Chebyshev System Research Articles

A modified version of the Methfessel‒Paxton method

A modified version of the method for calculating integrals over the Brillouin zone, which relies on the use of the orthogonal function series, is discussed. It is shown that adoption of the system of Chebyshev’s polynomials makes it possible, while retaining all the advantages of the original scheme, to overcome a number of its disadvantages and to improve the numerical stability in the case of high-degree polynomial approximation, to reduce the number of parameters in the method, and to rule out the development of regions with negative values of the density of states.

On linear homogeneous differential equation of Chebyshev type

Let L[y] = y(n)(z)+gn-1(z)y(n-1)(z)+. . .+g1(z)y(1)(z)+g0(z)y(z) = 0 be a differential equation of nth order with analytic in circle |z| < R coefficients. We will call above equation by equation of Chebyshev type in
 |z| < R, if fundamental system of its solution is a Chebyshev system in circle |z| < R . In present
 paper the conditions with the fulfillment of which the equation L[y] = 0 is of Chebyshev type.

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vn is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S1 that takes f to a function L2-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of \({\mathbb{RP}}^1\) that takes f to the Schwarzian derivative of a function on \({\mathbb{RP}}^1\). We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere.

Slepian functions on the sphere, generalized Gaussian quadrature rule

Denote by K the operator of ‘time–band–time’ limiting on the surface of the sphere and consider the problem of computing singular vectors of K. This problem can be reduced to a simpler task of computing eigenfunctions of a differential operator, if a differential operator, which commutes with K and has a simple spectrum, can be exhibited. In Grünbaum et al (1982 SIAM J. Appl. Math. 42 941–55) such a second-order differential operator commuting with K on the appropriate subspaces was constructed. In this paper, this algebraic property of commutativity is used to produce an efficient numerical scheme for computing a convenient basis for the space of singular vectors of K. The basis forms an extended Chebyshev system, and a generalized Gaussian quadrature rule for such a basis is presented.

Quadrature rules for qualocation

AbstractQualocation is a method for the numerical treatment of boundary integral equations on smooth curves which was developed by Chandler, Sloan and Wendland (1988‐2000) [1,2]. They showed that the method needs symmetric J–point–quadrature rules on [0, 1] that are exact for a maximum number of 1–periodic functions$$ G _{\alpha} (x) \ggleich \sum ^{\infty} _{k=1} k ^{-\alpha} \cos (2 \pi kx), \qquad \alpha > {1 \over {2}}. $$The existence of 2–point–rules of that type was proven by Chandler and Sloan. For J ∈ {3, 4} such formulas have been calculated numerically in [2]. We show that the functions Gα form a Chebyshev–system on [0, 1/2] for arbitrary indices á and thus prove the existence of such quadrature rules for any J.

Chebyshev systems and estimation theory for discrete distributions

In this paper we construct unbiased matrix estimators and estimators minimizing the bias for discrete distributions families, whose probabilities as functions of the parameter form a Chebyshev system.

Decompositions of metrizable compact spaces admittingT-systems of complex-valued functions

Upper semicontinuous decompositions into continua of a metrizable compact space admitting a Chebyshev system of continuous complex-valued functions are considered. It is proved that the cyclic elements of the Moore decomposition space can be embedded in the two-dimensional sphere.

Convergence of an algorithm for constructing snakes

We investigate an algorithm for constructing snakes (extremal polynomials introduced by S. Karlin) suggested by Dzyadyk. It is proved that, in the general case, this algorithm is linearly convergent. In the case where the basis functions of the Chebyshev system belong to the classC 2, this algorithm is quadratically convergent.

Generalized divided differences, with applications to generalized B-splines

Within the theory of spline functions, there has always been great interest in the study of B-splines, i.e. splines with a finite support. In addition to the classical polynomial case, there also exist various approaches to the definition of B-splines from other function classes, such as trigonometric or hyperbolic ones (cf. section 0). In the present paper we define B-splines from a rather general function space, with covers almost all existing approaches as special cases. The only condition that the spaces under consideration must satisfy is that of being translation invariant, a fundamental property which will be defined in the paper.Our definition of generalized B-splines is based on generalized divided differences, which go back to Popoviciu [14] and were further investigated by Mühlbach [10, 11]. In the first section of the paper, we therefore study these operators and prove new results, such as a contour integral representation and a multistep-formula; the latter expresses—in closed form—a generalized divided difference of order m+j using those of order m, for arbitrary j ∈ ℕ. This makes it possible to compute generalized divided differences recursively, even if the underlying function space is spanned by anon-complete Chebyshev system.

