Polynomials Of Best Uniform Approximation Research Articles

Investigation on deformation of single-sided stringer parts based on fluctuant initial residual stress

Deformation has a significant effect on the machining accuracy of the monolithic components. Fluctuant initial residual stress is one of the factors causing deformation. In this paper, initial residual stresses of six specimens of 7050-T7451 aluminum alloy plates with 65 mm thickness and 70 mm thickness are measured respectively. Fluctuant initial residual stresses are fitted by segmented polynomials of best uniform approximation. An analytical model in terms of the deformation and initial residual stress is established for single-sided stringer parts. The results show that the weight of web-induced deformation in the whole deformation of the part is about 74%–98.5% and the weight sidewall-induced deformation is about 1.5%–26%. Therefore, minimizing the web-induced deformation is the key to minimize the whole part deformation. The condition number calculated by coefficients matrix of initial residual stress inside the web is defined in the model. The value of condition number is determined by initial residual stress inside the web and its fluctuant degree, which is positively correlated with web-induced deformation when the structural stiffness is invariant. By selecting the blank with a small fluctuant amplitude of stress or changing the workpiece position, condition number will be decreased, and the deformation is minimized.

A note on uniform approximation of functions having a double pole

AbstractWe consider the classical problem of finding the best uniform approximation by polynomials of$1/(x-a)^2,$where$a>1$is given, on the interval$[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

Polynomial of Best Uniform Approximation to and Smoothing in Two-level Methods

AbstractWe derive defect correction scheme for constructing the sequence of polynomials of best approximation in the uniform norm to 1/x on a finite interval with positive endpoints. As an application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

Polynomials of the best uniform approximation to sgn(x) on two intervals

We describe polynomials of best uniform approximation to sgn(x) on the union of two intervals [−A,−1] ⊂ [1, B] in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error in this approximation.

On a Conjecture Concerning Strong Unicity Constants

Let f∈C[−1, 1] be real-valued. We consider the sequence of strong unicity constants (γn(f))n induced by the polynomials of best uniform approximation of f. It is proved that liminfn→∞γn(f)=0, whenever f is not a polynomial.

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

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.

A Discrepancy Theorem on Quasiconformal Curves

In [7], Blatt and Mhaskar estimated the Erdős-Turan type discrepancy of a signed Borel measure σ on a sufficiently smooth Jordan curve or arc L in terms of the logarithmic potential of σ on a curve enclosing L. We extend this result to a measure σ on an arbitrary quasiconformal curve. As applications, estimates for the distribution of simple zeros of monic polynomials, Fekete points, extreme points of polynomials of best uniform approximation are obtained.

On discrepancy theorems with applications to approximation theory

We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turan on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.

Local limiting behavior of the zeros of approximating polynomials

Let f be a piecewise analytic (but not analytic) function in C k [ a, b], k ⩾ 0, and let p * n be the sequence of polynomials of best uniform approximation to f on [ a, b]. It is well known that every point of [ a, b] is a limit point of the zeros of the p * n . Let x\\gE [ a, b], and suppose that f is analytic at x and f( x) ≠ 0. The main purpose of this paper is to show that there exists a constant γ (which depends only on x) such that there is no zero of p * n within the circle of radius (γ/ n) log n centered at x, for all sufficiently large values of n.

Indicators of Growth of Polynomials of Best Uniform Approximation to Holomorphic Functions on Compacta in CN

Let E be a compact and L-regular subset of C N. Siciak has shown that a function ƒ on E has a holomoprhic extension to E R—the interior of the level curve of the Siciak extremal function—if and only if lim sup n → ∞ (sup E |ƒ − p n | 1/ n ) ≤ 1/ R ( R > 1), where p n is a best approximating polynomial to ƒ of degree not greater than n. The aim of this paper is to show that ƒ has a holomorphic extension to E R if for some sequence { p n } of the polynomials of best approximation to ƒ [formula] and if ƒ has such an extension, for all { p n }, there holds [formula]. Here [formula] denotes a norm on the homogeneous terms of degree n in p n and c m ( E), d( E) are some multidimensional counterparts of the logarithmic capacity and the Chebyshev constant, respectively.

Polynomial of best uniform approximation in an iterative method of solving systems of linear algebraic equations

Consider a system of equations ( a I A ) z = f , ( 1 ) where a is some number, I is the unity matrix, A is a symmetric matrix, z is the sought vector, and F is a known vector. A solution has the form z = ( a I A ) ' F . ( 2 ) A p p r o x i m a t i o n o f t h e i n v e r s e m a t r i x by an o p e r a t o r p o l y n o m i a l Pk(A) o f d e g r e e k y i e l d s an a p p r o x i m a t e s o l u t i o n z~ = P ~ ( A ) F . ( 3 ) The error ~k satisfies the inequality It~kll = Itz zh!l ~ II (al -A) i _ P~(A)11 ItF[I. ( 4 ) The no rm o f t h e s y m m e t r i c m a t r i x ( a I I ) -~ Pk (A) i s c o m p u t a b l e i n t h e m e t r i c o f Rn by means of the spectrum S(A) of the matrix A: E[ (aI -A)-* Ph (:1)][ = m,qx l (a )0 -1 Pk (k) I" ( 5 ) ~s(A) The best estimate of ]lekJ] is yielded by that operator polynomial Pk(A) which minimizes the quantity max I(a--k) -1-1p~(k)[ on the spectrum of the operator A. Usually the spectrum S(A) is unknown, and it is impossible to construct that polynomial. The error (4) can be minimized also under the conditions of incomplete information on the spectrum S(A). Let m, M be lower and upper estimates of the bounds of the spectrum S(A), and let a ~ [m, M]. From S ( A ) ~ [ m , M ] i t f o l l o w s t h a t m a x I (a ) . ) l P ~ ( X ) I < ~ max I ( a z ) -1 P h ( ~ ) [ . ( 6 ) ~.~S(A) x~[m,Ml Denote

