Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

scholarly journals Concentration of the error between a function and its polynomial of best uniform approximation

1998 ◽  
Vol 41 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Uniform Approximation ◽  
Algebraic Polynomial ◽  
General Class ◽  
Supremum Norm ◽  
Best Uniform Approximation ◽  
Class A ◽  
Image Position ◽  
Class Of Functions

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following propertyfor some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions f for whichinfinitely often.

Classical Approximation

1995 ◽  
Author(s):  
Marek A. Kowalski ◽  
Krzystof A. Sikorski ◽  
Frank Stenger
Keyword(s):  
20Th Century ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Inner Product ◽  
Normed Spaces ◽  
Remez Algorithm ◽  
Best Uniform Approximation ◽  
Converse Theorems ◽  
Inner Product Spaces ◽  
Finite Dimensional

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.

Uniform Approximation on the Graph of a Smooth Map in Cn

1980 ◽  
Vol 32 (6) ◽  
pp. 1390-1396 ◽  
Author(s):  
Barnet M. Weinstock
Keyword(s):  
Uniform Approximation ◽  
Compact Set ◽  
Continuous Complex ◽  
Smooth Map ◽  
Image Position ◽  
Complex Valued

Let X be a compact set in Cn, and let ƒ1, …, ƒm, m ≧ n, be continuous, complex-valued functions on X which have C1 extensions to some neighborhood of X. We wish to describe the algebra A of continuous complex-valued functions on X which can be approximated uniformly by polynomials in the functions z1 …, zn, ƒ1 …, ƒm. For this purpose we introduce the setsand

On the Behavior of Zeros of Polynomials of Best and Near-Best Approximation

1991 ◽  
Vol 43 (5) ◽  
pp. 1010-1021
Author(s):  
K. G. Ivanov ◽  
E. B. Saff ◽  
V. Totik
Keyword(s):  
Uniform Approximation ◽  
Best Approximation ◽  
Cluster Point ◽  
Zeros Of Polynomials ◽  
Closed Disk ◽  
Best Uniform Approximation

AbstractAssume ƒ is continuous on the closed disk D1 : |z| ≤ 1, analytic in |z| ≤ 1, but not analytic on D1. Our concern is with the behavior of the zeros of the polynomials of best uniform approximation to ƒ on D1. It is known that, for such ƒ, every point of the circle |z| = 1 is a cluster point of the set of all zeros of Here we show that this property need not hold for every subsequence of the Specifically, there exists such an f for which the zeros of a suitable subsequence all tend to infinity. Further, for near-best polynomial approximants, we show that this behavior can occur for the whole sequence. Our examples can be modified to apply to approximation in the Lq-norm on |z|= 1 and to uniform approximation on general planar sets (including real intervals).

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

Optimal Order Picking in Carousels Storage System

2012 ◽  
Vol 601 ◽  
pp. 347-353
Author(s):  
Xiong Zhi Wang ◽  
Guo Qing Wang
Keyword(s):  
Approximation Algorithm ◽  
General Problem ◽  
Storage System ◽  
Order Picking ◽  
Total Order ◽  
Experimental Results ◽  
Optimal Order ◽  
Np Hard ◽  
Special Case

We study the order picking problem in carousels system with a single picker. The objective is to find a picking scheduling to minimizing the total order picking time. After showing the problem being strongly in NP-Hard and finding two characteristics, we construct an approximation algorithm for a special case (two carousels) and a heuristics for the general problem. Experimental results verify that the solutions are quickly and steadily achieved and show its better performance.

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