Showing papers on "Discrete orthogonal polynomials published in 1982"

Approximation results for orthogonal polynomials in Sobolev spaces

Abstract: We analyze the approximation properties of some interpolation operators and some L2-orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function o(x1, . . ., Xd), d > 1. The error estimates for the Legendre system and the Chebyshev system of the first kind are given in the norms of the Sobolev spaces H'. These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods. 0. Introduction. Spectral methods are a classical and largely used technique to solve differential equations, both theoretically and numerically. During the years they have gained new popularity in automatic computations for a wide class of physical problems (for instance in the fields of fluid and gas dynamics), due to the use of the Fast Fourier Transform algorithm. These methods appear to be competitive with finite difference and finite element methods and they must be decisively preferred to the last ones whenever the solution is highly regular and the geometric dimension of the domain becomes large. Moreover, by these methods it is possible to control easily the solution (filtering) of those numerical problems affected by oscillation and instability phenomena. The use of spectral and pseudo-spectral methods in computations in many fields of engineering has been matched by deeper theoretical studies; let us recall here the pioneering works by Orszag [25], [26], Kreiss and Oliger [14] and the monograph by Gottlieb and Orszag [13]. The theoretical results of such works are mainly concerned with the study of the stability of approximation of parabolic and hyperbolic equations; the solution is assumed to be infinitely differentiable, so that by an analysis of the Fourier coefficients an infinite order of convergence can be achieved. More recently (see Pasciak [27], Canuto and Quarteroni [10], [11], Maday and Quarteroni [20], [211, [22], Mercier [23]), the spectral methods have been studied by the variational techniques typical of functional analysis, to point out the dependence of the approximation error (for instance in the L2-norm, or in the energy norm) on the regularity of the solution of continuous problems and on the discretization parameter (the dimension of the space in which the approximate solution is sought). Indeed, often the solution is not infinitely differentiable; on the other hand, sometimes even if the solution is smooth, its derivatives may have very Received August 9, 1980; revised June 12, 1981. 1980 Mathematics Subject Classification. Primary 41A25; Secondary 41A 10, 41A05. ? 1982 American Mathematical Society 0025-571 8/82/0000-0470/$06.00 (67 This content downloaded from 207.46.13.111 on Tue, 09 Aug 2016 06:29:39 UTC All use subject to http://about.jstor.org/terms 68 C. CANUTO AND A. QUARTERONI large norms which affect negatively the rate of convergence (for instance in problems with boundary layers). Both spectral and pseudo-spectral methods are essentially Ritz-Galerkin methods (combined with some integration formulae in the pseudo-spectral case). It is well known that when Galerkin methods are used the distance between the exact and the discrete solution (approximation error) is bounded by the distance between the exact solution and its orthogonal projection upon the subspace (projection error), or by the distance between the exact solution and its interpolated polynomial at some suitable points (interpolation error). This upper bound is often realistic, in the sense that the asymptotic behavior of the approximation error is not better than the one of the projection (or even the interpolation) error. Even more, in some cases the approximate solution coincides with the projection of the true solution upon the subspace (for