Erdös-Turán theorems on a system of Jordan curves and arcs

Erdos and Turan established in [4] a qualitative result on the distribution of the zeros of a monic polynomial, the norm of which is known on [−1, 1]. We extend this result to a polynomial bounded on a systemE of Jordan curves and arcs. If all zeros of the polynomial are real, the estimates are independent of the number of components ofE for any regular compact subsetE ofR. As applications, estimates for the distribution of the zeros of the polynomials of best uniform approximation and for the extremal points of the optimal error curve (generalizations of Kadec's theorem) are given.

On the Behavior of Zeros of Polynomials of Best and Near-Best 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).

Behavior of Polynomials of Best Uniform Approximation

We investigate the asymptotic behavior of the polynomials $\{ {P_n}(f)\} _0^\infty$ of best uniform approximation to a function $f$ that is continuous on a compact set $K$ of the complex plane ${\mathbf {C}}$ and analytic in the interior of $K$, where $K$ has connected complement. For example, we show that for "most" functions $f$, the error $f - {P_n}(f)$ does not decrease faster at interior points of $K$ than on $K$ itself. We also describe the possible limit functions for the normalized error $(f - {P_n}(f))/{E_n}$, where ${E_n}: = ||f - {P_n}(f)|{|_K}$, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to $f$ on $K$ exist that converge more rapidly at the interior points of $K$.

Behavior of polynomials of best uniform approximation

We investigate the asymptotic behavior of the polynomials { P n ( f ) } 0 ∞ \{ {P_n}(f)\} _0^\infty of best uniform approximation to a function f f that is continuous on a compact set K K of the complex plane C {\mathbf {C}} and analytic in the interior of K K , where K K has connected complement. For example, we show that for "most" functions f f , the error f − P n ( f ) f - {P_n}(f) does not decrease faster at interior points of K K than on K K itself. We also describe the possible limit functions for the normalized error ( f − P n ( f ) ) / E n (f - {P_n}(f))/{E_n} , where E n := | | f − P n ( f ) | | K {E_n}: = ||f - {P_n}(f)|{|_K} , and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to f f on K K exist that converge more rapidly at the interior points of K K .

Behavior of zeros of polynomials of near best approximation

The purpose of this paper is to study the asymptotic behavior of the zeros of polynomials of near best approximation to continuous functions f on a compact set E in the case when f is analytic on the interior of E but not everywhere on the boundary. For example, suppose E is a finite union of compact intervals of the real line and f is a continuous function on E, but is not analytic on E; then we show (cf. Corollary 2.2) that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E. This fact answers a question posed by P. Borwein who showed that, for the case when E is a single interval and f is real-valued, then the above hypotheses on f imply that at least one point of E is the limit point of zeros of such polynomials.

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[?

Suppose thatk, rez+, W o r H[ω]C= {f∶f is a 2π-periodic function,fe Cr [−π, π], ω (f(r), δ) ⩽ω (δ)}, Tk is the space of trigonometric polynomials of order k, pk(f)eTk is the polynomial of best uniform approximation to f, and Ek(f) is the error of the best approximation. It is shown that for an arbitrary e > 0 we have , where for 0 0.R (e) is the root of the equation $$R = (\varepsilon ^l R)^{r^l (2k)} \omega ((\varepsilon \cdot R)^{1(2k)} )$$ , and for k = 0 or e > ω(1) we have R(e)=e.

Negative Theorems on Monotone Approximation

In this paper we show that for $f$ continuous on $[ - 1, + 1]$ and satisfying $(f({x_2}) - f({x_1}))/({x_2} - {x_1}) \geqq \delta > 0$, it is possible to have infinitely many of the polynomials of best uniform approximation to $f$ not increasing on $[ - 1, + 1]$.

Properties of best approximation with interpolatory and restricted range side conditions

An alternation property of polynomials of best uniform approximation to a function | ϵ C[ a, b] having restricted ranges of some of their derivatives is proven. For this purpose, the problem of best uniform approximation to continuous functions by polynomials having restricted ranges and satisfying interpolatory conditions on their derivatives is discussed. The method is an improved version of the one used in [3] and provides an easily computed lower bound for the number of alternations.

An Estimate of $E_n (f)$ for Large n

An algorithm for estimating $E_n (f)$, the maximum error occurring when a continuous function $f(x)$ is approximated by its polynomial of best uniform approximation of degree $ \leqq n$, is described and analyzed.