differential equations y"(x) + fix)y(x) = 0; Y"(x) + F(x) Y(x) = 0 and let them have consecutive zeros at xx, x2, .. . , xm and Xx, X2, . . . , Xm respectively on an interval (a, b). Suppose that fix) and F(x) are continuous, that fix) < F(x), a < x < Xm, and that (0 lim+ [y'(x)Y(x) y(x)Y'(x)] = 0. x—>cr -1 Then Xk < xk, k = 1, 2, . . . , m. Proof of the theorem. The function «(x) = (1 x2)A/2+1/4P^(x) Received by the editors March 6, 1981. 1980 Mathematics Subject Classification. Primary 33A45; Secondary 34A50.

Abstract: The two-dimensional subsonic, piecewise continuous-kernel function method used for studying either oscillatory or steady flows is extended in the present work to three-dimensi onal problems involving finite-span wings. The work treats questions associated with the choice of spanwise pressure polynomials, spanwise collocation points, and numerical integration techniques that need be faced by this method. A subsequent paper contains results which confirm the accuracy of the method, its rapid convergence, and its very high efficiency in terms of computational time. The method is tested for a limited class of geometrical discontinuities (i.e., at the wing root only). A third paper contains additional results which relate to a wider class of problems associated with geometrical discontinuities. EOMETRICAL discontinuities have become common in modern airplane wings, including not only control surface deflections but also wing chord discontinuities such as those existing at the root of a delta wing (discontinuity in the first derivative of the chord along the span), at leading-edge extensions, or at wing surface break points. Since these geometrical discontinuities lead to pressure singularities at the same geometrical locations, it is necessary to know the exact form of these pressure singularities if the use of the kernel function method (KFM) is contemplated. The lattice methods (i.e., the vortex or doublet) can successfully cope with unknown pressure singularities if their location is known, but they require a relatively large number of unknowns ("boxes") for convergence which at times leads to a relatively large residual error at the converged values.1'2 In Refs. 1 and 2, a different method is proposed which represents the pressure distribution by a set of piecewise continuous polynomials spanning the regions between adjoining singularities (also referred to as "boxes") and employs the KFM for solution of the pressure coefficients. It is shown in Refs. 1 and 2 where this two-dimensional problem was treated that such an ap- proach, referred to as the piecewise continuous-kernel func- tion method (PCKFM), has the ability to treat pressure discontinuities in a manner similar to the doublet-lattice method, with the added accuracy and rapid convergence characteristics of the kernel function method. In addition, it is not essential to determine the nature or form of the pressure singularities that might exist along some of the boundaries forming each box. However, to accelerate convergence, pressure singularities are assumed to be known only along the boundaries of the wing; or more specifically, the form of the leading-edge (LE), trailing-edge (TE), and wing-tip pressure singularities are assumed to be known and are treated in the analysis. All other pressure singularities are ignored during the analysis and their consideration is limited to the deter- mination of the boundaries for the different boxes. The basic problems associated with the two-dimensional PCKFM were treated in Refs. 1 and 2. These problems in- cluded the determination of orthogonal pressure polynomials for boxes with different known pressure singularities along their boundaries, the determination of the collocation points associated with the assumed pressure polynomials, the determination of the desired number of boxes, and the number of orthogonal polynomials required in each box. These problems will be addressed again while attempting to extend the PCKFM to wings with finite spans. Additional problems arising in three-dimensi onal flow configurations involve the formulation of numerical techniques which are required for the successful application of the method. These techniques, which are useful beyond the methods described in this work, will be developed herein. Wing-Tip Pressure Singularity and Associated

Abstract: We give the relation between Hermite–Pade approximants and vector orthogonal polynomials. An algorithm for calculating vector orthogonal polynomials near the diagonal is described. An exact multiple integral formula for vector orthogonal polynomials is proved. An example is given showing how the recurrence relations of Pade may be used to calculate Hermite–Pade approximants of degrees $m,\mu ,\mu $, with $\mu $ increasing.

Abstract: The solution of the trigonometric moment problem involves the computation of a (0/n) Laurent-Pade approximant for a positive real function on the complex unit circle. The incoming scheme is equivalent with the recursion for Szego's orthogonal polynomials, while the outgoing scheme is equivalent to the schur recursion for contractions of the unit disc. The numerical stability of both algorithms is proved under certain conditions via a backward error analysis.

Abstract: In this paper we obtain the solution of a class of nonlinear filtering problems in the form of a series expansion in terms of multiple Wiener integrals. The solution is explicit in the sense that the kernels of the integrals in the expansion are explicitly determine.

Abstract: Let the Radon transform Rf of a function f be given in the directions ωo,…, ωp ɛ S1. Based on the characterization of the range of R, the so-called consistency conditions, an approximation Rfp in terms of orthogonal polynomials is calculated. It is shown that the L2-error of fp is of the order p -αif the function f belongs to the Sobolev space H о α . The method is applied for reconstructing an object from its x-ray projections in arbitrary directions. In order to use the fast reconstruction algorithms approximations of the data in equally distributed directions are provided. The complexity of the algorithm and numerical limitations are studied, numerical experiments show the usefulness of the procedure.