Bernstein's Weighted Approximation on Still Has Problems

Japan Journal of Industrial and Applied Mathematics, 1984

This work is motivated by the analysis of stability and convergence of spectral methods using Chebyshev and Legendre polynomials. We investigate the properties of the polynomial approximation to a functionu in the norms of the weightedL wp (− 1, 1) spaces. p is any real number between 1 and∞, andw (x) is either the Chebyshev or the Legendre weight. The estimates are given in terms of the degreeN of the polynomials and of the smoothness ofu. They include and generalize some theorems of Jackson. Some Bernstein ...

Demonstratio Mathematica

In this paper we prove basic results in the approximation of vector-valued functions by polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain : estimates in terms of Ditzian-Totik L p-moduli of smoothness for approximation by Bernstein-Kantorovich generalized polynomials and by other kinds of operators like the Szasz-Mirakian operators, Baskakov operators, Post-Widder operators and their Kantorovich analogues and inverse theorems for these operators. Applications to approximation of random functions and of fuzzy-number-valued functions are given.

Journal of Computational and Applied Mathematics, 1992

Jansche, S. and R.L. Stens, Best weighted polynomial approximation on the real line; a functional-analytic approach, Journal of Computational and Applied Mathematics 40 (1992) 199-213.

1999

2. The Weierstrass approximation theorem 3. Estimates for the Bernstein polynomials 4. Weierstrass' original proof 5. The Stone-Weierstrass approximation theorem 6. Chebyshev's theorems 7. Approximation by polynomials and trigonometric polynomials 8. The nonexistence of a continuous linear projection 9. Approximation of functions of higher regularity 10. Inverse theorems References Introductory remarks These notes comprise the main part of a course on approximation theory presented at Uppsala University in the Fall of 1998, viz. the part on polynomial approximation. The material is mainly classical. As sources I used Cheney [1966], Dzjadyk [1977], Korovkin [1959], and Lorentz [1953], as well as papers listed in the bibliography. The emphasis is on explaining the main ideas behind the most important techniques. The last part of the course was on rational approximation and is not included here. I followed mainly Cheney [1966, Chapter 5, pp. 150-167]. I also discussed Padé approximation briefly, following Cheney [1966, Chapter 5, pp. 173-177] and the introduction in Rudälv [1998]. I am grateful to Tsehaye Kahsu Araaya for remarks to the manuscript. 2 a k 2 − d(0, A) 2 , since 1 2 a j + 1 2 a k belongs to A in view of the convexity. We see that the right-hand side tends to zero as j, k → ∞. This implies that (a j) is a Cauchy sequence, and it must therefore have a limit in H. The limit cannot depend on the sequence, for if we take two sequences (a j) and (b j) and mix them, the new sequence (a 0 , b 0 , a 1 , b 1 , a 2 , ...) must converge by the same argument. We call the limit π(0); by translation we define π(x) ∈ A. Exercise 1.3. Prove that the mapping π: H → A is continuous; more precisely that π(x) − π(y) x − y , x, y ∈ H. Prove also that the set A is contained in a half-space as soon as x / ∈ A: in the real case every a ∈ A must satisfy (a − x|π(x) − x) π(x) − x 2. What about the complex case? Exercise 1.4. Prove that if A is a closed linear subspace, then x−π(x) is orthogonal to π(x). Prove that we get two idempotent mappings π and I − π, and determine all possible relations between the subspaces ker π, ker(I − π), im π, im(I − π). So Hilbert space is an easy case where the best approximant is unique. However, there are other interesting cases when we can prove uniqueness of the best approximant. Now we may perhaps dare to say that approximation theory is the study of approximating sequences, best approximants and their uniqueness or nonuniqueness in cases where X is a space of interesting functions, and A is some subspace of nice functions, like polynomials, trigonomentric polynomials,... 2. The Weierstrass approximation theorem The best starting point for these lectures is the classical Weierstrass 1 approximation theorem. It says that for any continuous real-valued function f on the interval [0, 1] and any integer k 1 there is a polynomial p k such that |f (x) − p k (x)| 1/k for all

