Copositive Approximation Research Articles

Copositive approximation by Hermits extension

In this paper, we have found a new expansion of the Hermits polynomial of degree ≤ 3n + 2, n ≥ 1, and by this expansion we have studied the copositive approximation of the functions f in space Lψ ,p (I), 0 < p < 1, and by using the quasi normed space, we presented a development of Whitney theory for the copositive approximation using the modulus of smoothness and average modulus of smoothness τ 3n+3(f,|I|,I) ψ ,p ,n ≥ 1 by using the Hermits polynomial expansion.

Copositive spline approximation in weighted space

In this paper, we estimate the error of best approximation of Lψ ,p (I) –approximation 0 < p < 1,ψ(x) > 0, to the function f ∈ Lψ ,p (I) ∩ Δ0 (JN ) by spline in terms of usual modulus of smoothness for quasi-normed space and Ditzian-Totik modulus for Chebyshev knots.

Pointwise Estimation of the Almost Copositive Approximation of Continuous Functions by Algebraic Polynomials

In the case where a function f continuous on a segment changes its sign at s points y i : − 1 < y s < y s−1 < ... < y 1 < 1, for any n ∈ ℕ greater than a certain constant N(k, y i ) that depends only on k ∈ ℕ and $$ \underset{i=1,\dots, s-1}{\min}\left\{{y}_i-{y}_{i+1}\right\}, $$ we determine an algebraic polynomial P n of degree ≤ n such that: P n has the same sign as f everywhere except, possibly, small neighborhoods of the points $$ {y}_i:\left({y}_i-{\rho}_n\left({y}_i\right),{y}_i+{\rho}_n\left({y}_i\right)\right),\kern1em {\rho}_n(x):= 1/{n}^2+\sqrt{1-{x}^2}/n,\kern1em {P}_n\left({y}_i\right)=0, $$ and |f(x) − P n (x)| ≤ c(k, s)ω k (f, ρ n (x)), x ∈ [−1, 1], where c(k, s) is a constant that depends only on k and s and ω k (f, ·) is the modulus of continuity of the function f of order k.

On Copositive Approximation in Spaces of Continuous Functions II: The Uniqueness of Best Copositive Approximation

On Copositive Approximation in Spaces of Continuous Functions II: The Uniqueness of Best Copositive Approximation

On Copositive Approximation in Spaces of Continuous Functions I: The Alternation Property of Copositive Approximation

On Copositive In this paper the author writes a simple characterization for the best copositiveapproximation to elements of C（Q） by elements of finite dimensional strict Chebyshevsubspaces of C（Q） in the case when Q is any compact subset of real numbers. Atthe end of the paper the author applies this result for different classes of Q.

Copositive approximation by rational functions with prescribed numerator degree

The paper proves that, if f(x) ∈ L[−1,1]p, 1 ≤ p < ∞, changes sign l times in (−1, 1), then there exists a real rational function r(x) ∈ Rn(2μ−1)l which is copositive with f(x), such that the following Jackson type estimate $$ \left\| {f - r} \right\|_p \leqslant C_\delta l^{2\mu } \omega _\rho \left( {f,\frac{1} {n}} \right)_p $$ holds, where μ is a natural number ≥ 3/2 + 1/p, and Cδ is a positive constant depending only on δ.

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y i ∈ Y:= {y i } i∈ℤ such that for x ∈ [y i , y i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P n of order ≦ n, changing its sign at the same points y i ∈ Y as f, and $$ \left\| {f - P_n } \right\| \leqq c(s)\omega _3 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω 3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.

On copositive approximation in some classical spaces of sequences

In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces and shows that it is unique.

The similarity between the best approximation and the best copositive approximation

In this paper the author studies the copositive approximation in C(ϱ) by elements of finite dimensional Chebyshev subspaces in the general case when ϱ is any totally ordered compact space. He studies the similarity between me behavior of the ordinary best approximation and the behavior pf the copositive best approximation. At the end of this paper, the author isolates many cases at which the two behaviors are the same.

Copositive pointwise approximation

We prove that if a functionf ∈C (1) (I),I: = [−1, 1], changes its signs times (s ∈ ℕ) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ℕ, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k − 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,x ∈I, and |f(x) −P n (x)| ≤c(k)ω k (f;n −2 +n −1 √1 −x 2),x ∈I.

