Haar Condition Research Articles

A unified approach to methods for the simultaneous computation of all zeros of generalized polynomials

Applying Newton's method to a particular system of nonlinear equations we derive methods for the simultaneous computation of all zeros of generalized polynomials. These generalized polynomials are from a function space satisfying a condition similar to Haar's condition. By this approach we bring together recent methods for trigonometric and exponential polynomials and a well-known method for ordinary polynomials. The quadratic convergence of these methods is an immediate consequence of our approach and needs not to be proved explicitly. Moreover, our approach yields new interesting methods for ordinary, trigonometric and exponential polynomials and methods for other functions occuring in approximation theory.

Satisfying the Haar condition in halfband FIR filter design

A reformulation of the FIR symmetric halfband filter design problem is described that allows elimination of the constraints on coefficients from the design, thereby facilitating the direct use of the Parks-McClellan design procedure. In this procedure, it is only necessary to use a single-frequency care-band for approximation with about one-half the filter order of the halfband filter. >

Tchebycheff approximation of continuous functions by harmonic polynomials on conic sections

The problem of finding a best Tchebycheff approximation to a given continuous function f, defined on a compact portion of a plane conic section, from the set of harmonic polynomials of degree n or less is studied. It is shown that the Haar condition is fulfilled by such harmonic polynomials. Interesting relationships which exist between this problem and certain classical approximation problems are explored. Numerical examples are given to illustrate the theory.

Discrete Chebyshev Approximation with Linear Constraints

The Chebyshev solution of an overdetermined system of linear equation is considered when linear constraints on the solution are present. Constraints considered are constraints on the given matrix of coefficients plus general linear constraints. No assumption on the matrix of coefficients, such as the Haar condition, is made; thus the most general discrete linear problem is considered. Use is made of the concept of H-sets to give characterization theorems. An ascent exchange algorithm is given with numerical examples. Convergence of the algorithm when the problem has a solution is proven, and a computable condition is given for the case when the constraints make the problem infeasible.

An extension of the theory of orthogonal polynomials and Gaussian quadrature to trigonometric and hyperbolic polynomials

A class of functions was introduced by P. E. Koch (Research Report No. 72, Institute of Informatics, University of Oslo, 1982) that includes the algebraic, trigonometric, and hyperbolic polynomials, but retains sufficient similarities to the algebraic polynomials so that it was possible to generalize the Jackson theorems. This class need not satisfy the Haar condition. Some of the properties of orthogonal polynomials are extended here to this class. In particular, the recurrence relation, the Erdös-Turan theorem, the Dini condition for pointwise convergence of orthogonal expansions, and some of the basic properties of Gaussian quadrature are generalized.

Saddle points and overdetermined complex equations

It is known that the best uniform norm solution of overdetermined complex valued systems of equations satisfying the Haar condition for matrices is also a best weighted l p norm solution for each p ⪖ 1, for some weight vector depending on p. This paper presents an alternative proof of this result which is valid for arbitrary matrices A. The proof relies on the fundamental theorem of game theory. It is shown that a saddle point ( z ∗, λ ∗ ) of a certain function gives a uniform norm solution, z ∗ , of Az = b and a weight vector λ ∗ of the equivalent weighted l p norm problem. With appropriate qualifications concerning the weights, it follows that the worst (i.e., largest) possible weighted least l p norm error is also the best (i.e., least) possible Chebyshev error. For p = 2, it is shown that the weight vector λ ∗ solves a nonlinear optimization problem which can be posed without reference to solution vectors of Az =. In other words, the problem of finding the best uniform norm solution of Az = b, when stated as a convex optimization problem, has a convex dual which for p = 2 can be posed independently of the primal variables z. The dual variables are the weights λ.

The Haar condition and multiplicity of zeros

In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situate at regular points), aq-fold zero (in the sense that its geometric multiplicity equalsq), counts forq distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.

A cutting-plane algorithm with linear and geometric rates of convergence

This paper presents a cutting-plane algorithm for nonlinear programming which, under suitable conditions, exhibits a linear or geometric global rate of convergence. Other known rates of convergence for cutting-plane algorithms are no better than arithmetic for problems not satisfying a Haar condition. The feature responsible for this improved rate of convergence is the addition at each iteration of a new cut for each constraint, rather than adding only one new cut corresponding to the most violated constraint as is typically the case. Certain cuts can be dropped at each iteration, and there is a uniform upper bound on the number of old cuts retained. Geometric convergence is maintained if the subproblems at each iteration are approximated, rather than solved exactly, so the algorithm is implementable. The algorithm is flexible with respect to the point used to generate new cuts.

Haar intervals for trigonometric polynomials

Intervals are specified on which generalized trigonometric polynomials satisfy the Haar condition.

Uniform approximation by generalised polynomials

We here describe methods both numerical and analytic to solve the problem of finding the best uniform approximations to a continuous function by a finite dimensional linear space of functions which does not necessarily satisfy the Haar condition. We show how a knowledge ofH-sets is essential and how the theory is simplified by the use of this concept.

H-sets in linear approximation

This paper studies the role H-sets play in finding the best linear Tchebycheff approximation to a given continuous function. A simple definition is given for H-sets and the algebraic theory for linear approximation is developed. We find that many of the theorems where the Haar condition is supposed can be generalized in terms of H-sets; thus a general framework for Linear Tchebycheff Approximation is made.

