Order Of Best Approximation Research Articles

Approximation of fuzzy numbers by max-product Bernstein operators

In this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear Bernstein operators of max-product kind, by proving that their order of uniform approximation is the same as in the particular case of the unit interval. Then, similarly to the particular case of the unit interval, we will prove that these operators preserve the quasi-concavity too. Since these properties will help us to generate in a simple way fuzzy numbers of the same support, it turns out that these results are very suitable in the approximation of fuzzy numbers. Moreover, these operators approximate the (non-degenerate) segment core with a good order of approximation. In addition, in the case when the fuzzy numbers are given in parametric form, the Bernstein max-product operator generates a sequence of fuzzy numbers whose widths, expected intervals, ambiguities and values, approximate with a convergent rate the width, the expected interval, the ambiguity and the value of the approximated fuzzy number. Finally, we obtain a quantitative approximation with respect to a well-known integral type metric for some subclasses of functions, leading to applications to the approximation of some subclasses of fuzzy numbers.

On best approximations by analogs of “proper” and “improper” hyperbolic crosses

The exact approximation orders for the classes H q Ω are calculated for the case in which Ω(t) contains both power and logarithmic multipliers. For these classes, the exact orders of best approximation by analogs of “improper” hyperbolic crosses are also obtained.

Theorem of coincidence of classes for a generalised shift operator

In this paper, for a generalised shift operator introduced earlier, we prove theorem of coincidence of classes of functions defined by the order of best approximation by algebraical polynomials and the generalised Lipschitz classes defined by the generalised shift operator. Mathematics Subject Classification: Primary 41A35, Secondary 41A50, 42A16

Approximation of Sobolev classes by polynomials and ridge functions

Let W p r ( B d ) be the usual Sobolev class of functions on the unit ball B d in R d , and W p ∘ , r ( B d ) be the subclass of all radial functions in W p r ( B d ) . We show that for the classes W p ∘ , r ( B d ) and W p r ( B d ) , the orders of best approximation by polynomials in L q ( B d ) coincide. We also obtain exact orders of best approximation in L 2 ( B d ) of the classes W p ∘ , r ( B d ) by ridge functions and, as an immediate consequence, we obtain the same orders in L 2 ( B d ) for the usual Sobolev classes W p r ( B d ) .

Approximation by polynomials and ridge functions of classes of s-monotone radial functions

We obtain estimates on the order of best approximation by polynomials and ridge functions in the spaces L q of classes of s-monotone radial functions which belong to the space L p , 1 ⩽ q ⩽ p ≤ ∞ .

On the Orders of Nonlinear Approximations for Classes of Functions of Given Form

Suppose that Δ + s is the set of functions x: I → ℝ on a finite interval I such that the divided differences [x; t 0,..., t s ] of order s ∈ ℕ of these functions are nonnegative for all collections from (s +1) different points t 0,..., t s ∈ I. For all s ∈ ℕ and 1 ≤ p ≤ ∞, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of L p for classes $$\Delta _ + ^s B_p : = \Delta _ + ^s \cap B_p$$ , where B p is the unit ball in L p . We also establish the asymptotics of pseudodimensional widths in L p of these classes of functions.

On the order of approximation for the rational interpolation to |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are\(x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0)\), it is proved that the exact order of approximation is\(O\left( {\frac{1}{n}} \right),O\left( {\frac{1}{{n\log n}}} \right) and O\left( {\frac{1}{{n^r }}} \right)\), respectively, corresponding to 0 1.

The Order of Best Approximation in Some Classes of Functions

Starting from the equivalence between the Ditzian–Totik modulus\(\omega _1^\varphi \left( {f;\delta } \right)_\infty \) and\(\omega _1 \left( {g;\delta } \right)_\infty \), where\(g\left( t \right) = f\left( {\cos t} \right)\), in this article large classes of functions\(f\) are introduced for which the modulus \(\omega _1 \left( {g;\delta } \right)_\infty \)can be easily calculated. As a consequence, very good estimates for the bestapproximation are obtained. The attempts to estimate or calculate themodulus \(\omega _{_1 }^\varphi \left( {f;\delta } \right)_\infty \) can be a very intricateproblem.

Weak Type Inequalities for Best Simultaneous Approximation in Banach Spaces

For arbitrary Banach spaces Butzer and Scherer in 1968 showed that the approximation order of best approximation can characterized by the order of certain K-functionals. This general theorem has many applications such as the characterization of the best approximation of algebraic polynomials by moduli of smoothness involving the Legendre, Chebyshev, or more general the Jacobi transform. In this paper we introduce a family of seminorms on the underlying approximation space which leads to a generalization of the Butzer–Scherer theorems. Now the characterization of the weighted best algebraic approximation in terms of the so-called main part modulus of Ditzian and Totik is included in our frame as another particular application. The goal of the paper is to show that for the characterization of the orders of best approximation, simultaneous approximation (in different spaces), reduction theorems, and K-functionals one has (essentially) only to verify three types of inequalities, namely inequalities of Jackson-, Bernstein-type and an equivalence condition which guarantees the equivalence of the seminorm and the underlying norm on certain subspaces. All the results are given in weak-type estimates for almost arbitrary approximation orders, the proofs use only functional analytic methods.

Best Approximation of Functions like |x|λ exp(−A|x|−α)

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions likeϕλ, α(x)=|x|λexp(−A|x|−α). In particular, it is shown thatE(ϕλ, α, Pn, Lp(−1, 1))∼n−(2λp+αp+2)/2p(1+α)×exp(−(1+α−1)(Aα)1/(1+α)cosαπ/2(1+α)nα/(1+α)), whereE(ϕλ, α, Pn, Lp(−1, 1)) denotes best polynomial approximation ofϕλ, αinLp(−1, 1),λ∈R,α∈(0, 2],A>0, 1⩽p⩽∞. The problem, concerning the exact order of decrease ofE(ϕ0, 2, Pn, L∞(−1, 1)), has been posed by S. N. Bernstein.

On Simultaneous Approximation by Optimal Algebraic Polynomials

In this paper the order of bestapproximation by algebraic polynomials is related to the order of weighted simultaneous approximation. A special case of a Markov-Bernstein type inequality proved by Ditzian and Totik in a very general fashion is used.

An elementary construction of periodic singular integrals achieving the order of best approximation

An elementary construction of periodic singular integrals achieving the order of best approximation

SPLINE INTERPOLATION AND THE HIGHER DERIVATIVES

Edmund Landau was very much interested in the theory of approximation. It was he who inspired Dunham Jackson to do his fundamental work on the orders of best approximation. His proof of Weierstrass’ theorem by means of Landau’s singular integral is now classical. A new generalization of Weierstrass’ theorem found by a graduate student at the University of Wisconsin, Martin Marsden, plays an essential role in the present paper.

The unrestricted Hartree-Fock method

The unrestricted Hartree-Fock method is important for discussing several effects, including the spin density produced in the core of an atom by exchange with an unfilled valence electron shell. The method allows electrons withα andβ spins to have slightly different orbitals due to the differing exchange interaction, but ignores the fact that this automatically results in gaining also some in-out correlation energy. The validity of the method must therefore be questioned, since correlation is in general the larger effect. It is shown that in atomic calculations the method gives results which are in fact correct to a good order of approximation. However, the interplay of spin polarisation and correlation is such that in some molecular or other systems the method would fail.