The Degree of Copositive Approximation and a Computer Algorithm

The main results are as follows. (1) Let $f \in {\bf C}[0,1]$ change its sign a finite number of times; then the degree of copositive approximation of f by splines with n equally spaced knots is bounded by $C\omega _3 (f,{1 / n})$ for n large enough. This rate is the best in the sense that $\omega _3 $ cannot be replaced by $\omega _4 $. (2) An algorithm is developed basedon the proof. (3) The first result above holds for a copositive polynomial approximation of f. (4) If $f \in C^1 [0,1]$, then the degree of approximation by copositive splines of order r is bounded by $Cn^{ - 1} \omega _{r - 1} (f',{1 / n})$. The results on $f \in {\bf C}[0,1]$ fill a gap left by S. P Zhou [Israel J. Math., 78 (1992), pp. 75–83], and Y K. Hu, D. Leviatan, and X. M. Yu [J. Anal., 1(1993), pp. 85–90; J. Approx. Theory, 80 (1995), pp. 204–218].

On copositive approximation by algebraic polynomials

Для функцииf ∈C[−1, 1] с ог раниченным числом пе ремен знака строится последовательность многочленовр п , коположительных сf (т.е.f(x)p n (x)≥0, −1≤х<1) и таких, что $$\left\| {f - p_n } \right\|_\infty \leqslant C\omega _\varphi ^3 (f,n^{ - 1} ),$$ гдеω ϕ 3 (f, δ) — модуль непр ерывности Дитциана-Т отика третьего порядка. Изв естно, чтоω ϕ 3 нельзя заменить ни наω ϕ 4 , ни на ω4. Таким образом, приведенная оценка точна в некотором смы сле. В качестве следст вия установлена эквивал ентность соотношений $$E_n (f) = O(n^{ - \alpha } )\user2{}E_n^{(0)} (f,r) = O(n^{ - \alpha } )\user2{}0< \alpha< 3.$$

Copositive Polynomial and Spline Approximation

We prove that if a function f ∈ C [0, 1] changes sign finitely many times, then for any n large enough the degree of copositive approximation to f by quadratic spliners with n−1 equally spaced knots can be estimated by Cω2(f, 1/n), where C is an absolute constant. We also show that the degree of copositive polynomial approximation to f ∈ C1[0, 1] can be estimated by Cn−1ωr(f′, 1/n), where the constant C depends only on the number and position of the points of sign change. This improves the results of Leviatan (1983, Proc. Amer. Math. Soc.88, 101-105) and Yu (1989, Chinese Ann. Math.10, 409-415), who assumed that for some r ≥ 1, f ∈ Cr[0, 1]. In addition, the estimates involved Cn−rω(fr, 1/n) and the constant C dependended on the behavior of f in the neighborhood of those points. One application of the results is a new proof to our previous ω2 estimate of the degree of copositive polynomia approximation of f ∈ C[0, 1], and another shows that the degree of copositive spline approximation cannot reach ω4, just as in the case of polynomials.

On copositive approximation

The present note investigates the divergence phenomenon of copositive approximation by polynomials in Lp spaces. We show that the Jackson type estimates cannot hold, in general, for moduli of smoothness of higher degree.

Best Copositive Approximation

A theory of best copositive approximation including a "zero in the convex hull" characterization, alternation theorem, and strong uniqueness theorem is developed for a general n-dimensional Chebyshev subspace in C[a, b]. In addition, a computational algorithm is discussed.

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.

General characterization of best approximation and its application to uniform approximation with restricted ranges

Let R be a normed linear space, K be an arbitrary convex subset of an n-dimensional subspace Φn⊂R. This paper first gives a general charactaerization for a best approximation from K in form of “zero in the convex hull”. Applying it to the uniform approximation by generalized polynomials with restricted ranges, we get further an alternation characterization. Our results ocntains the special cases of interpolatory approximation, positive approximation, copositive approximation, and the classical characterizations in forms of convex hull and alternation in approximation without restriction.

The Degree of Copositive Approximation by Polynomials

The Degree of Copositive Approximation by Polynomials

The degree of copositive approximation by polynomials

Jackson type theorems are established for the approximation of a function f f that changes sign finitely many times in [ − 1 , 1 ] [ - 1,1] by polynomials p n {p_n} which are copositive with it f p n ⩾ 0 on [ − 1 , 1 ] f{p_n} \geqslant 0{\text { on }}[ - 1,1] . The results yield the rate of nonconstrained approximation and are thus best possible in the same sense as in the nonconstrained case.

Copositive rational approximation

The purpose of this paper is to develop a theory for best uniform copositive rational approximation of continuous functions. In Section 2 the basic definitions and notations needed for the problem are presented. Existence and characterization of best copositive rational approximants on a closed interval are discussed in Section 3 and uniqueness and strong uniqueness are developed in Section 4. The continuity of the best copositive rational approximation operator is discussed in Section 5 and finally, in Section 6, the interval [a, b] is replaced by a finite subset of it and some discretization results are given. This paper generalizes the work of [3]. 2.