Exponential Models, Maximum Likelihood Estimation, and the Haar Condition

Abstract Let f(x) be a probability density function with respect to the non-atomic measure μ, over the set χ. Suppose f(x) = exp [Σ m i = 1 τiϕi(x) − ψ m (τ)] for × ∈ χ, where ψ m (τ) = ψ m (τ1, τ2, …, τm ) is well-defined by exp [ψ m (τ)] = ∫ exp [Σ m i = 1, τiϕi,(x)] dμ(x) when the right side is finite, and the integration is over χ. Let ϕ0(x) ≡ 1 on χ and assume {ϕi(X)} m i = 0 is a collection of functions which satisfy the Haar Condition. Using convexity properties, we obtain some results on the almost sure existence or nonexistence of the MLE for τ.

Exponential Models, Maximum Likelihood Estimation, and the Haar Condition

Abstract Let f(x) be a probability density function with respect to the non-atomic measure μ, over the set χ. Suppose f(x) = exp [Σ m i = 1 τiϕi(x) − ψ m (τ)] for × ∈ χ, where ψ m (τ) = ψ m (τ1, τ2, …, τm ) is well-defined by exp [ψ m (τ)] = ∫ exp [Σ m i = 1, τiϕi,(x)] dμ(x) when the right side is finite, and the integration is over χ. Let ϕ0(x) ≡ 1 on χ and assume {ϕi(X)} m i = 0 is a collection of functions which satisfy the Haar Condition. Using convexity properties, we obtain some results on the almost sure existence or nonexistence of the MLE for τ.

The Remez exchange algorithm for approximation with linear restrictions

This paper demonstrates a Remez exchange algorithm applicable to approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions. As special cases are included the notion of restricted derivatives approximation (examples of which are monotone and convex approximation and restricted range approximation) and the notion of approximation with restrictions at poised Birkhoff data (examples of which are bounded coefficients approximation, ε \varepsilon -interpolator approximation, and polynomial approximation with restrictions at Hermite and “Ferguson-Atkinson-Sharma” data and pyramid matrix data). Furthermore the exchange procedure is completely simplified in all the cases of approximation with restrictions at poised Birkhoff data. Also results are obtained in the cases of general linear restrictions where the Haar condition prevails. In the other cases (e.g., monotone approximation) the exchange in general requires essentially a matrix inversion, although insight into the exchange is provided and partial alternation results are obtained which lead to simplifications.

Complex Chebyshev approximation with the local Haar condition

Complex Chebyshev approximation with the local Haar condition

Chebyshev approximation for two-dimensional nonrecursive digital filters

The Remez exchange algorithm is extended for the design of two-dimensional nonrecursive digital filters approximating circularly symmetrical low-pass specifications according to a weighted Chebyshev error norm. Since the approximating function does not satisfy the Haar condition, the optimal solution is not necessarily unique and a straightforward extension of the one-dimensional exchange method may fail to converge. It is shown how the algorithm has to be complemented with a perturbation technique in order to force convergence under all circumstances. In the case of nonuniqueness the solution provided by the algorithm is a vertex of the polyhedron containing all optimal solutions, and a method is given which allows one to compute an adjacent vertex located on the same edge, thereby allowing the successive determination of all the vertices defining the polyhedron. For this case also, a procedure is described which selects, among all optimal solutions, the best one according to some additional criterion. Finally, the efficiency and accuracy aspects of the algorithm are considered and practical conclusions are drawn as an aid for the designer.

Convergence of Stiefel’s Exchange Method on an Infinite Set

Stiefel’s exchange method is used to compute a best Chebyshev approximation on a finite set by a linear family satisfying the Haar condition. The method is extended to approximation on an infinite set by a linear family which may not satisfy the Haar condition. Convergence is guaranteed if the Haar condition is satisfied on all critical point sets of the best approximation and the starting deviation is sufficiently high.

On the discrete linear L1 approximation and L1 solutions of overdetermined linear equations

Usow's algorithm for solving the discrete linear L 1 approximation problem is generalized so that it can also solve an Overdetermined system of linear equations in the L 1 norm. It is then shown that this algorithm is completely equivalent to a dual simplex algorithm applied to a linear programming problem in nonnegative bounded variables. However, one iteration in the former is equivalent to one or more iterations in the latter. A dual simplex algorithm is described which seems to be the most efficient and capable method for solving these two problems. Its efficiency is due to the absence of artificial variables and to its simplicity. Its capability is due to the fact that the Haar condition associated with Usow's method is completely relaxed. Numerical results are given.

Chebyshev Approximation with the Local Haar Condition

Chebyshev approximation by nonlinear families on a general compact space is studied. Attention is restricted to approximants satisfying a local Haar condition. A necessary and sufficient condition for the approximant to be locally best is given. A linear approximation problem is given which is equivalent to the nonlinear problem of locally best approximation. The existence of a minimal set on which a locally best approximation is locally best is shown. An alternation result is given for approximation on an interval.

A method for solving the linear tchebycheff approximation problem when the haar condition is not satisfied

Die vorliegende Arbeit gibt eine Verallgemeinerung desRemes-Algorithmus zur linearenTschebyscheff-Approximation bei nicht erfullterHaarscher Bedingung an. Gegenuber fruheren Verfahren wird unter schwacheren voraussetzungen eine Folge von Approximationen konstruiert, die Minimalfolge des Approximationsproblems ist und sich mit weitgehend reduziertem Speicherbedarf und Rechenaufwand berechnen last.