instance when linear problems with constant coefficients are approximated by spectral methods). This motivates the interest in evaluating the projection and the interpolation errors in differently weighted Sobolev norms. So we must face a situation different from the one of the classical approximation theory where the properties of approximation of orthogonal function systems, polynomial and trigonometric, are studied in the LP-norms, and mostly in the maximum norm (see, e.g., Butzer and Berens [6], Butzer and Nessel [7], Nikol'skiT [24], Sansone [291, Szego [30], Triebel [31], Zygmund [32]; see also Bube [5]). Approximation results in Sobolev norms for the trigonometric system have been obtained by Kreiss and Oliger [15]. In this paper we consider the systems of Legendre orthogonal polynomials, and of Chebyshev orthogonal polynomials of the first kind in dimension d > 1. The reason for this interest must be sought in the applications to spectral approximations of boundary value problems. Indeed, if the boundary conditions are not periodic, Legendre approximation seems to be the easiest to be investigated (the weight w is equal to 1). On the other hand, the Chebyshev approximation is the most effective for practical computations since it allows the use of the Fast Fourier Transform algorithm. The techniques used to obtain our results are based on the representation of a function in the terms of a series of orthogonal polynomials, on the use of the so-called inverse inequality, and finally on the operator interpolation theory in Banach spaces. For the theory of interpolation we refer for instance to Calderon [8], Lions [17], Lions and Peetre [19], Peetre [28]; a recent survey is given, e.g., by Bergh and Lofstrom [4]. An outline of the paper is as follows. In Section 1 some approximation results for the trigonometric system are recalled; the presentation of the results to the interpolation is made in the spirit of what will be its application to Chebyshev polynomials. In Section 2 we consider the La-projection operator upon the space of polynomials of degree at most N in any variable (w denotes the Chebyshev or Legendre weight). In Section 3 a general interpolation operator, built up starting by integration formulas which are not necessarily the same in different spatial dimensions, is considered, and its approximation properties are studied. In [22] Maday and Quarteroni use the results of Section 2 to study the approximation properties of some projection operators in higher order Sobolev norms. Recently, an interesting method which lies inbetween finite elements and This content downloaded from 207.46.13.111 on Tue, 09 Aug 2016 06:29:39 UTC All use subject to http://about.jstor.org/terms ORTHOGONAL POLYNOMIALS IN SOBOLEV SPACES 69 spectral methods has been investigated from the theoretical point of view by Babuska, Szabo and Katz [3]. In particular they obtain approximation properties of polynomials in the norms of the usual Sobolev spaces. Acknowledgements. Some of the results of this paper were announced in [9]; we thank Professor J. L. Lions for the presentation to the C. R. Acad. Sci. of Paris. We also wish to express our gratitude to Professors F. Brezzi and P. A. Raviart for helpful suggestions and continuous encouragement. Notations. Throughout this paper we shall use the following notations: I will be an open bounded interval c R, whose variable is denoted by x; Q the product Id C Rd (d integer > 1) whose variable is denoted by x = (x(.')I_ d; for a multi-integer k E Zd, we set ikV = jd X I'12 and IkloK = m x 1, Dj = a/ax@). The symbol X'J=p (q eventually + oo) will denote the summation over all integral k such that p 0 in U. Set L2(Q) = ({: Q -C I 0 is measurable and ( 0, set Hs ( () = C E L(Q) I 1111ksI, < +?}, where /d 2 11I412I= kENd f DI L/)4 D w dx.

