Restricted Range Approximation Research Articles

Limit theory of restricted range approximations of complex-valued continuous functions

This paper is concerned with the problem of the best restricted range approximations of complex-valued continuous functions. Several properties for the approximating set PΩ such that the classical characterization results and/or the uniqueness results of the best approximations hold are introduced. Under the very mild conditions, we prove that these properties are equivalent that P is a Haar subspce.

On best restricted range approximation in continuous complex-valued function spaces

To provide a Kolmogorov-type condition for characterizing a best approximation in a continuous complex-valued function space, it is usually assumed that the family of closed convex sets in the complex plane used to restrict the range satisfies a strong interior-point condition, and this excludes the interesting case when some Ω t is a line-segment or a singleton. The main aim of the present paper is to remove this restriction by virtue of a study of the notion of the strong CHIP for an infinite system of closed convex sets in a continuous complex-valued function space.

On Constraint Qualification for an Infinite System of Convex Inequalities in a Banach Space

For a general infinite system of convex inequalities in a Banach space, we study the basic constraint qualification and its relationship with other fundamental concepts, including various versions of conditions of Slater type, the Mangasarian--Fromovitz constraint qualification, as well as the Pshenichnyi--Levin--Valadier property introduced by Li, Nahak, and Singer. Applications are given in the restricted range approximation problem, constrained optimization problems, as well as in the approximation problem with constraints by conditionally positive semidefinite functions.

Constraint Qualification, the Strong CHIP, and Best Approximation with Convex Constraints in Banach Spaces

Several fundamental concepts such as the basic constraint qualification (BCQ), the strong conical hull intersection property (CHIP), and the perturbations for convex systems of inequalities in Banach spaces (over $\mathbb{R}$ or $\mathbb{C}$) are extended and studied; here the systems are not necessarily finite. Their relationships with each other in connection with the best approximations are investigated. As applications, we establish results on the unconstrained reformulation of best approximations with infinitely many constraints in Hilbert spaces; also we give several characterizations of best restricted range approximations in $C(Q)$ under quite general constraints.

On best uniform restricted range approximation in complex-valued continuous function spaces

We investigate the problem of best restricted range approximation of complex-valued continuous functions for a very general system of restrictions. Our results, including the characterizations, uniqueness and strong uniqueness, extend all recent results due to Smirnovs.

A GENERAL CHARACTERIZATION OF BEST RESTRICTED RANGE APPROXIMATIONS

A general characterization of best restricted range approximations is established under an arbitrary norm on C [a, b]. Consequently, many previous characterizations of various best constrained approximations can be treated as special cases of this general characterization theorem.

Best restricted range approximation

A theory of best restricted range approximation is developed for an extended n-dimensional Chebyshev subspace of C[a,b] of order n without restricting the upper and lower restraining functions. This theory includes a “zero in the convex hull” characterization, an alternation theorem, and a strong uniqueness theorem.

A characterization of best approximations with restricted ranges

Restricted range approximation in uniform norm from an extended Haar space of a certain order is an important and widely applicable problem in restricted approximations. In the past 20 years or more, many authors have investigated the characterization of best approximations in various special cases of restricted range approximation, which include approximation with interpolatory constraints, one-sided approximation, and copositive approximation. But the characterization in the general case is still an open question. The paper gives a general characterization theorem in the form of convex hull and alternation. Many known important results are exactly its special cases.

Restricted range polynomial interpolation

Let f be a real-valued continuous function with 0≤f(x)≤1 for all x in [0,1]. We show that, for each positive integer n, there is a polynomial p of degree ≤n that interpolates f at n+1 distinct points in [0,1], and such that 0≤p(x)1 for all x in [0,1]. (For certain f there is a restriction on the parity of n.). We use a perturbation of p*, the best restricted range approximation to f in the uniform norm on [0,1], combined with an alternation theorem. It follows easily that if f is differentiable on (0,1), then p* is a Hermite interpolant to f

𝐿¹-approximation with constraints

In this paper we study problems of best L 1 {L^1} -approximation to continuous functions from finite-dimensional subspaces under a variety of constraints. Included are problems of bounded coefficient approximation, approximation with interpolation, restricted range approximation, and restricted range and derivative approximation. Emphasis is placed on problems of uniqueness.

The limits of a Chebyshev-type theory of restricted range approximation

The Chebyshev-type theory of restricted range approximation includes existence, alternation, characterization, and uniqueness. In this paper a detailed study is made of the limits of this theory.

Restricted range approximate solutions of nonlinear differential systems with boundary conditions

This paper develope uniform approximate solutions to systems of nonlinear differential equations with boundary conditions using restricted range approximations. In the process of deriving the approximations an existence theorem is generated for solutions of nonlinear equations. Also, an algorithm is provided for computing these approximations.

Restricted Range Approximation by Splines and Variational Inequalities

Error estimates are obtained for the approximation of nonnegative functions from below by nonnegative splines. The maximum norm results are of the same order as the Jackson-type estimates for unconstrained approximation by splines. The $L_2 $ results are used to estimate the rate of convergence of the Ritz method for certain variational inequalities. The more general problem of approximating functions f in the set $W = \{ {g \in C[ {a,b} ]:l(x) \leqq g(x) \leqq u(x),x \in [ {a,b} ]} \}$ by splines in W is also discussed.

Uniqueness in restricted range approximation with betweenness

Uniqueness in restricted range approximation with betweenness

Restricted Range Approximation of k-Convex Functions in Monotone Norms

Restricted range approximation of k-convex functions by k-concave functions in monotone norms (norms for which $|f(x)| \leqq |g(x)|,a \leqq x \leqq b$, implies $||f|| \leqq ||g||$) is treated. It is proved that the class of best approximants contains a polynomial of degree $ \leqq k - 1$. The results generalize those of Kimchi and Richter-Dyn [J. Math. Anal. Appl.,1976] where the special case of approximation in the sup-norm without restrictions on the ranges is discussed. The general approach used here includes the study of some properties of monotone norms and of polynomials of best approximation in such norms. The results indicate that in restricted derivative approximation in monotone norms, it is impossible to approximate arbitrarily closely a function with one derivative outside the range.

Restricted range approximation with side conditions

Restricted range approximation with side conditions

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.

Restricted Range Approximation and Its Application to Digital Filter Design

Restricted Range Approximation and Its Application to Digital Filter Design

Kriterien erster und zweiter Ordnung f�r lokal beste Approximationen bei Problemen mit Nebenbedingungen

In a previous paper [6] we considered the problem of approximating (in the Chebyshev-norm) a real-valued functionf(x) on a compact subsetX<? m ,m?1, by an element of a set of functionsa(p, x),p?P,P<? n an open set. Both necessary and sufficient conditions of the second order for ana(p 0,x) to be a locally best approximation were derived. In this paper we generalize these results to problems where the setP of admitted parameters is constrained by some inequality. Included are subjects as monotone, one-sided or restricted range approximation.

Restricted range approximation and its application to digital filter design

The multiple exchange algorithm for restricted range approximation is discussed. Efficient formulas are derived for the numerical implementation of the method. Discretization effects are analyzed mathematically. The method is applied to a certain problem arising in digital filter design.