Positive Weight Function Research Articles

On mean convergence of extended Lagrange interpolation

Lagrange interpolation to any continuous function on [-1,1] at the zeros of orthogonal polynomials is known to converge in the mean. Here, following Bellen, we study mean convergence of Lagrange interpolation on an extended set of nodes that includes, in addition to the n zeros of the orthagonal (relative to some positive weight function w) polynomial π n of degree n, other n+1 nodes, which in turn are zeros of an orthogonal polynomial \\ ̂ gp n+1 of degree n+1 corresponding to the weight function w ̂ n=π 2 nw . A sufficient criterion of Bellen for mean convergence (as n→∞) of such extended Lagrange interpolation, for arbitrary continuous functions, is shown to fail for Chebyshev weight functions of the first, third and fourth kind. (It holds trivially for Chebyshev weights of the second kind.) Based on extensive computations, it is conjectured, on the other hand, that the criterion is satisfied for certain Jacobi weights with parameters α and β suitably restricted. Necessary conditions for mean convergence, due to Erdős and Turán, are shown to be violated for the three kinds of Chebyshev weights mentioned above. For smooth functions, a comparison is made of the speed of convergence of simple vs. extended Lagrange interpolation.

Generalized Kronrod Patterson type imbedded quadratures

We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.

Behavior of the Bergman projection on the Diederich-Fornæss worm

w 1. Introduction In this paper we show that the Bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by Sobolev norms. Let Q be a smooth bounded pseudoconvex in C and let p~t) denote the orthogonal projection from L2(f~) onto the Bergman subspace B(ff~)=L2(~) f3 r with respect to the weighted norm I{fll~ J'olf(z)l 2 e-t{lzll2dV)l/2. Let wk(t)) denote the Sobolev space consisting of functions whose derivatives of order ~ -to(k, g2). On the other hand, there is a large collection of results implying that for certain types of domains the unweighted Bergman projection p=pr preserves W k for all k~0. (See [FK] for the strictly pseudoconvex case; for results on weakly pseudoconvex domains the reader may consult [Ko2], [Ca], [Si] and the recent [BStl], [Ch] as well as the references cited therein. Most of these results are focused on the a-Neumann operator rather than P; see [BSt2] for the connection. Also, except for [Kol], positive results in this area are typically valid for any choice of smooth positive weight function on 0.) The question of whether or not P is similarly well-behaved for all weakly pseudoconvex domains has remained open for many years. In this paper we show that this is not the case; in fact when Q is the so-called domain of Diederich and Forna~ss then P does not map W* to W k when k>~zr/(total amount of winding). This latter quantity is explained in section 4 below, where the construction of the worm is reviewed and the main result is proved. The proof depends on computations for a piecewise Levi-flat model depending in turn on certain one-dimensional computations; these are treated in sections 3 and 2, respectively. Section 5 contains additional remarks and questions.

Algorithms for the quickest path problem and the enumeration of quickest paths

Let N = ( V, A, c, l) be an input network with node set V, arc set A, positive arc weight function c and nonnegative arc weight function l. Let σ be the amount of data to be transmitted. The quickest path problem is to find a routing path in N to transmit the given amount of data in minimum time. In a recent paper, Chen and Chin proposed this problem and developed algorithms for the single pair quickest path problem with time complexity O( re + rn log n), where n = ¦V¦, e = ¦A¦, and r is the number of distinct capacity values of N. In this paper, we first develop an alternative algorithm for the single pair quickest path problem with same time complexity and less space requirement. We then study the constrained quickest path problem and propose an O( re + rn log n) time algorithm. Finally, we develop an algorithm to enumerate the first m quickest paths to send a given amount of data from one node to another with time complexity O( rmne + rmn 2 log n).

On Zeros of a System of Polynomials and Application to Sojourn Time Distributions of Birth-and-Death Processes

Zeros of the following system of polynomials are considered: \[ \left \{ \begin {gathered} {P_0}(x) = 1, \hfill \\ {P_1}(x) = {B_0} - {A_0}x, \hfill \\ {P_{n + 1}}(x) = ({B_n} - {A_n}x){P_n}(x) - {C_n}{P_{n - 1}}(x)\quad {\text {for}}\;n \geqslant 1. \hfill \\ \end {gathered} \right .\] Numbers of positive and negative zeros are determined and a separation property of the zeros of ${P_m}(x)$ and ${P_n}(x)$ is proved under the condition that ${C_n} > 0$ and ${P_n}(0) > 0$ for every $n$. No condition is imposed on ${A_n}$. These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.

Necessary conditions for uniqueness in L1-approximation

Let K be a compact subset of R m with K = int K . Necessary conditions on an n-dimensional subspace U n of C( K) are given so that for each ƒ ϵ C(K) there exists a unique best L 1(w) -approximation U n , for every fixed positive weight function w.

On zeros of a system of polynomials and application to sojourn time distributions of birth-and-death processes

Zeros of the following system of polynomials are considered: \[ { P 0 ( x ) = 1 , P 1 ( x ) = B 0 − A 0 x , P n + 1 ( x ) = ( B n − A n x ) P n ( x ) − C n P n − 1 ( x ) for n ⩾ 1. \left \{ \begin {gathered} {P_0}(x) = 1, \hfill \\ {P_1}(x) = {B_0} - {A_0}x, \hfill \\ {P_{n + 1}}(x) = ({B_n} - {A_n}x){P_n}(x) - {C_n}{P_{n - 1}}(x)\quad {\text {for}}\;n \geqslant 1. \hfill \\ \end {gathered} \right . \] Numbers of positive and negative zeros are determined and a separation property of the zeros of P m ( x ) {P_m}(x) and P n ( x ) {P_n}(x) is proved under the condition that C n > 0 {C_n} > 0 and P n ( 0 ) > 0 {P_n}(0) > 0 for every n n . No condition is imposed on A n {A_n} . These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.