On Generating Orthogonal Polynomials

Abstract: We consider the problem of numerically generating the recursion coefficients of orthogonal polynomials, given an arbitrary weight distribution of either discrete, continuous, or mixed type We discuss two classical methods, respectively due to Stieltjes and Chebyshev, and modern implementations of them, placing particular emphasis on their numerical stability properties The latter are being studied by analyzing the numerical condition of appropriate finite-dimensional maps A number of examples are given to illustrate the strengths and weaknesses of the various methods and to test the theory developed for them

Orthogonal Polynomials, Duality and Association Schemes

Abstract: This paper reconstructs and characterizes the Askey–Wilson orthogonal polynomials as those having duals (in the sense of Delsarte) which are also orthogonal. It introduces the concepts of eigenvalues and Delsarte’s duality to the study of orthogonal polynomials and provides those interested in P- and Q-polynomial association schemes with a closed form for their parameters.

Reducible polynomials in more than one variable

Abstract: Polynomials in more than one variable arise frequently in multidimensional signal processing applications. Unlike polynomials in a single variable, multidimensional polynomials cannot, in general, be factored. In this note, it is shown that the set of factorable multi-dimensional polynomials is extremely small in the sense that almost all polynomials in two or more variables are irreducible.

Scattering Theory and Matrix Orthogonal Polynomials on the Real Line

Abstract: The techniques of scattering and inverse scattering theory are used to investigate the properties of matrix orthogonal polynomials. The discrete matrix analog of the Jost function is introduced and its properties investigated. The matrix distribution function with respect to which the polynomials are orthonormal is constructed. The discrete matrix analog of the Marchenko equation is derived and used to obtain further results on the matrix Jost function and the distribution function.

The krall polynomials: a new class of orthogonal polynomials

Abstract: A new set of orthogonal polynomials is found that are solutions to a sixth order formally self adjoint differential equation. These polynomials are shown to generalize the Legendre and Legendre type polynomials. We also show that these polynomials satisfy many properties shared by the classical orthogonal polynomials of Jacobi, Laguerre and Hermite.

The Recursion Formulas for Orthogonal Polynomials in n Variables

Abstract: In this paper we introduce a very promising matrix-vector notation for orthogonal polynomials in n variables. This enables us to prove some new properties of the related recursion formulas.

Orthogonality and Recursion Formulas for Polynomials in n Variables

Abstract: In this paper we obtain a necessary and sufficient condition for a sequence of polynomials in n variables to be orthogonal A comparison criterion of orthogonality for these polynomials is also established In conclusion an integral representation for a corresponding quasi-inner product is given

Krawtchouk Polynomials, a Unification of Two Different Group Theoretic Interpretations

Abstract: The canonical matrix elements of irreducible unitary representations of $SU(2)$ are written as Krawtchouk polynomials, with the orthogonality being the row orthogonality for the unitary representation matrix. Dunkl’s interpretation of Krawtchouk polynomials as spherical functions on wreath products of symmetric groups is generalized to the case of intertwining functions. A conceptual unification is given of these two group theoretic interpretations of Krawtchouk polynomials.

Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings

Abstract: We first study a family of invariant transformations for the integer moment problem. The fixed point of these transformations generates a positive measure with support on a Cantor set depending on a parameter q. We analyze the structure and properties of the set of orthogonal polynomials with respect to this measure. Among these polynomials, we find the iterates of the canonical quadratic mapping: F(x)=(x−q)2, q≧2. It appears that the measure is invariant with respect to this mapping. Algebraic relations among these polynomials are shown to be analytically continuable below q=2, where bifurcation doubling among stable cycles occurs. As the simplest possible consequence we analyze the neighborhood of q=2 (transition region) for q<2.

Vector polynomials orthogonal to the gradient of Zernike polynomials.

Abstract: A set of vector polynomials is constructed, and it is shown that they are orthonormal to the gradient of the Zernike polynomials. Such a set can be used to obtain directly the Zernike decomposition of the wave front from the measurements involving the gradient of the wave front.

Equivalence, adjoints and symmetry of quasi-differential expressions with matrix-valued coefficients and polynomials in them

Abstract: Quasi-differential expressions with matrix-valued coefficients, which generalize those of Shin and Zettl, are considered with regard to equivalence, adjoints and symmetry. The characterization results imply that in the scalar case the class of quasi-differential expressions considered here coincides with that of Shin and is equivalent to that of Zettl. Furthermore polynomials in quasi-differential expressions are defined as expressions of the same kind and shown to coincide with the usual ones. Finally it is indicated that the known general results for the deficiency indices carry over to quasi-differential expressions.

Some characterizations of appell andℊ-appell polynomials

Abstract: Several characterizations are given for the wellknown Appell polynomials and for their basic analogues: the ℊ-Appell polynomials defined by Equation (3.3)below. The main results contained in Theorems 1, 2and 3of the present paper, and the applications considered in Section 2,are believed to be new. Some interesting connections with earlier results are also indicated.

Nonclassical orthogonal polynomials as solutions to second order differential equations

Abstract: Abstract One of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation: In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

On the Cayley-Hamilton theorem for two-dimensional systems

Abstract: An alternate proof of the 2-D Cayley-Hamilton theorem is supplied together with an extension to derived polynomials.