Best interpolation with polynomials lacking the linear term

Interpolation of minimal norm into the space of polynomials spanned by the monomials 1, t 2, …, t n + 1 on an interval [ a, b], with 0 < a < b, is shown here to be characterized by the equioscillation properties (Bernstein-Erdős conditions) which characterize minimal norm interpolation into several other spaces, including in particular the classical case of the space spanned by the consecutive monomials 1, t, …, t n on any underlying interval. A natural conjecture is that the Bernstein and Erd ' ́ os conditions in fact characterize minimal norm interpolation in practically any space spanned by an extended Chebyshev system (Markov system). Thus far, this conjecture eludes proof. It is hoped that the techniques employed here, in a concrete case, may be of help in this more general problem.

A method of determining all the zeros of a generalized polynomial with respect to an arbitrary Chebyshev system

An iterational method for simultaneously finding all the zeros of a certain generalized polynomial with respect to an arbitrary Chebyshev system, if the multiplicities of the zeros are given, is considered. The method is more general than the ones previously described, which relate only to algebraic, trigonometric, and exponential polynomials. Its quadratic convergence is established, and the method is realized on a computer.

Optimal input for discrimination of autoregressive models under output amplitude constraints

Optimal input design is considered for discriminating effectively between two rival autoregressive models when the amplitude of the system output has to be regulated within a certain tolerance limit with high certainty. This constraint is more appropriate than a power constraint when an extremely large system output may cause hazardous conditions in the system. First, conditions for the optimal input are derived based on the Ds-criterion, which corresponds to the power of the likelihood ratio test. Then two approaches are presented to construct the optimal input satisfying the conditions: one is based on the idea of a Chebyshev system, and the other is an autoregressive recursion approach. Numerical simulations illustrate the applicability of the proposed optimal input for autoregressive model discrimination.

Piecewise Chebyshev approximation of experimental data

The problem of approximating experimental data with an a priori fixed error by means of piecewise polynomial functions (with respect to a Chebyshev system) with free conjunction nodes is considered. The properties of the approximation with the minimum number of nodes are studied. A computational scheme for constructing the optimal approximation is proposed.

Chebyshev Systems of Minimal Degree

Let $F = \{ {f_i } \}_{i = 0}^n $ be a set of continuous functions on $[a,b]$, and let $F^ * = \{ {f_i f_j } \}_{i,j = 0}^n $. We determine conditions on F which are necessary and sufficient for the set $F^ * $ to be a Chebyshev system on $[a,b]$ consisting of exactly $2n + 1$ distinct functions. The results have applications in the field of experimental design.

Concertina-like movement in the absence of a Chebyshev system

Concertina-like movement in the absence of a Chebyshev system

Lipschitz constants and moduli of continuity for the Chebyshev projection

It is shown how to construct Lipschitz constants and moduli of continuity for the Chebyshev projection of C [ 0 , 1 ] C[0,1] onto the finite-dimensional subspace spanned by a Chebyshev system.

Approximation results for orthogonal polynomials in Sobolev spaces

We analyze the approximation properties of some interpolation operators and some L ω 2 L_\omega ^2 -orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function ω ( x 1 , … , x d ) \omega ({x_1}, \ldots ,{x_d}) , d ⩾ 1 d \geqslant 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 ω s H_\omega ^s . These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods.

The mühlbach-neville-aitken algorithm and some extensions

A new proof of the Mühlbach-Neville-Aitken algorithm for interpolation by a linear family of functions forming a Chebyshev system is given. This proof is based on Sylvester's identity for determinants. The algorithm is then applied to the general interpolation problem, and applications to orthogonal polynomials and Padé-type approximants are treated. Finally the extension to rational interpolation is also studied.

A Chebyshev system approach to optimal input design

This paper considers the problem of generating test signals for estimating P parameters in linear single-input single-output stochastic systems using a frequency domain approach. The input signals are power constrained and are optimal in the sense of maximizing system information where the criterion employed is a scalar function of the inverse Fisher information matrix. Minimal properties of test signals are discussed and a necessary and sufficient condition for local identifiability is proved based on the concept of an input design index. Conditions are sought under which optimality is achieved by test signals whoso design index is the minimum necessary to ensure identifiability. It is shown that the theory of Chebyshev systems ( T -systems) furnishes a fruitful approach in which the set of information matrices can be represented as a convex set (moment space) induced in Rpby a T -system. These considerations lead to the concept of canonical and principal spectral representations of test signals with the required minimal properties and to sufficient conditions for the existence of optimal designs of the desired type, placing only weak restrictions on the choice of optimality criterion. A number of examples are presented that illustrate the usefulness of a geometrical approach.

Linear Programming Algorithms for the Chebyshev Solution to a System of Consistent Linear Equations

Efficient linear programming algorithms for the solution of a set of overdetermined linear equations in the $l_1 $ norm are employed in order to obtain a minimum $l_\infty $ solution to a set of consistent linear equations. In addition, the solution is proved to be of a particular form in case the set of equations is a Chebyshev system.