Use of adaptive methods in premixed combustion

AbstractThis paper investigates the application of spatially adaptive numerical methods in the solution of one‐dimensional premixed combustion problems. The selection of grid points by equidistribution of positive weight functions across consecutive mesh intervals is considered. The method is applied in the solution of a steady premixed laminar flame problem and a time‐dependent freely propagating flame problem. The adaptive numerical results are compared with corresponding equispaced calculations.

One-Sided L1-Approximation

AbstractLet Un be an n-dimensional subspace of C[0, 1]. We prove that if n ≥ 2, and Un contains a function which is strictly positive on (0, 1), then there exists an f ∈ C[0, 1] which has more than one best one-sided L '-approximation from Un. We also characterize those Un with the property that each f ∈ C[0, 1] has a unique best one-sided L1(w)-approximation from Un with respect to every strictly positive continuous weight function w.

Two-dimensional fully adaptive solutions of solid-solid alloying reactions

Solid-solid alloying reactions occur in a variety of pyrotechnical applications. They arise when a mixture of powders composed of appropriate oxidizing and reducing agents is heated. The large quantity of heat evolved produces a self-propagating reaction front that is often very narrow with sharp changes in both the temperature and the concentrations of the reacting species. Solution of problems of this type with an equispaced or mildly nonuniform grid can be extremely inefficient. In this paper we develop a two-dimensional fully adaptive method for solving problems of this class. The method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval in each direction at each time level. We monitor the solution from one time level to another to ensure that the local error per unit step associated with the time differencing method is below some specified tolerance. The method is applied to several examples involving exothermic, diffusion-controlled, self-propagating reactions in packed bed reactors.

Fully Adaptive Solutions of One-Dimensional Mixed Initial-Boundary Value Problems with Applications to Unstable Problems in Combustion

We develop a fully adaptive method for solving one-dimensional mixed initial-boundary value problems. The method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval at each time level. In addition, when the solution can be described accurately by a fixed number of points, the mesh is moved by extrapolating the nodal positions from earlier time levels. We monitor the solution from one time level to another to insure that the local error per unit step associated with the time differencing method is below some specified tolerance. The method is applied to several unstable problems in combustion.

On the strong limit-point and Dirichlet properties of second order differential expressions

SynopsisSecond order differential expressions of the form w−1(−(pf′)′ + qf) are considered at a singular end-point. Some of the known relationships between properties such as Dirichlet, weak Dirichlet and strong limit-point, are extended to incorporate an arbitrary, positive weight function and complexvalued coefficients.

On bounds for Titchmarsh–Weyl m-coefficients and for spectral functions for second-order differential operators

SynopsisOrder-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L2 w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

Equidistributing Meshes with Constraints

Adaptive methods which “equidistribute” a given positive weight function are now used fairly widely for selecting discrete meshes. The disadvantage of such schemes is that the resulting mesh may not be smoothly varying. In this paper a technique is developed for equidistributing a function subject to constraints on the ratios of adjacent steps in the mesh. Given a weight function $f \geqq 0$ on an interval $[a,b]$ and constants c and K, the method produces a mesh with points $x_0 = a,x_{j + 1} = x_j + h_j ,j = 0,1, \cdots ,n - 1$ and $x_n = b$ such that \[ \int_{xj}^{x_{j + 1} } {f \leqq c\quad {\text{and}}\quad \frac{1} {K}} \leqq \frac{{h_{j + 1} }} {{h_j }} \leqq K\quad {\text{for}}\, j = 0,1, \cdots ,n - 1 . \] A theoretical analysis of the procedure is presented, and numerical algorithms for implementing the method are given. Examples show that the procedure is effective in practice. Other types of constraints on equidistributing meshes are also discussed. The principal application of the procedure is to the solution of boundary value problems, where the weight function is generally some error indicator, and accuracy and convergence properties may depend on the smoothness of the mesh. Other practical applications include the regrading of statistical data.

Error estimates for a class of quadrature formulae

The problem considered is that of estimating the error of a class of quadrature formulae for ∫ −1 1 w r (x)f(x)dx, (w r (x) being a positive weight-function), where only values off(x) in (−1,1) and off(x) and its derivatives at the end-points of the interval are considered.

Zur Existenz optimaler Quadraturformeln mit freien Knoten bei Integration analytischer Funktionen

In the present paper we discuss the optimal quadrature rules for integration with positive continuous weight function in Hardy space H2 of functions analytic in a circle of the complex plane. The new representations of the optimal weights and the norm of the error functional as functions of the nodes are obtained. On this basis we give an elementary proof for the existence of the optimal quadrature formula with free nodes.

Cubature formulae and orthogonal polynomials

IN [1] the proposition (Theorem 2' ) was stated that k1 common zeros of two orthogonal polynomials of degree k of a plane domain and with a positive weight function, can be taken as the nodes of a cubature formula exact for polynomials of degree 2 k — 1. Here this statement is proved in a somewhat stronger form: the common zeros of the orthogonal polynomials are not assumed to be different.