Abstract and Applied Analysis, 1996

Abstract. Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials. ... (1.2) Bn(f,x) − f(x) = O (( 1 √ n δn(x) )α) ⇐⇒ ω1(f,t) = O(tα),

Journal of Approximation Theory, 1989

In the article are studied some problems of approximation theory in the spaces $S^p$ $(1 \leqslant p < \infty)$ introduced by A.I.\,Stepanets. It is obtained the exact values of extremal characteristics of a special form which connect the values of best polynomial approximations of functions $\e_{n-1}(f)_{S^p}$ with expressions which contain modules of continuity of functions $f(x)\in S^p$. We have obtained the asymptotically sharp inequalities of Jackson type that connect the best polynomial approximations $\e_{n-1}(f)_{S^p}$ with modules of continuity of functions $f(x)\in S^p$ $(1 \leqslant p < \infty)$. Exact values of Kolmogorov, linear, Bernstein, Gelfand and projection $n$-widths in the spaces $S^p$ are obtained for some classes of functions $f(x)\in S^p$. The upper bound of the Fourier coefficients are found for some classes of functions.

Eprint Arxiv Math 0701099, 2007

Let W : R → (0, 1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → f W L∞(R). The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typically even, and supported on the whole real line, so we exclude Laguerre type weights on [0, ∞). Nor do we discuss Saff's weighted approximation problem, nor the asymptotics of orthogonal polynomials. MSC: 41A10, 41A17, 41A25, 42C10 Keywords: Weighted approximation, polynomial approximation, Jackson-Bernstein theorems Contents 1 Bernstein's Approximation Problem 2 Some Ideas for the Resolution of Bernstein's Problem 3 Weighted Jackson and Bernstein Theorems 4 Methods for Proving Weighted Jackson Theorems 19 4.

Diyala Journal For Pure Science, 2018

In this paper, we introduce a new norm and modulus in weighted spaces )𝑳 𝒑,𝜷 (𝑿)) of order 𝒌. Via these modulus, we prove the direct and inverse spline approximation inequalities of unbounded functions in weighted spaces )𝑳 𝒑,𝜷 (𝑿) ( ; 0 < 𝑝 ≤ 1 which is the main results of our paper.

Journal of Approximation Theory, 1968

2003

Journal of Approximation Theory

We obtain matching direct and inverse theorems for the degree of weighted L papproximation by polynomials with the Jacobi weights (1-x) α (1+x) β . Combined, the estimates yield a constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials. * AMS classification: 41A10, 41A17, 41A25. Keywords and phrases: Approximation by polynomials in weighted L p -norms, Degree of approximation, direct and inverse theorems, Jacobi weights, moduli of smoothness, characterization of smoothness classes.

Journal of Approximation Theory, 1969

Let {,5,) be a sequence of linear operators on C [0, l] into C [0, 11. Evaluation of the remainder is useful in the investigation of the approximation properties of the operators&.

We estimate the constants sup x∈(0,1) sup f ∈C[0,1]\Π 1 |Bn(f,x)−f (x)| ω 2 f, Õ x(1−x) n and inf x∈(0,1) sup f ∈C[0,1]\Π 1 |Bn(f,x)−f (x)| ω 2 f, Õ x(1−x) n , where B n is the Bernstein operator of degree n and ω 2 is the second order modulus of continuity.

Journal of Mathematical Analysis and Applications, 2019

We investigate properties of the m-th error of approximation by polynomials with constant coefficients D m (x) and with modulus-constant coefficients D * m (x) introduced by Berná and Blasco (2016) to study greedy bases in Banach spaces. We characterize when lim inf m D m (x) and lim inf m D * m (x) are equivalent to x in terms of the democracy and superdemocracy functions, and provide sufficient conditions ensuring that lim m D * m (x) = lim m D m (x) = x , extending previous very particular results.